From 809c8e313185624c7a3d5ed977e2cc41f5c6b8ed Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 03:38:50 +0100 Subject: [PATCH 01/40] MCA: refute affine-span incidence compiler --- agents.md | 5 +- experimental/agents-log.md | 25 +++ experimental/grande_finale.tex | 175 +++++++----------- ...affine_span_incidence_counterexample_v1.md | 142 ++++++++++++++ ...affine_span_incidence_counterexample_v1.py | 116 ++++++++++++ ...incidence_counterexample_v1_independent.py | 82 ++++++++ 6 files changed, 434 insertions(+), 111 deletions(-) create mode 100644 experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md create mode 100644 experimental/verify_mca_affine_span_incidence_counterexample_v1.py create mode 100644 experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py diff --git a/agents.md b/agents.md index 75a01e4b6..e9f03f2a5 100644 --- a/agents.md +++ b/agents.md @@ -1,7 +1,7 @@ # AGENTS.md — RS–MCA Resolution Protocol -> **Updated:** 2026-07-29 -> **State snapshot:** `main@b13de8113a03f06b6fc22bbd2f289a8abcdf7e95` +> **Updated:** 2026-08-13 +> **State snapshot:** `main@93fba1be90b7e82d0d02911900b6da1c1c192b24` > **Supersedes:** all older priority lists in this file. Edit this workboard in place. Never append another “current focus”, “highest priority”, or competing task list. @@ -80,6 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20 23" +status: COUNTEREXAMPLE +impact: ARCHITECTURE_ROUTE_CUT +--- + +# Affine-span MCA incidence counterexample v1 + +## Result + +The affine-span transverse MCA compiler in +`experimental/grande_finale.tex`, formerly labelled +`thm:affine-span-mca`, is false even under its printed direction-separation +hypothesis. + +Over `F=GF(1009)`, take + +```text +D={0,...,99}, +C=RS[F,D,1], +(n,K,m,w,s)=(100,1,21,20,1). +``` + +There is one received line with 31 distinct selected slopes such that: + +1. the selected explanations lie in the one-dimensional constant code; +2. every selected maximal agreement support has size exactly 21; +3. every such support is same-support pair-noncontained; +4. `max_(c in C) agr(r_1,c)=20<21=m`; and +5. the theorem's two terms both floor to 23. + +Thus `31>23` refutes the theorem. + +## Construction + +Partition the domain into + +```text +C0={0,...,19}, E={20,...,49}, T={50,...,70}, W={71,...,99}. +``` + +Put `(r_0,r_1)=(0,0)` on `C0`. On the `i`-th point of `E`, for +`1<=i<=30`, put + +```text +(r_0,r_1)=(-i^2,i). +``` + +For each slope `i`, select the zero codeword. It agrees exactly on the +20-point core and the `i`-th point of `E`. + +On `T`, put `r_0=1` and choose 21 distinct fresh nonzero direction values +avoiding `1+i r_1=0` for all `1<=i<=30`. At slope zero, select the constant +one codeword. It agrees exactly on `T`. + +On `W`, choose fresh direction values and choose each base value outside + +```text +{0,1} union {-i r_1:1<=i<=30}. +``` + +The field has ample room for every choice. The verifier uses the first +legal value at each step, so the record is deterministic. + +The direction values are zero 20 times and otherwise pairwise distinct. +Since the code consists of constants, its maximum codeword agreement is +exactly 20. The direction-separation hypothesis therefore holds. + +## Pair noncontainment + +On a zero-explanation support, the 20 core coordinates force any simultaneous +constant explanation pair to be `(0,0)`, while the remaining coordinate has +nonzero direction value. On `T`, the base is constant one but all 21 +direction values are distinct. No selected support is pair-contained. + +## Claimed bound + +At `s=1`, both terms in the compiler are + +```text +floor(100*99/(21*20))=23. +``` + +The selected family has size 31. + +## Proof gap + +Direction separation does force the incident normals on each selected +support to span the two-dimensional parameter space. It does not supply the +proper-subspace occupancy bound used to count incident ordered bases. + +For a zero-explanation support, 20 normals lie on one line and only one is +transverse. There are only `2*20=40` ordered bases, while the rejected proof +charges the support for + +```text +m*w=21*20=420 +``` + +ordered bases. This is the exact failed step. + +## Scope + +The counterexample refutes the MCA affine-span compiler and every active-row +payment that uses its denominator. It does not refute: + +- the ordinary affine-span LIST compiler; +- common-core cancellation; +- the directional Johnson ray compiler; +- codeword-direction gauge equivalence; +- the selector-free all-LineRay error-affine-core set-pair theorem; or +- the target KoalaBear MCA inequality itself. + +The distinction matters for the common-core PR stack: its fixed-core +codeword-affine-span payments inherit the refuted denominator, while its +separate all-LineRay payment through error affine rank three does not. + +## Replay + +```bash +python3 experimental/verify_mca_affine_span_incidence_counterexample_v1.py +python3 experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py +``` + +Expected output: + +```text +MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_PASS slopes=31 bound=23 direction_max=20 mutations=4/4 +MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_INDEPENDENT_PASS slopes=31 support_checks=31 field_values=1009 bound=23 +``` diff --git a/experimental/verify_mca_affine_span_incidence_counterexample_v1.py b/experimental/verify_mca_affine_span_incidence_counterexample_v1.py new file mode 100644 index 000000000..3355975c6 --- /dev/null +++ b/experimental/verify_mca_affine_span_incidence_counterexample_v1.py @@ -0,0 +1,116 @@ +#!/usr/bin/env python3 +"""Verify the exact GF(1009) affine-span MCA counterexample.""" + +from __future__ import annotations + +import copy +from collections import Counter + + +class Reject(ValueError): + pass + + +def construct(p: int) -> tuple[list[int], list[int], list[tuple[int, int]]]: + base = [0] * 100 + direction = [0] * 100 + slopes = list(range(1, 31)) + + for i, x in enumerate(range(20, 50), 1): + base[x] = (-i * i) % p + direction[x] = i + + used = set(range(31)) + candidate = 31 + for x in range(50, 71): + while candidate in used or any( + (1 + slope * candidate) % p == 0 for slope in slopes + ): + candidate += 1 + base[x] = 1 + direction[x] = candidate + used.add(candidate) + candidate += 1 + + for x in range(71, 100): + while candidate in used: + candidate += 1 + direction[x] = candidate + used.add(candidate) + forbidden = {0, 1} | { + (-slope * candidate) % p for slope in slopes + } + base[x] = next(value for value in range(2, p) if value not in forbidden) + candidate += 1 + + selected = [(slope, 0) for slope in slopes] + [(0, 1)] + return base, direction, selected + + +def validate(expected: dict[str, int]) -> dict[str, int]: + p = 1009 + base, direction, selected = construct(p) + if any(value == 0 for value in direction[20:]): + raise Reject("nonzero direction tail") + if len(set(direction[20:])) != 80: + raise Reject("distinct direction tail") + + explanations = set() + for slope, explanation in selected: + support = [ + x for x in range(100) + if (base[x] + slope * direction[x] - explanation) % p == 0 + ] + if len(support) != expected["support_size"]: + raise Reject("maximal support") + if ( + len({base[x] for x in support}) == 1 + and len({direction[x] for x in support}) == 1 + ): + raise Reject("pair containment") + explanations.add(explanation) + + direction_max = max(Counter(direction).values()) + bound = max(100 * 99 // (21 * 20), 100 * 99 // (20 * 21)) + observed = { + "slopes": len(selected), + "support_size": 21, + "affine_rank": len(explanations) - 1, + "direction_max": direction_max, + "bound": bound, + } + if observed != expected: + raise Reject("expected values") + if not direction_max < 21 or not len(selected) > bound: + raise Reject("theorem violation") + return observed + + +def main() -> None: + expected = { + "slopes": 31, + "support_size": 21, + "affine_rank": 1, + "direction_max": 20, + "bound": 23, + } + result = validate(expected) + mutations = 0 + for key in ("slopes", "support_size", "direction_max", "bound"): + changed = copy.deepcopy(expected) + changed[key] += 1 + try: + validate(changed) + except Reject: + mutations += 1 + if mutations != 4: + raise AssertionError("mutation controls") + print( + "MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_PASS " + f"slopes={result['slopes']} bound={result['bound']} " + f"direction_max={result['direction_max']} mutations={mutations}/4" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py b/experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py new file mode 100644 index 000000000..a44721088 --- /dev/null +++ b/experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py @@ -0,0 +1,82 @@ +#!/usr/bin/env python3 +"""Independently reconstruct and audit the affine-span MCA counterexample.""" + +from __future__ import annotations + + +class Reject(ValueError): + pass + + +def main() -> None: + p = 1009 + base: list[int | None] = [None] * 100 + direction: list[int | None] = [None] * 100 + + for x in range(20): + base[x], direction[x] = 0, 0 + for slope in range(1, 31): + x = 19 + slope + base[x], direction[x] = (-slope * slope) % p, slope + + forbidden_direction = { + (-pow(slope, -1, p)) % p for slope in range(1, 31) + } + available = iter( + value for value in range(31, p) if value not in forbidden_direction + ) + used = set(range(31)) + for x in range(50, 71): + value = next(value for value in available if value not in used) + base[x], direction[x] = 1, value + used.add(value) + + candidate = max(used) + 1 + for x in range(71, 100): + while candidate in used: + candidate += 1 + direction[x] = candidate + used.add(candidate) + forbidden_base = {0, 1} | { + (-slope * candidate) % p for slope in range(1, 31) + } + base[x] = next( + value for value in range(2, p) if value not in forbidden_base + ) + candidate += 1 + + if any(value is None for value in base + direction): + raise Reject("incomplete construction") + r0 = [int(value) for value in base] + r1 = [int(value) for value in direction] + selected = [(slope, 0) for slope in range(1, 31)] + [(0, 1)] + + for slope, explanation in selected: + support = tuple( + x + for x in range(100) + if (r0[x] + slope * r1[x]) % p == explanation + ) + if len(support) != 21: + raise Reject("selected support size") + if len({r0[x] for x in support}) == 1 and len( + {r1[x] for x in support} + ) == 1: + raise Reject("same-support pair containment") + + direction_max = max( + sum(value == constant for value in r1) for constant in range(p) + ) + affine_bound = (100 * 99) // (21 * 20) + if (len(selected), direction_max, affine_bound) != (31, 20, 23): + raise Reject("claimed invariants") + + print( + "MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_INDEPENDENT_PASS " + "slopes=31 support_checks=31 field_values=1009 " + "bound=23" + ) + + +if __name__ == "__main__": + main() From 75c61ae450eaaea2815c85caa1ba1fc9919a9349 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 04:01:51 +0100 Subject: [PATCH 02/40] MCA: add corrected occupancy compiler --- agents.md | 2 +- experimental/agents-log.md | 8 +- experimental/grande_finale.tex | 93 +++++++++++ ...affine_span_incidence_counterexample_v1.md | 35 ++++ ...a_proper_subspace_occupancy_compiler_v1.py | 157 ++++++++++++++++++ 5 files changed, 292 insertions(+), 3 deletions(-) create mode 100644 experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py diff --git a/agents.md b/agents.md index e9f03f2a5..2c3430499 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `200\), and let +\(\mathcal A=c_0+C'\subseteq C\) have \(\dim C'=s\), where +\(1\le s\le K\). Let \(Z\subseteq\F\) be a set of distinct slopes such +that for every \(\gamma\in Z\), some \(c_\gamma\in\mathcal A\) has +agreement at least \(m\), and its maximal agreement support is +same-support pair-noncontained. Define +\[ + e=\min_{b\in C}\wt(r_1-b),\qquad + L=\max\{1,e-(n-m)\}. +\] +Then +\begin{equation} + |Z|\le\left\lfloor\frac1L\max\left\{ + \frac{n^{\underline{s+1}}} + {m(w+1)^{\overline{s-1}}}, + \frac{(n-K+s)^{\underline{s+1}}} + {(w+1)^{\overline s}} + \right\}\right\rfloor . + \label{eq:proper-subspace-mca} +\end{equation} +For \(s=1\), the empty rising product in the first denominator is one. +\end{theorem} + +\begin{proof} +Choose a basis \(c_1,\ldots,c_s\) of \(C'\). As in the rejected proof, +the agreement hyperplane at coordinate \(x\) has normal +\[ + v_x=(r_1(x),-c_1(x),\ldots,-c_s(x))\in\F^{s+1}. +\] +The normals incident with every selected parameter point span +\(\F^{s+1}\). Indeed, an annihilator with zero slope coordinate would +give a nonzero codeword with at least \(m>K-1\) roots. An annihilator with +nonzero slope coordinate would give simultaneous codeword explanations for +\(r_0\) and \(r_1\) on the same maximal support, contrary to +pair noncontainment. + +Let \(W<\F^{s+1}\) have dimension \(jd=67472\) and +\(882143>d=67448\), respectively, so this compiler does not pay those +cells. The exact replay is in +\texttt{experimental/verify\_mca\_proper\_subspace\_occupancy\_compiler\_v1.py}. +\end{remark} + \begin{theorem}[Directional Johnson ray compiler] \label{thm:directional-johnson-ray} Retain the hypotheses and notation of \cref{thm:fixed-union-ray}, but allow diff --git a/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md b/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md index c995cb282..11fbc97c0 100644 --- a/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md +++ b/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md @@ -127,11 +127,45 @@ The distinction matters for the common-core PR stack: its fixed-core codeword-affine-span payments inherit the refuted denominator, while its separate all-LineRay payment through error affine rank three does not. +## Correct replacement + +The proved replacement is `thm:proper-subspace-mca`. For affine explanation +rank `s`, direction-coset support + +```text +e=min_(b in C) wt(r_1-b), t=n-m, L=max(1,e-t), +``` + +it gives + +```text +|Z| <= floor((1/L) max{ + n_fall_(s+1)/(m (w+1)_rise_(s-1)), + (n-K+s)_fall_(s+1)/(w+1)_rise_s +}). +``` + +The factor `L` is the exact safe repair: pair noncontainment gives one final +transverse normal, while direction-coset distance can raise that count. It +is not replaced by `w` without proof. + +At the first common-core-shortened rows, the corrected compiler pays all +KoalaBear ranks through 9 and Mersenne rank 1 for every `e`. Its next exact +support walls are + +```text +KoalaBear q=10,11,12,13: e>=981108,981153,981861,992852; +Mersenne q=2,3,4,5: e>=981144,981363,984779,1037876. +``` + +The top ranks remain unpaid by this compiler. + ## Replay ```bash python3 experimental/verify_mca_affine_span_incidence_counterexample_v1.py python3 experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py +python3 experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py ``` Expected output: @@ -139,4 +173,5 @@ Expected output: ```text MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_PASS slopes=31 bound=23 direction_max=20 mutations=4/4 MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_INDEPENDENT_PASS slopes=31 support_checks=31 field_values=1009 bound=23 +MCA_PROPER_SUBSPACE_OCCUPANCY_COMPILER_V1_PASS zero_normal_cases=616 adjacent=8 regression=471 toy_selections=540 ``` diff --git a/experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py b/experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py new file mode 100644 index 000000000..ee46de851 --- /dev/null +++ b/experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py @@ -0,0 +1,157 @@ +#!/usr/bin/env python3 +"""Verify the exact finite walls of the corrected MCA occupancy compiler.""" + +from __future__ import annotations + +from fractions import Fraction +from itertools import product as cartesian_product +from math import prod + + +def falling(value: int, length: int) -> int: + result = 1 + for offset in range(length): + result *= value - offset + return result + + +def bound(R: int, d: int, K: int, rank: int, factor: int) -> int: + N, m = R + K, d + K + middle = prod(d + offset for offset in range(1, rank)) + values = [] + for zero_normals in range(K - rank + 1): + for global_normals in range(zero_normals + 1): + values.append( + Fraction( + falling(N - zero_normals, rank + 1), + (m - global_normals) * middle * factor, + ) + ) + maximum = max(values) + return maximum.numerator // maximum.denominator + + +def first_factor(R: int, d: int, K: int, rank: int, budget: int) -> int: + low = high = 1 + while bound(R, d, K, rank, high) > budget: + high *= 2 + while low < high: + middle = (low + high) // 2 + if bound(R, d, K, rank, middle) <= budget: + high = middle + else: + low = middle + 1 + return low + + +def exhaustive_rank_one_toy() -> int: + p, N, K, m = 3, 3, 1, 2 + words = list(cartesian_product(range(p), repeat=N)) + code = [(constant,) * N for constant in range(p)] + checks = 0 + for base in words: + for direction in words: + e = min( + sum(value != constant for value in direction) + for constant in range(p) + ) + options: list[list[int | None]] = [] + for slope in range(p): + received = tuple( + (base[x] + slope * direction[x]) % p for x in range(N) + ) + candidates: list[int | None] = [None] + for explanation, word in enumerate(code): + support = tuple( + x for x in range(N) if received[x] == word[x] + ) + if len(support) < m: + continue + pair_contained = any( + all( + base[x] == first and direction[x] == second + for x in support + ) + for first in range(p) + for second in range(p) + ) + if not pair_contained: + candidates.append(explanation) + options.append(candidates) + corrected = bound(N - K, m - K, K, 1, max(1, e - (N - m))) + for choice in cartesian_product(*options): + selected = [value for value in choice if value is not None] + if len(selected) < 2 or len(set(selected)) < 2: + continue + if len(selected) > corrected: + raise ValueError("rank-one toy violation") + checks += 1 + return checks + + +def main() -> None: + rows = ( + { + "name": "KoalaBear", + "R": 1048576, + "d": 67472, + "K": 14, + "budget": 274980728111395087, + "factors": (1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 49, 757, 11748, 182530), + "walls": (981108, 981153, 981861, 992852), + "top_bound": 743896698428332665, + }, + { + "name": "Mersenne-31", + "R": 1048576, + "d": 67448, + "K": 6, + "budget": 16777215, + "factors": (1, 16, 235, 3651, 56748, 882143), + "walls": (981144, 981363, 984779, 1037876), + "top_bound": 219426634, + }, + ) + zero_normal_cases = 0 + adjacent_checks = 0 + for row in rows: + factors = tuple( + first_factor(row["R"], row["d"], row["K"], rank, row["budget"]) + for rank in range(1, row["K"] + 1) + ) + if factors != row["factors"]: + raise ValueError(row["name"] + " factors") + conditional = factors[len(factors) - 5:-1] + walls = tuple(row["R"] - row["d"] + factor for factor in conditional) + if walls != row["walls"]: + raise ValueError(row["name"] + " walls") + for offset, factor in enumerate(conditional, len(factors) - 4): + paid = bound(row["R"], row["d"], row["K"], offset, factor) + previous = bound(row["R"], row["d"], row["K"], offset, factor - 1) + if not paid <= row["budget"] < previous: + raise ValueError(row["name"] + " adjacent crossing") + adjacent_checks += 1 + if bound(row["R"], row["d"], row["K"], row["K"], row["d"]) != row["top_bound"]: + raise ValueError(row["name"] + " top-rank wall") + if not factors[-1] > row["d"]: + raise ValueError(row["name"] + " top-rank feasibility") + zero_normal_cases += sum( + (row["K"] - rank + 1) * (row["K"] - rank + 2) // 2 + for rank in range(1, row["K"] + 1) + ) + + regression = bound(99, 20, 1, 1, 1) + if regression != 471 or not 31 <= regression: + raise ValueError("GF(1009) regression") + toy_checks = exhaustive_rank_one_toy() + if toy_checks != 540: + raise ValueError("toy census") + print( + "MCA_PROPER_SUBSPACE_OCCUPANCY_COMPILER_V1_PASS " + f"zero_normal_cases={zero_normal_cases} adjacent={adjacent_checks} " + f"regression={regression} toy_selections={toy_checks}" + ) + + +if __name__ == "__main__": + main() From 0a4960f63f8735b1b2a50003629242e44304da54 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 04:19:19 +0100 Subject: [PATCH 03/40] MCA: split full explanation rank by lifted rank --- agents.md | 2 +- experimental/agents-log.md | 12 +- experimental/grande_finale.tex | 82 +++++++ ...affine_span_incidence_counterexample_v1.md | 30 +++ ...lanation_lifted_rank_gauge_dichotomy_v1.py | 226 ++++++++++++++++++ 5 files changed, 348 insertions(+), 4 deletions(-) create mode 100644 experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py diff --git a/agents.md b/agents.md index 2c3430499..60131a3ad 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne; the full-lift middle cells remain unpaid. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. - Grande Finale v4 explicitly supersedes the old v3 `prob:saturated-bc` status: primitive one-pencil MCA BC is proved; higher-dimensional MCA BC remains jointly governed by spread-component, large-owner, and exception routing; and the list-interior clause is governed by row-sharp list completion. The spread-abundance and `prob:next` statements are intermediate forms, not additional terminal inputs. - `experimental/proximity_prize_results_v4.tex` is the current compact synthesis of proved partial results. It is exposition/status guidance, not a replacement proof source for a live atom or adjacent-row closure. - A schema/hash pass is structural preflight only. The trusted-source registry is empty; parsing a manifest does not prove an atom. diff --git a/experimental/agents-log.md b/experimental/agents-log.md index 5491940e8..cace093e0 100644 --- a/experimental/agents-log.md +++ b/experimental/agents-log.md @@ -30,7 +30,7 @@ Keep entries concise and link to the relevant files. ## Entries -### 2026-08-13 - Affine-span MCA compiler counterexample +### 2026-08-13 - Affine-span MCA compiler repair and top-rank split - **Agent/model:** Codex. - **Files added or changed:** `experimental/grande_finale.tex`, @@ -38,8 +38,9 @@ Keep entries concise and link to the relevant files. `experimental/verify_mca_affine_span_incidence_counterexample_v1.py`, `experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py`, `experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py`, + `experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py`, `agents.md`, and `experimental/agents-log.md`. -- **Status:** PROVED COUNTEREXAMPLE / ROUTE CUT. +- **Status:** PROVED COUNTEREXAMPLE / PROVED REPLACEMENT / ROUTE CUT. - **What is being added:** An exact `GF(1009)` Reed--Solomon received line with 31 direction-separated MCA-bad slopes against the affine-span compiler's claimed bound 23. Two independently structured verifiers @@ -51,7 +52,12 @@ Keep entries concise and link to the relevant files. incident normals. The corrected `thm:proper-subspace-mca` proves the valid bound with final factor `max(1,e-(n-m))`, fully paying the first KoalaBear ranks through nine and Mersenne rank one and printing the exact - remaining support walls. + remaining support walls. The full-explanation lifted-rank dichotomy then + proves that each top cell has only two branches: lifted rank `K`, where an + affine hyperplane of codeword gauges drops explanation rank to `K-1`, and + lifted rank `K+1`, where no gauge drops rank. This pays the rank-drop + suffixes at `e>=992852` and `e>=1037876` while isolating the full-lift + middle intervals as the true remaining top-rank residuals. - **What to do next:** Replace the false ordered-basis denominator by a sharper aggregate theorem, or route the remaining top-rank K3 cells through a different slope compiler. The common-core cancellation and diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 7d4fd3c42..01b579104 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1549,6 +1549,88 @@ \section{Higher-codimension support-flag recursion} \texttt{experimental/verify\_mca\_proper\_subspace\_occupancy\_compiler\_v1.py}. \end{remark} +\begin{theorem}[Full-explanation lifted-rank gauge dichotomy] +\label{thm:full-explanation-lifted-rank-dichotomy} +Let \(C\leq\F^D\) have dimension \(K\), let +\(r_\gamma=r_0+\gamma r_1\), and let \(Z\subseteq\F\) contain distinct +slopes. For every \(\gamma\in Z\), choose \(c_\gamma\in C\), and assume +that the \(c_\gamma\) have full affine rank \(K\). Fix +\(\gamma_0\in Z\) and put +\[ + V=\operatorname{span}\{(\gamma-\gamma_0, + c_\gamma-c_{\gamma_0}):\gamma\in Z\} + \leq \F\oplus C. +\] +Then \(\dim V\in\{K,K+1\}\), and exactly one of the following holds. +\begin{enumerate} +\item If \(\dim V=K\), there is a unique nonzero linear functional + \(\ell:C\to\F\) such that + \(V=\{(\ell(u),u):u\in C\}\). The gauge + \[ + r_1\longmapsto r_1-b, + \qquad c_\gamma\longmapsto c_\gamma-\gamma b + \] + drops the explanation affine rank to exactly \(K-1\) precisely for + \(b\in C\) satisfying \(\ell(b)=1\). Thus the rank-dropping gauges form + an affine hyperplane in \(C\). +\item If \(\dim V=K+1\), then \(V=\F\oplus C\), and every \(b\in C\) + leaves the explanation affine rank equal to \(K\). +\end{enumerate} +If, in addition, every chosen maximal agreement support is same-support +pair-noncontained, then \(r_1\notin C\), and the selected error vectors +\(r_\gamma-c_\gamma\) have affine rank exactly \(\dim V\). +\end{theorem} + +\begin{proof} +Projection of \(V\) to \(C\) is onto because the explanations have full +affine rank. Hence \(K\leq\dim V\leq K+1\). If \(\dim V=K\), projection +is an isomorphism and \(V\) is the graph of a unique functional \(\ell\). +It is nonzero because the slopes are not all equal. On anchored +differences, the gauge is the map +\[ + T_b:V\longrightarrow C,\qquad T_b(a,u)=u-ab. +\] +For \(V=\{(\ell(u),u):u\in C\}\), this map is +\(u\mapsto u-\ell(u)b\). It has rank \(K-1\) exactly when +\(\ell(b)=1\), in which case its image is \(\ker\ell\); otherwise it is +invertible. If \(\dim V=K+1\), then \(V=\F\oplus C\), and \(T_b\) is +surjective for every \(b\). + +Suppose now that \(r_1\in C\). On any selected agreement support, +\(r_1\) explains the direction and \(c_\gamma-\gamma r_1\in C\) explains +the base, contradicting pair noncontainment. Thus \(r_1\notin C\). The +linear map +\[ + (a,u)\longmapsto ar_1-u +\] +is consequently injective on \(\F\oplus C\), and it sends the anchored +lifted differences to the anchored error-vector differences. Their affine +rank is therefore \(\dim V\). +\end{proof} + +\begin{remark}[Exact top-rank split at the first shortened rows] +\label{rem:full-explanation-top-rank-split} +The gauge preserves agreement supports, pair noncontainment, and +\(e=\min_{b\in C}\wt(r_1-b)\). Therefore the \(\dim V=K\) branch may use +\cref{thm:proper-subspace-mca} at explanation rank \(K-1\). Combining +this with the already separate low-support and recursive-direction gates +gives the following exact routing table; \(q\) is explanation affine rank +and \(h=\dim V\), equivalently the error affine rank. +\[ +\begin{array}{c|c|c|c} +\text{row}&q&h&\text{currently paid support range}\\ \hline +\text{KoalaBear}&14&14&e\le5\ \text{or}\ e\ge992852\\ +\text{KoalaBear}&14&15&e\le5\ \text{or}\ e\ge1044239\\ +\text{Mersenne-31}&6&6&e\le1\ \text{or}\ e\ge1037876\\ +\text{Mersenne-31}&6&7&e\le1\ \text{or}\ e\ge1044242 +\end{array} +\] +The improvement is only in the \(h=K\) rows. The full-lift \(h=K+1\) +middle intervals remain unpaid and are the wider top-rank residuals. A +finite-field audit and the exact adjacent arithmetic are in +\texttt{experimental/verify\_mca\_full\_explanation\_lifted\_rank\_gauge\_dichotomy\_v1.py}. +\end{remark} + \begin{theorem}[Directional Johnson ray compiler] \label{thm:directional-johnson-ray} Retain the hypotheses and notation of \cref{thm:fixed-union-ray}, but allow diff --git a/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md b/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md index 11fbc97c0..fad0a649f 100644 --- a/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md +++ b/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md @@ -160,12 +160,41 @@ Mersenne q=2,3,4,5: e>=981144,981363,984779,1037876. The top ranks remain unpaid by this compiler. +## Full-explanation top-rank split + +The proved lifted-rank gauge dichotomy now resolves the internal structure of +each top-rank cell. If the selected explanations have affine rank `K`, fix an +anchor and let + +```text +V=span{(gamma-gamma_0,c_gamma-c_gamma0)} <= F direct_sum C. +``` + +Then `dim(V)` is `K` or `K+1`. In the rank-`K` branch, `V` is the graph of +a nonzero functional `ell:C->F`, and exactly the gauges `ell(b)=1` drop the +explanation rank to `K-1`. In the rank-`K+1` branch, every gauge retains +rank `K`. Pair noncontainment also forces `r_1 notin C`, so `dim(V)` equals +the affine rank of the selected error vectors. + +Consequently, the corrected compiler at rank `K-1` pays the top-rank +rank-drop branches at + +```text +KoalaBear: e>=992852 when lifted/error rank is 14; +Mersenne-31: e>=1037876 when lifted/error rank is 6. +``` + +The full-lift branches, of error ranks 15 and 7 respectively, do not receive +this payment. Their current high-support suffixes remain `e>=1044239` and +`e>=1044242`, so their middle-support intervals are still open. + ## Replay ```bash python3 experimental/verify_mca_affine_span_incidence_counterexample_v1.py python3 experimental/verify_mca_affine_span_incidence_counterexample_v1_independent.py python3 experimental/verify_mca_proper_subspace_occupancy_compiler_v1.py +python3 experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py ``` Expected output: @@ -174,4 +203,5 @@ Expected output: MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_PASS slopes=31 bound=23 direction_max=20 mutations=4/4 MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_INDEPENDENT_PASS slopes=31 support_checks=31 field_values=1009 bound=23 MCA_PROPER_SUBSPACE_OCCUPANCY_COMPILER_V1_PASS zero_normal_cases=616 adjacent=8 regression=471 toy_selections=540 +MCA_FULL_EXPLANATION_LIFTED_RANK_GAUGE_DICHOTOMY_V1_PASS gauges=625 hostile=1 adjacent=2 ``` diff --git a/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py b/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py new file mode 100644 index 000000000..e4d0a82fc --- /dev/null +++ b/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py @@ -0,0 +1,226 @@ +#!/usr/bin/env python3 +"""Audit the lifted-rank gauge dichotomy and its two deployed rank-drop walls.""" + +from __future__ import annotations + +from fractions import Fraction +from itertools import product +from math import prod + + +def matrix_rank(rows: list[tuple[int, ...]], prime: int) -> int: + work = [[entry % prime for entry in row] for row in rows] + rank = 0 + width = len(work[0]) if work else 0 + for column in range(width): + pivot = next( + (index for index in range(rank, len(work)) if work[index][column]), + None, + ) + if pivot is None: + continue + work[rank], work[pivot] = work[pivot], work[rank] + inverse = pow(work[rank][column], -1, prime) + work[rank] = [(inverse * value) % prime for value in work[rank]] + for index in range(len(work)): + if index == rank or work[index][column] == 0: + continue + scale = work[index][column] + work[index] = [ + (value - scale * basis) % prime + for value, basis in zip(work[index], work[rank]) + ] + rank += 1 + return rank + + +def dot(left: tuple[int, ...], right: tuple[int, ...], prime: int) -> int: + return sum(a * b for a, b in zip(left, right)) % prime + + +def gauge_image_rank( + lifted_basis: list[tuple[int, ...]], + gauge: tuple[int, ...], + prime: int, +) -> int: + image = [ + tuple((row[index + 1] - row[0] * gauge[index]) % prime + for index in range(len(gauge))) + for row in lifted_basis + ] + return matrix_rank(image, prime) + + +def audit_small_field() -> tuple[int, int]: + prime, dimension = 5, 2 + gauges = list(product(range(prime), repeat=dimension)) + standard = [ + tuple(int(index == coordinate) for index in range(dimension)) + for coordinate in range(dimension) + ] + gauge_checks = 0 + + for functional in product(range(prime), repeat=dimension): + if not any(functional): + continue + graph_basis = [ + (functional[index], *standard[index]) + for index in range(dimension) + ] + if matrix_rank(graph_basis, prime) != dimension: + raise ValueError("graph lifted rank") + histogram: dict[int, int] = {} + for gauge in gauges: + observed = gauge_image_rank(graph_basis, gauge, prime) + expected = dimension - int(dot(functional, gauge, prime) == 1) + if observed != expected: + raise ValueError("graph gauge classification") + histogram[observed] = histogram.get(observed, 0) + 1 + gauge_checks += 1 + expected_histogram = { + dimension - 1: prime ** (dimension - 1), + dimension: prime**dimension - prime ** (dimension - 1), + } + if histogram != expected_histogram: + raise ValueError("graph gauge histogram") + error_basis = [ + (row[0], *tuple(-value % prime for value in row[1:])) + for row in graph_basis + ] + if matrix_rank(error_basis, prime) != dimension: + raise ValueError("graph error rank") + + full_basis = [ + tuple(int(index == coordinate) for index in range(dimension + 1)) + for coordinate in range(dimension + 1) + ] + if matrix_rank(full_basis, prime) != dimension + 1: + raise ValueError("full lifted rank") + for gauge in gauges: + if gauge_image_rank(full_basis, gauge, prime) != dimension: + raise ValueError("full-lift gauge rank") + gauge_checks += 1 + full_errors = [ + (row[0], *tuple(-value % prime for value in row[1:])) + for row in full_basis + ] + if matrix_rank(full_errors, prime) != dimension + 1: + raise ValueError("full-lift error rank") + + graph_basis = [(1, 1, 0), (0, 0, 1)] + hostile = graph_basis + [(0, 1, 0)] + if matrix_rank(hostile, prime) != dimension + 1: + raise ValueError("hostile perturbation lifted rank") + if any( + gauge_image_rank(hostile, gauge, prime) != dimension + for gauge in gauges + ): + raise ValueError("hostile perturbation gauge rank") + return gauge_checks, 1 + + +def falling(value: int, length: int) -> int: + return prod(value - offset for offset in range(length)) + + +def occupancy_bound( + residual: int, + distance: int, + dimension: int, + rank: int, + factor: int, +) -> int: + length, agreement = residual + dimension, distance + dimension + middle = prod(distance + offset for offset in range(1, rank)) + candidates = [] + for zero_normals in range(dimension - rank + 1): + for global_normals in range(zero_normals + 1): + candidates.append( + Fraction( + falling(length - zero_normals, rank + 1), + (agreement - global_normals) * middle * factor, + ) + ) + maximum = max(candidates) + return maximum.numerator // maximum.denominator + + +def first_factor( + residual: int, + distance: int, + dimension: int, + rank: int, + budget: int, +) -> int: + low = high = 1 + while occupancy_bound(residual, distance, dimension, rank, high) > budget: + high *= 2 + while low < high: + middle = (low + high) // 2 + if occupancy_bound(residual, distance, dimension, rank, middle) <= budget: + high = middle + else: + low = middle + 1 + return low + + +def audit_deployed_walls() -> int: + rows = ( + { + "name": "KoalaBear", + "R": 1048576, + "d": 67472, + "K": 14, + "budget": 274980728111395087, + "drop_high": 992852, + "low": 5, + "frontier_j": 4337, + "full_high": 1044239, + }, + { + "name": "Mersenne-31", + "R": 1048576, + "d": 67448, + "K": 6, + "budget": 16777215, + "drop_high": 1037876, + "low": 1, + "frontier_j": 4334, + "full_high": 1044242, + }, + ) + adjacent = 0 + for row in rows: + rank = row["K"] - 1 + factor = first_factor( + row["R"], row["d"], row["K"], rank, row["budget"] + ) + if row["R"] - row["d"] + factor != row["drop_high"]: + raise ValueError(row["name"] + " rank-drop wall") + if not ( + occupancy_bound(row["R"], row["d"], row["K"], rank, factor) + <= row["budget"] + < occupancy_bound( + row["R"], row["d"], row["K"], rank, factor - 1 + ) + ): + raise ValueError(row["name"] + " adjacent crossing") + if row["R"] - row["frontier_j"] != row["full_high"]: + raise ValueError(row["name"] + " full-lift arithmetic") + if not row["low"] < row["drop_high"] < row["full_high"]: + raise ValueError(row["name"] + " support ordering") + adjacent += 1 + return adjacent + + +def main() -> None: + gauges, hostile = audit_small_field() + adjacent = audit_deployed_walls() + print( + "MCA_FULL_EXPLANATION_LIFTED_RANK_GAUGE_DICHOTOMY_V1_PASS " + f"gauges={gauges} hostile={hostile} adjacent={adjacent}" + ) + + +if __name__ == "__main__": + main() From f966e38c54d698a1fc273858de46630a22fd978c Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 04:39:42 +0100 Subject: [PATCH 04/40] MCA: identify full lift as near-MDS extension --- agents.md | 2 +- experimental/agents-log.md | 6 +- experimental/grande_finale.tex | 59 +++++++++- ...affine_span_incidence_counterexample_v1.md | 23 +++- ...lanation_lifted_rank_gauge_dichotomy_v1.py | 104 +++++++++++++++++- 5 files changed, 185 insertions(+), 9 deletions(-) diff --git a/agents.md b/agents.md index 60131a3ad..5c6439996 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne; the full-lift middle cells remain unpaid. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. Those full-lift middle cells therefore require extension-specific or row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. - Grande Finale v4 explicitly supersedes the old v3 `prob:saturated-bc` status: primitive one-pencil MCA BC is proved; higher-dimensional MCA BC remains jointly governed by spread-component, large-owner, and exception routing; and the list-interior clause is governed by row-sharp list completion. The spread-abundance and `prob:next` statements are intermediate forms, not additional terminal inputs. - `experimental/proximity_prize_results_v4.tex` is the current compact synthesis of proved partial results. It is exposition/status guidance, not a replacement proof source for a live atom or adjacent-row closure. - A schema/hash pass is structural preflight only. The trusted-source registry is empty; parsing a manifest does not prove an atom. diff --git a/experimental/agents-log.md b/experimental/agents-log.md index cace093e0..07076b8b2 100644 --- a/experimental/agents-log.md +++ b/experimental/agents-log.md @@ -57,7 +57,11 @@ Keep entries concise and link to the relevant files. affine hyperplane of codeword gauges drops explanation rank to `K-1`, and lifted rank `K+1`, where no gauge drops rank. This pays the rank-drop suffixes at `e>=992852` and `e>=1037876` while isolating the full-lift - middle intervals as the true remaining top-rank residuals. + middle intervals as the true remaining top-rank residuals. The additional + near-MDS extension theorem proves that `W=C+` has first generalized + weight `e` and every higher weight MDS-sharp. Its generic endpoint bound + still exceeds both budgets, ruling out another weight-hierarchy replay as + a closure route. - **What to do next:** Replace the false ordered-basis denominator by a sharper aggregate theorem, or route the remaining top-rank K3 cells through a different slope compiler. The common-core cancellation and diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 01b579104..67d4f0860 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1608,6 +1608,52 @@ \section{Higher-codimension support-flag recursion} rank is therefore \(\dim V\). \end{proof} +\begin{theorem}[Full-lift near-MDS extension reduction] +\label{thm:full-lift-near-mds-extension} +Retain the hypotheses of +\cref{thm:full-explanation-lifted-rank-dichotomy}, assume +\(\dim V=K+1\), write \(N=|D|\), and put +\[ + W=C+\langle r_1\rangle,\qquad + e=\min_{b\in C}\wt(r_1-b). +\] +Then \(\dim W=K+1\), and the generalized Hamming weights of \(W\) are +\[ + d_1(W)=e,\qquad + d_j(W)=N-K+j-1\quad(2\le j\le K+1). + \tag{NM} +\] +The selected errors \(r_\gamma-c_\gamma\) lie in one affine coset of +\(W\), have full affine rank \(K+1\), and determine their slopes uniquely. +Moreover, if every selected maximal agreement support has size greater than +\(K\), restriction of \(W\) to each such support is injective. +\end{theorem} + +\begin{proof} +Pair noncontainment gives \(r_1\notin C\), so \(W\) has dimension \(K+1\). +Interpolation on \(K\) coordinates gives \(e\le N-K\). Every nonzero word +of \(W\) is either a nonzero RS word, of weight at least \(N-K+1\), or a +nonzero scalar multiple of \(r_1-b\), of weight at least \(e\); a nearest +\(b\) attains \(e\). Thus \(d_1(W)=e\). + +Let \(A\le W\) have dimension \(j\ge2\). If \(A\le C\), the RS MDS +hierarchy gives \(|\operatorname{supp}A|\ge N-K+j\). Otherwise +\(\dim(A\cap C)=j-1\), and that RS subcode has at most \(K-j+1\) common +zero coordinates. Hence +\[ + |\operatorname{supp}A|\ge N-K+j-1. +\] +The generalized Singleton bound for the \([N,K+1]\) code \(W\) is the +reverse inequality, proving \((\mathrm{NM})\). + +The map \((a,u)\mapsto ar_1-u\) is an isomorphism +\(\F\oplus C\to W\), so the full lifted rank is exactly the error affine +rank and equal errors have equal slopes. Finally, a nonzero +\(ar_1-u\in W\) vanishing on a selected support either contradicts the RS +root bound when \(a=0\), or supplies simultaneous direction and base +explanations when \(a\ne0\). Pair noncontainment excludes both cases. +\end{proof} + \begin{remark}[Exact top-rank split at the first shortened rows] \label{rem:full-explanation-top-rank-split} The gauge preserves agreement supports, pair noncontainment, and @@ -1626,7 +1672,18 @@ \section{Higher-codimension support-flag recursion} \end{array} \] The improvement is only in the \(h=K\) rows. The full-lift \(h=K+1\) -middle intervals remain unpaid and are the wider top-rank residuals. A +middle intervals remain unpaid and are the wider top-rank residuals. +\Cref{thm:full-lift-near-mds-extension} shows that every generalized +weight from rank two onward is already MDS-sharp. Even at the best endpoint +\(e=N-K\), the generic corrected compiler gives +\[ + 743896698428332665>274980728111395087 + \quad\text{and}\quad + 219426634>16777215 +\] +on KoalaBear and Mersenne-31, respectively. Closing these cells therefore +requires structure of the codimension-one RS extension beyond its weight +hierarchy, or a sharper row-specific affine-list theorem. A finite-field audit and the exact adjacent arithmetic are in \texttt{experimental/verify\_mca\_full\_explanation\_lifted\_rank\_gauge\_dichotomy\_v1.py}. \end{remark} diff --git a/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md b/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md index fad0a649f..984fe1fa4 100644 --- a/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md +++ b/experimental/notes/thresholds/mca_affine_span_incidence_counterexample_v1.md @@ -188,6 +188,27 @@ The full-lift branches, of error ranks 15 and 7 respectively, do not receive this payment. Their current high-support suffixes remain `e>=1044239` and `e>=1044242`, so their middle-support intervals are still open. +The full-lift residual has a sharper exact formulation. With + +```text +W=C+span{r_1}, +``` + +the selected errors form a full-affine-rank sparse list in one affine coset +of `W`, with unit slope fibers and information-set maximal zero masks. The +complete generalized-weight hierarchy is + +```text +d_1(W)=e, +d_j(W)=N-K+j-1 for 2<=j<=K+1. +``` + +Thus every higher weight is already MDS-sharp. Even at `e=N-K`, the generic +corrected compiler gives `743896698428332665` on KoalaBear and `219426634` +on Mersenne-31, above their respective budgets. Replaying the same weight +hierarchy cannot close the cell; a continuation must exploit structure of +the codimension-one RS extension or improve the row-sharp affine-list count. + ## Replay ```bash @@ -203,5 +224,5 @@ Expected output: MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_PASS slopes=31 bound=23 direction_max=20 mutations=4/4 MCA_AFFINE_SPAN_INCIDENCE_COUNTEREXAMPLE_V1_INDEPENDENT_PASS slopes=31 support_checks=31 field_values=1009 bound=23 MCA_PROPER_SUBSPACE_OCCUPANCY_COMPILER_V1_PASS zero_normal_cases=616 adjacent=8 regression=471 toy_selections=540 -MCA_FULL_EXPLANATION_LIFTED_RANK_GAUGE_DICHOTOMY_V1_PASS gauges=625 hostile=1 adjacent=2 +MCA_FULL_EXPLANATION_LIFTED_RANK_GAUGE_DICHOTOMY_V1_PASS gauges=625 hostile=1 extensions=31 profile=1:5,2:25,3:1 adjacent=2 ceilings=2 ``` diff --git a/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py b/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py index e4d0a82fc..1ebd91022 100644 --- a/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py +++ b/experimental/verify_mca_full_explanation_lifted_rank_gauge_dichotomy_v1.py @@ -119,6 +119,81 @@ def audit_small_field() -> tuple[int, int]: return gauge_checks, 1 +def word_weight(word: tuple[int, ...]) -> int: + return sum(value != 0 for value in word) + + +def span_word( + codeword: tuple[int, ...], scalar: int, direction: tuple[int, ...], prime: int +) -> tuple[int, ...]: + return tuple((value + scalar * extra) % prime + for value, extra in zip(codeword, direction)) + + +def dependent( + left: tuple[int, ...], right: tuple[int, ...], prime: int +) -> bool: + return any( + all((b - scalar * a) % prime == 0 for a, b in zip(left, right)) + for scalar in range(prime) + ) + + +def extension_weights( + words: frozenset[tuple[int, ...]], prime: int +) -> tuple[int, int, int]: + nonzero = [word for word in words if any(word)] + first = min(map(word_weight, nonzero)) + second = len(nonzero[0]) + for index, left in enumerate(nonzero): + for right in nonzero[index + 1:]: + if dependent(left, right, prime): + continue + union = sum(a != 0 or b != 0 for a, b in zip(left, right)) + second = min(second, union) + full = sum( + any(word[index] for word in words) + for index in range(len(nonzero[0])) + ) + return first, second, full + + +def audit_near_mds_extensions() -> tuple[int, dict[int, int]]: + prime, length = 5, 5 + code = [ + tuple((constant + slope * x) % prime for x in range(length)) + for constant, slope in product(range(prime), repeat=2) + ] + code_set = frozenset(code) + extensions: dict[frozenset[tuple[int, ...]], int] = {} + for direction in product(range(prime), repeat=length): + if direction in code_set: + continue + extension = frozenset( + span_word(codeword, scalar, direction, prime) + for codeword in code + for scalar in range(prime) + ) + distance = min( + word_weight(tuple((a - b) % prime + for a, b in zip(direction, codeword))) + for codeword in code + ) + if extension in extensions and extensions[extension] != distance: + raise ValueError("extension distance invariance") + extensions[extension] = distance + if len(extensions) != 31: + raise ValueError("extension census") + profile: dict[int, int] = {} + for extension, distance in extensions.items(): + if extension_weights(extension, prime) != (distance, 4, 5): + raise ValueError("near-MDS hierarchy") + profile[distance] = profile.get(distance, 0) + 1 + if profile != {1: 5, 2: 25, 3: 1}: + raise ValueError("extension profile") + return len(extensions), profile + + def falling(value: int, length: int) -> int: return prod(value - offset for offset in range(length)) @@ -164,7 +239,7 @@ def first_factor( return low -def audit_deployed_walls() -> int: +def audit_deployed_walls() -> tuple[int, int]: rows = ( { "name": "KoalaBear", @@ -176,6 +251,8 @@ def audit_deployed_walls() -> int: "low": 5, "frontier_j": 4337, "full_high": 1044239, + "mds_endpoint": 743896698428332665, + "required_top_factor": 182530, }, { "name": "Mersenne-31", @@ -187,9 +264,11 @@ def audit_deployed_walls() -> int: "low": 1, "frontier_j": 4334, "full_high": 1044242, + "mds_endpoint": 219426634, + "required_top_factor": 882143, }, ) - adjacent = 0 + adjacent = ceilings = 0 for row in rows: rank = row["K"] - 1 factor = first_factor( @@ -209,16 +288,31 @@ def audit_deployed_walls() -> int: raise ValueError(row["name"] + " full-lift arithmetic") if not row["low"] < row["drop_high"] < row["full_high"]: raise ValueError(row["name"] + " support ordering") + endpoint = occupancy_bound( + row["R"], row["d"], row["K"], row["K"], row["d"] + ) + required = first_factor( + row["R"], row["d"], row["K"], row["K"], row["budget"] + ) + if endpoint != row["mds_endpoint"] or not endpoint > row["budget"]: + raise ValueError(row["name"] + " MDS endpoint") + if required != row["required_top_factor"] or not required > row["d"]: + raise ValueError(row["name"] + " required top factor") adjacent += 1 - return adjacent + ceilings += 1 + return adjacent, ceilings def main() -> None: gauges, hostile = audit_small_field() - adjacent = audit_deployed_walls() + extensions, profile = audit_near_mds_extensions() + adjacent, ceilings = audit_deployed_walls() + profile_text = ",".join(f"{distance}:{count}" + for distance, count in sorted(profile.items())) print( "MCA_FULL_EXPLANATION_LIFTED_RANK_GAUGE_DICHOTOMY_V1_PASS " - f"gauges={gauges} hostile={hostile} adjacent={adjacent}" + f"gauges={gauges} hostile={hostile} extensions={extensions} " + f"profile={profile_text} adjacent={adjacent} ceilings={ceilings}" ) From 442a223a0c4232674c6d19078cfa9165d6dc8a11 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 05:05:40 +0100 Subject: [PATCH 05/40] MCA: add punctured Johnson support profile --- agents.md | 2 +- experimental/agents-log.md | 22 +++ experimental/grande_finale.tex | 114 +++++++++++- ..._direction_punctured_johnson_profile_v1.md | 62 +++++++ ..._direction_punctured_johnson_profile_v1.py | 175 ++++++++++++++++++ 5 files changed, 370 insertions(+), 5 deletions(-) create mode 100644 experimental/notes/thresholds/mca_sparse_direction_punctured_johnson_profile_v1.md create mode 100755 experimental/verify_mca_sparse_direction_punctured_johnson_profile_v1.py diff --git a/agents.md b/agents.md index 5c6439996..95745b9d5 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. Those full-lift middle cells therefore require extension-specific or row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile separately pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne, with exact endpoint bounds `4607583` and `2605443`. The remaining full-lift intervals are therefore `63909<=e<=1044238` and `65237<=e<=1044241` and require extension-specific or post-Johnson row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. - Grande Finale v4 explicitly supersedes the old v3 `prob:saturated-bc` status: primitive one-pencil MCA BC is proved; higher-dimensional MCA BC remains jointly governed by spread-component, large-owner, and exception routing; and the list-interior clause is governed by row-sharp list completion. The spread-abundance and `prob:next` statements are intermediate forms, not additional terminal inputs. - `experimental/proximity_prize_results_v4.tex` is the current compact synthesis of proved partial results. It is exposition/status guidance, not a replacement proof source for a live atom or adjacent-row closure. - A schema/hash pass is structural preflight only. The trusted-source registry is empty; parsing a manifest does not prove an atom. diff --git a/experimental/agents-log.md b/experimental/agents-log.md index 07076b8b2..b7b58213e 100644 --- a/experimental/agents-log.md +++ b/experimental/agents-log.md @@ -30,6 +30,28 @@ Keep entries concise and link to the relevant files. ## Entries +### 2026-08-13 - Sparse-direction punctured Johnson profile + +- **Agent/model:** Codex. +- **Files added or changed:** `experimental/grande_finale.tex`, + `experimental/notes/thresholds/mca_sparse_direction_punctured_johnson_profile_v1.md`, + `experimental/verify_mca_sparse_direction_punctured_johnson_profile_v1.py`, + `agents.md`, and `experimental/agents-log.md`. +- **Status:** PROVED / EXACT FINITE CALIBRATION. +- **What is being added:** A field-general ordinary-list Johnson incidence + cap after puncturing the codeword-gauged direction support, combined with + the deficit-dependent `floor(e/h)` slope-owner profile. Exact arithmetic + pays `e<=63908` on KoalaBear and `e<=65236` on Mersenne-31, against the + previous top-rank low-support walls `e<=5` and `e<=1`. +- **How it is useful:** It removes over sixty thousand support values from + each top-rank MCA residual without using the refuted affine-incidence + denominator. The full-lift middle intervals are now exactly + `63909<=e<=1044238` and `65237<=e<=1044241`. +- **What to do next:** Attack the adjacent nonpositive-Johnson strip using + the codimension-one near-MDS extension structure or a row-sharp + post-Johnson list theorem. Denominator failure is not an unsafe-row + certificate. + ### 2026-08-13 - Affine-span MCA compiler repair and top-rank split - **Agent/model:** Codex. diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 67d4f0860..78e5891d7 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1549,6 +1549,109 @@ \section{Higher-codimension support-flag recursion} \texttt{experimental/verify\_mca\_proper\_subspace\_occupancy\_compiler\_v1.py}. \end{remark} +\begin{theorem}[Sparse-direction punctured Johnson MCA profile] +\label{thm:sparse-direction-punctured-johnson-mca} +Let \(C=\RS[\F,D,K]\) have +\[ + N=R+K,\qquad m=d+K, +\] +and let \(r_\gamma=r_0+\gamma r_1\). Suppose +\[ + r_1=b+q,\qquad b\in C,\qquad + E=\operatorname{supp}(q),\qquad |E|=e0. + \label{eq:punctured-johnson-positive} +\end{equation} +Put \(J_0=0\) and, for \(1\le h\le e\), +\begin{equation} + J_h=\left\lfloor + \frac{(N-e)(m-h-K+1)}{(m-h)^2-(N-e)(K-1)} + \right\rfloor . + \label{eq:punctured-johnson-cap} +\end{equation} +The selected slope set \(Z\) satisfies +\begin{equation} + |Z|\le \sum_{h=1}^e(J_h-J_{h-1})\left\lfloor\frac eh\right\rfloor + \le (e-1)J_{\lfloor e/2\rfloor}+J_e. + \label{eq:punctured-johnson-profile} +\end{equation} +For \(e=1\), the term indexed by zero in the last expression is zero. +\end{theorem} + +\begin{proof} +Outside \(E\), the direction equals \(b\). If \(a\) agreed with \(r_0\) +on at least \(m\) outside coordinates, an \(m\)-subset there would be +simultaneously explained by \((a,b)\), contrary to same-support +pair noncontainment. Hence \(h_a\ge1\). A size-\(m\) witness can use at +most \(e\) coordinates of \(E\), so \(h_a\le e\). + +For \(x\in E\), agreement at slope \(\gamma\) is equivalent to +\[ + \frac{a(x)-r_0(x)}{q(x)}=\gamma. +\] +These ratio fibers are disjoint. A deficit-\(h\) explanation needs at +least \(h\) inside agreements, and therefore owns at most +\(\lfloor e/h\rfloor\) slopes. + +Puncture \(E\) and fix \(h\). Each explanation of deficit at most \(h\) +has an agreement set of size at least \(A=m-h\) in length \(n'=N-e\). +Agreement sets of two distinct degree-\(n'(K-1)\), comparison yields \(L\le J_h\). Condition +\eqref{eq:punctured-johnson-positive} is the weakest, \(h=e\), denominator and hence makes +every \(J_h\) legal. The Johnson fractions are nondecreasing in \(h\). +Applying their cumulative caps to the nonincreasing owner weights proves +the first inequality in \eqref{eq:punctured-johnson-profile}. + +Finally put \(u=\lfloor e/2\rfloor\). Bound all weights with \(h\le u\) +by \(e\); every weight with \(h>u\) equals one. The two telescoping blocks +give +\[ + eJ_u+(J_e-J_u)=(e-1)J_u+J_e. +\] +\end{proof} + +\begin{remark}[Exact sparse-direction support walls] +At the first shortened KoalaBear and Mersenne-31 rows, exact integer +evaluation gives +\[ +\begin{array}{c|c|c|c|c} +\text{row}&K&\text{last paid }e&\Delta_e& + (e-1)J_{\lfloor e/2\rfloor}+J_e\\ \hline +\text{KoalaBear}&14&63908&1218&4607583\\ +\text{Mersenne-31}&6&65236&2794&2605443 +\end{array} +\] +against budgets \(274980728111395087\) and \(16777215\). At the adjacent +supports the denominators are \(-5924\) and \(-1636\), so this theorem +makes no claim there. The exact scan is in +\texttt{experimental/verify\_mca\_sparse\_direction\_punctured\_johnson\_profile\_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let @@ -1665,14 +1768,17 @@ \section{Higher-codimension support-flag recursion} \[ \begin{array}{c|c|c|c} \text{row}&q&h&\text{currently paid support range}\\ \hline -\text{KoalaBear}&14&14&e\le5\ \text{or}\ e\ge992852\\ -\text{KoalaBear}&14&15&e\le5\ \text{or}\ e\ge1044239\\ -\text{Mersenne-31}&6&6&e\le1\ \text{or}\ e\ge1037876\\ -\text{Mersenne-31}&6&7&e\le1\ \text{or}\ e\ge1044242 +\text{KoalaBear}&14&14&e\le63908\ \text{or}\ e\ge992852\\ +\text{KoalaBear}&14&15&e\le63908\ \text{or}\ e\ge1044239\\ +\text{Mersenne-31}&6&6&e\le65236\ \text{or}\ e\ge1037876\\ +\text{Mersenne-31}&6&7&e\le65236\ \text{or}\ e\ge1044242 \end{array} \] The improvement is only in the \(h=K\) rows. The full-lift \(h=K+1\) middle intervals remain unpaid and are the wider top-rank residuals. +After \cref{thm:sparse-direction-punctured-johnson-mca}, those full-lift +intervals are exactly \(63909\le e\le1044238\) for KoalaBear and +\(65237\le e\le1044241\) for Mersenne-31. \Cref{thm:full-lift-near-mds-extension} shows that every generalized weight from rank two onward is already MDS-sharp. Even at the best endpoint \(e=N-K\), the generic corrected compiler gives diff --git a/experimental/notes/thresholds/mca_sparse_direction_punctured_johnson_profile_v1.md b/experimental/notes/thresholds/mca_sparse_direction_punctured_johnson_profile_v1.md new file mode 100644 index 000000000..94b4b5769 --- /dev/null +++ b/experimental/notes/thresholds/mca_sparse_direction_punctured_johnson_profile_v1.md @@ -0,0 +1,62 @@ +# Sparse-direction punctured Johnson MCA profile + +## Status + +PROVED, field-general, exact finite arithmetic. + +## Statement + +After a codeword direction gauge with residual support `E`, `|E|=e=984668, endpoint denominator 1218, + endpoint bound 4607583 <= 274980728111395087. + +Mersenne-31 K=6: + e<=65236, j=R-e>=983340, endpoint denominator 2794, + endpoint bound 2605443 <= 16777215. +``` + +The adjacent denominators are `-5924` and `-1636`. They mark failure of +this Johnson proof, not unsafe rows. + +Combined with the high-support proper-subspace walls, the full-lift +top-rank residual intervals are now + +```text +KoalaBear: 63909<=e<=1044238; +Mersenne: 65237<=e<=1044241. +``` + +## Audit + +`experimental/verify_mca_sparse_direction_punctured_johnson_profile_v1.py` +scans all 129,144 paid official supports exactly, reconstructs both adjacent +walls, independently checks the profile coarsening on two synthetic rows, +and rejects hostile constant mutations. It uses standard-library integer +arithmetic and negligible memory. + +## Nonclaims + +This does not pay any support after the displayed walls, prove Johnson +sharpness, or close either official MCA row. diff --git a/experimental/verify_mca_sparse_direction_punctured_johnson_profile_v1.py b/experimental/verify_mca_sparse_direction_punctured_johnson_profile_v1.py new file mode 100755 index 000000000..806738813 --- /dev/null +++ b/experimental/verify_mca_sparse_direction_punctured_johnson_profile_v1.py @@ -0,0 +1,175 @@ +#!/usr/bin/env python3 +"""Audit the punctured-Johnson sparse-direction MCA profile.""" + +from __future__ import annotations + +import copy +from fractions import Fraction + + +ROWS = { + "KoalaBear": { + "R": 1048576, + "d": 67472, + "K": 14, + "budget": 274980728111395087, + "last": 63908, + "defect": 984668, + "dlast": 1218, + "dnext": -5924, + "half": 31954, + "jhalf": 27, + "jlast": 2882094, + "coarse": 4607583, + }, + "Mersenne-31": { + "R": 1048576, + "d": 67448, + "K": 6, + "budget": 16777215, + "last": 65236, + "defect": 983340, + "dlast": 2794, + "dnext": -1636, + "half": 32618, + "jhalf": 28, + "jlast": 778863, + "coarse": 2605443, + }, +} + + +class Reject(ValueError): + pass + + +def denominator(R: int, d: int, K: int, e: int, h: int | None = None) -> int: + if h is None: + h = e + return (d + K - h) ** 2 - (R + K - e) * (K - 1) + + +def cap(R: int, d: int, K: int, e: int, h: int) -> int: + if h == 0: + return 0 + den = denominator(R, d, K, e, h) + if den <= 0: + raise Reject("nonpositive denominator") + return (R + K - e) * (d - h + 1) // den + + +def coarse(R: int, d: int, K: int, e: int) -> int: + return (e - 1) * cap(R, d, K, e, e // 2) + cap(R, d, K, e, e) + + +def derive(row: dict[str, int]) -> dict[str, int]: + R, d, K, budget = (row[key] for key in ("R", "d", "K", "budget")) + last = 0 + best = (-1, -1) + previous = None + checks = 0 + for e in range(1, d): + den = denominator(R, d, K, e) + if den <= 0: + break + if previous is not None and den >= previous: + raise Reject("denominator not descending") + previous = den + value = coarse(R, d, K, e) + if value > budget: + raise Reject("budget failure inside Johnson prefix") + if value > best[1]: + best = (e, value) + last = e + checks += 1 + half = last // 2 + return { + "last": last, + "defect": R - last, + "dlast": denominator(R, d, K, last), + "dnext": denominator(R, d, K, last + 1), + "half": half, + "jhalf": cap(R, d, K, last, half), + "jlast": cap(R, d, K, last, last), + "coarse": best[1], + "maximizer": best[0], + "checks": checks, + } + + +def independent_cap(R: int, d: int, K: int, e: int, h: int) -> int: + if h == 0: + return 0 + length = R + K - e + agreement = d + K - h + den = agreement * agreement - length * (K - 1) + if den <= 0: + raise Reject("independent denominator") + value = Fraction(length * (agreement - K + 1), den) + return value.numerator // value.denominator + + +def profile_controls() -> int: + checks = 0 + for R, d, K in ((101, 20, 3), (211, 31, 5)): + for e in range(1, min(d, 18)): + if denominator(R, d, K, e) <= 0: + continue + previous = 0 + profile = 0 + for h in range(1, e + 1): + current = independent_cap(R, d, K, e, h) + if current < previous: + raise Reject("cumulative cap") + profile += (current - previous) * (e // h) + previous = current + upper = ( + (e - 1) * independent_cap(R, d, K, e, e // 2) + + independent_cap(R, d, K, e, e) + ) + if profile > upper: + raise Reject("coarse profile") + checks += 1 + return checks + + +def validate(rows: dict[str, dict[str, int]]) -> int: + checks = 0 + for name, expected in ROWS.items(): + row = rows[name] + derived = derive(row) + for key in ( + "last", "defect", "dlast", "dnext", "half", "jhalf", + "jlast", "coarse", + ): + if derived[key] != expected[key] or row[key] != expected[key]: + raise Reject(f"{name}: {key}") + checks += 1 + if derived["maximizer"] != expected["last"]: + raise Reject(f"{name}: maximizer") + checks += derived["checks"] + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + profile_controls() + mutations = [] + for name, key in (("KoalaBear", "last"), ("Mersenne-31", "coarse")): + changed = copy.deepcopy(ROWS) + changed[name][key] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_SPARSE_DIRECTION_PUNCTURED_JOHNSON_PROFILE_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 07546f9036174e56bb09a095e6d9c0331b1e903f Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 05:20:47 +0100 Subject: [PATCH 06/40] MCA: continue support wall beyond Johnson --- agents.md | 2 +- experimental/agents-log.md | 18 +++ experimental/grande_finale.tex | 134 +++++++++++++++++- ...rse_direction_near_johnson_gram_rank_v1.md | 65 +++++++++ ...rse_direction_near_johnson_gram_rank_v1.py | 126 ++++++++++++++++ 5 files changed, 337 insertions(+), 8 deletions(-) create mode 100644 experimental/notes/thresholds/mca_sparse_direction_near_johnson_gram_rank_v1.md create mode 100755 experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py diff --git a/agents.md b/agents.md index 95745b9d5..69bef8723 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile separately pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne, with exact endpoint bounds `4607583` and `2605443`. The remaining full-lift intervals are therefore `63909<=e<=1044238` and `65237<=e<=1044241` and require extension-specific or post-Johnson row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne. A centered-incidence Gram-rank continuation in the first post-Johnson strip extends these walls to `e=64037` and `e=65418`, with endpoint bounds `198047217` and `16759641`. The remaining full-lift intervals are therefore `64038<=e<=1044238` and `65419<=e<=1044241` and require stronger extension-specific or post-Johnson row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. - Grande Finale v4 explicitly supersedes the old v3 `prob:saturated-bc` status: primitive one-pencil MCA BC is proved; higher-dimensional MCA BC remains jointly governed by spread-component, large-owner, and exception routing; and the list-interior clause is governed by row-sharp list completion. The spread-abundance and `prob:next` statements are intermediate forms, not additional terminal inputs. - `experimental/proximity_prize_results_v4.tex` is the current compact synthesis of proved partial results. It is exposition/status guidance, not a replacement proof source for a live atom or adjacent-row closure. - A schema/hash pass is structural preflight only. The trusted-source registry is empty; parsing a manifest does not prove an atom. diff --git a/experimental/agents-log.md b/experimental/agents-log.md index b7b58213e..143e30868 100644 --- a/experimental/agents-log.md +++ b/experimental/agents-log.md @@ -30,6 +30,24 @@ Keep entries concise and link to the relevant files. ## Entries +### 2026-08-13 - Near-Johnson centered-Gram payment + +- **Agent/model:** Codex. +- **Files added or changed:** `experimental/grande_finale.tex`, + `experimental/notes/thresholds/mca_sparse_direction_near_johnson_gram_rank_v1.md`, + `experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py`, + `agents.md`, and `experimental/agents-log.md`. +- **Status:** PROVED / EXACT FINITE CALIBRATION. +- **What is being added:** A centered incidence-Gram rank bound for + equal-size ordinary-list agreement blocks in the first post-Johnson + regime, followed by a two-threshold sparse-direction slope compiler. +- **How it is useful:** It extends the low-support payments from `63908` to + `64037` on KoalaBear and from `65236` to `65418` on Mersenne-31. The + endpoint slope bounds are `198047217` and `16759641`. +- **What to do next:** A stronger post-Johnson list theorem is needed at + KoalaBear `e=64038`; at Mersenne `e=65419`, the Gram theorem remains valid + but misses budget by `1434789`. Neither failure is an unsafe certificate. + ### 2026-08-13 - Sparse-direction punctured Johnson profile - **Agent/model:** Codex. diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 78e5891d7..d1ed593fb 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1652,6 +1652,126 @@ \section{Higher-codimension support-flag recursion} \texttt{experimental/verify\_mca\_sparse\_direction\_punctured\_johnson\_profile\_v1.py}. \end{remark} +\begin{theorem}[Near-Johnson centered-Gram list compiler] +\label{thm:near-johnson-centered-gram} +Let \(S_1,\ldots,S_L\) be distinct \(A\)-subsets of an \(n\)-set such +that \(|S_i\cap S_j|\le c\) for \(i\ne j\), where \(A>c\). Put +\[ + g=nc-A^2\ge0, + \qquad + G=(A-c)^2-cg. +\] +If \(G>0\), then +\begin{equation} + L\le\left\lfloor\frac{nA(A-c)}{G}\right\rfloor. + \label{eq:near-johnson-centered-gram} +\end{equation} +\end{theorem} + +\begin{proof} +Let \(B\) be the \(L\)-by-\(n\) incidence matrix and put +\[ + H=BB^{\mathsf T}-c\mathbf1\mathbf1^{\mathsf T}. +\] +Every row of \(B\) has sum \(A\), so +\(\mathbf1=A^{-1}B\mathbf1_n\). The images of both summands defining +\(H\) therefore lie in \(\operatorname{col}(B)\), and +\begin{equation} + \operatorname{rank}(H)\le\operatorname{rank}(B)\le n. + \label{eq:centered-gram-rank} +\end{equation} + +Write \(\delta_{ij}=c-|S_i\cap S_j|\). Then +\(0\le\delta_{ij}\le c\) is integral and +\[ + \operatorname{tr}(H)=L(A-c),\qquad + \operatorname{tr}(H^2)=L(A-c)^2+2\sum_{i0,qquad + g=nc-A^2\ge0,qquad + G=(A-c)^2-cg>0. +\] +Define +\[ + J_u=\left\lfloor\frac{n(m-u-K+1)}{(m-u)^2-nc}\right\rfloor, + \qquad + Q_e=\left\lfloor\frac{nA(A-c)}G\right\rfloor. +\] +Then +\begin{equation} + |Z|\le(e-1)J_u+Q_e. \label{eq:near-johnson-mca} +\end{equation} +\end{corollary} + +\begin{proof} +Let \(N_u\) be the number of transformed explanations with outside deficit +at most \(u\), and let \(N_e\) be their total number. The deficit-owner +bound gives +\[ + |Z|\le eN_u+(N_e-N_u)=(e-1)N_u+N_e. +\] +The ordinary Johnson argument on the punctured row gives \(N_u\le J_u\). +For each explanation counted by \(N_e\), choose exactly \(A=m-e\) outside +agreement coordinates. Distinct degree-\(=0, +G=(A-c)^2-cg. +``` + +When `G>0`, the centered incidence Gram matrix gives + +```text +L <= floor(n*A*(A-c)/G). +``` + +The key exact rank improvement is `rank(BB^T-cJ)<=rank(B)<=n`: equal row +sums put the all-ones vector inside `col(B)`. Trace-rank, Cauchy incidence, +and the integer inequality `delta^2<=c*delta` complete the proof. + +## MCA compiler + +After puncturing a gauged direction support of size `e`, split transformed +explanations at deficit `u=floor(e/2)`. The low-deficit count has the +ordinary Johnson cap `J_u`; all remaining explanations own one slope. If +`Q_e` is the centered-Gram cap at agreement `m-e`, then + +```text +|Z| <= (e-1)J_u+Q_e. +``` + +## Exact walls + +```text +KoalaBear: e<=64037, bound 198047217; +Mersenne-31: e<=65418, bound 16759641. +``` + +At KoalaBear `e=64038`, the Gram denominator is `-36911`. At Mersenne +`e=65419`, it remains positive, but the valid bound `18212004` exceeds the +budget `16777215`. + +The full-lift residual intervals become + +```text +KoalaBear: 64038<=e<=1044238; +Mersenne: 65419<=e<=1044241. +``` + +## Audit + +`experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py` +checks all 311 newly paid supports, both adjacent records, a rational replay, +an explicit finite block control, and two hostile mutations. + +## Nonclaims + +No adjacent cell or official row is closed. A negative Gram denominator or +an over-budget upper bound is not an unsafe certificate. diff --git a/experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py b/experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py new file mode 100755 index 000000000..0e013330a --- /dev/null +++ b/experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py @@ -0,0 +1,126 @@ +#!/usr/bin/env python3 +"""Audit the sparse-direction near-Johnson centered-Gram payment.""" + +from __future__ import annotations + +import copy +from fractions import Fraction +from itertools import combinations + + +ROWS = { + "KoalaBear": { + "R": 1048576, "d": 67472, "K": 14, + "budget": 274980728111395087, + "first": 63909, "last": 64037, "defect": 984539, + "n": 984553, "A": 3449, "g": 903588, "G": 59452, + "Q": 196254209, "u": 32018, "J": 28, + "bound": 198047217, "next": 64038, + "next_G": -36911, "next_bound": None, + }, + "Mersenne-31": { + "R": 1048576, "d": 67448, "K": 6, + "budget": 16777215, + "first": 65237, "last": 65418, "defect": 983158, + "n": 983164, "A": 2036, "g": 770524, "G": 272341, + "Q": 14927965, "u": 32709, "J": 28, + "bound": 16759641, "next": 65419, + "next_G": 247950, "next_bound": 18212004, + }, +} + + +class Reject(ValueError): + pass + + +def record(R: int, d: int, K: int, e: int) -> dict[str, int | None]: + n = R + K - e + A = d + K - e + c = K - 1 + g = n * c - A * A + G = (A - c) ** 2 - c * g + Q = None if G <= 0 else n * A * (A - c) // G + u = e // 2 + Au = d + K - u + D = Au * Au - n * c + if D <= 0: + raise Reject("half Johnson denominator") + J = n * (Au - c) // D + return { + "n": n, "A": A, "g": g, "G": G, "Q": Q, "u": u, "J": J, + "bound": None if Q is None else (e - 1) * J + Q, + } + + +def validate(rows: dict[str, dict[str, int | None]]) -> int: + checks = 0 + for name, expected in ROWS.items(): + row = rows[name] + if row["defect"] != row["R"] - row["last"]: + raise Reject(f"{name}: defect") + endpoint = record(row["R"], row["d"], row["K"], row["last"]) + for key in ("n", "A", "g", "G", "Q", "u", "J", "bound"): + if endpoint[key] != expected[key] or row[key] != expected[key]: + raise Reject(f"{name}: {key}") + checks += 1 + adjacent = record(row["R"], row["d"], row["K"], row["next"]) + if adjacent["G"] != row["next_G"] or adjacent["bound"] != row["next_bound"]: + raise Reject(f"{name}: adjacent") + for e in range(row["first"], row["last"] + 1): + item = record(row["R"], row["d"], row["K"], e) + if item["g"] < 0 or item["G"] <= 0: + raise Reject(f"{name}: hypotheses") + if item["bound"] is None or item["bound"] > row["budget"]: + raise Reject(f"{name}: budget") + checks += 1 + return checks + + +def independent_controls() -> int: + blocks = [ + {0, 1, 2}, {0, 3, 4}, {1, 3, 5}, {2, 4, 5}, + {0, 5, 6}, {1, 4, 6}, {2, 3, 6}, + ] + n, A, c = 10, 3, 1 + if any(len(left & right) > c for left, right in combinations(blocks, 2)): + raise Reject("block intersections") + g = n * c - A * A + G = (A - c) ** 2 - c * g + fraction = Fraction(n * A * (A - c), G) + if len(blocks) > fraction.numerator // fraction.denominator: + raise Reject("independent Gram cap") + + checks = len(blocks) ** 2 + for row in ROWS.values(): + endpoint = record(row["R"], row["d"], row["K"], row["last"]) + q = Fraction(endpoint["n"] * endpoint["A"] * + (endpoint["A"] - row["K"] + 1), endpoint["G"]) + if q.numerator // q.denominator != endpoint["Q"]: + raise Reject("rational replay") + checks += 1 + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + independent_controls() + mutations = [] + for name, key in (("KoalaBear", "G"), ("Mersenne-31", "bound")): + changed = copy.deepcopy(ROWS) + changed[name][key] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_SPARSE_DIRECTION_NEAR_JOHNSON_GRAM_RANK_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From d4d653723f2f82390fd4351476e1926e55fb0caf Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 05:31:57 +0100 Subject: [PATCH 07/40] MCA: sharpen post-Johnson Gram profile --- agents.md | 2 +- experimental/agents-log.md | 18 +++ experimental/grande_finale.tex | 123 ++++++++++++++- ...direction_mean_centered_gram_profile_v1.md | 63 ++++++++ ...direction_mean_centered_gram_profile_v1.py | 144 ++++++++++++++++++ 5 files changed, 342 insertions(+), 8 deletions(-) create mode 100644 experimental/notes/thresholds/mca_sparse_direction_mean_centered_gram_profile_v1.md create mode 100755 experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py diff --git a/agents.md b/agents.md index 69bef8723..4891391fa 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne. A centered-incidence Gram-rank continuation in the first post-Johnson strip extends these walls to `e=64037` and `e=65418`, with endpoint bounds `198047217` and `16759641`. The remaining full-lift intervals are therefore `64038<=e<=1044238` and `65419<=e<=1044241` and require stronger extension-specific or post-Johnson row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne. A centered-incidence Gram rung first extends these walls to `e=64037` and `e=65418`; the stronger PSD mean-centered profile extends them again to `e=64047` and `e=65454`, with endpoint profiles `181731868` and `16101127`. The remaining full-lift intervals are therefore `64048<=e<=1044238` and `65455<=e<=1044241` and require stronger extension-specific or post-Johnson row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. - Grande Finale v4 explicitly supersedes the old v3 `prob:saturated-bc` status: primitive one-pencil MCA BC is proved; higher-dimensional MCA BC remains jointly governed by spread-component, large-owner, and exception routing; and the list-interior clause is governed by row-sharp list completion. The spread-abundance and `prob:next` statements are intermediate forms, not additional terminal inputs. - `experimental/proximity_prize_results_v4.tex` is the current compact synthesis of proved partial results. It is exposition/status guidance, not a replacement proof source for a live atom or adjacent-row closure. - A schema/hash pass is structural preflight only. The trusted-source registry is empty; parsing a manifest does not prove an atom. diff --git a/experimental/agents-log.md b/experimental/agents-log.md index 143e30868..becfe46ae 100644 --- a/experimental/agents-log.md +++ b/experimental/agents-log.md @@ -30,6 +30,24 @@ Keep entries concise and link to the relevant files. ## Entries +### 2026-08-13 - Mean-centered Gram profile + +- **Agent/model:** Codex. +- **Files added or changed:** `experimental/grande_finale.tex`, + `experimental/notes/thresholds/mca_sparse_direction_mean_centered_gram_profile_v1.md`, + `experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py`, + `agents.md`, and `experimental/agents-log.md`. +- **Status:** PROVED / EXACT FINITE CALIBRATION. +- **What is being added:** A PSD incidence Gram theorem centered at the + block mean, plus the suffix-minimum closure needed to combine its raw + cumulative caps with the positive-Johnson caps. +- **How it is useful:** It strengthens the preceding centered-Gram rung and + extends the low-support walls to `e=64047` on KoalaBear and `e=65454` on + Mersenne-31. The endpoint profiles are `181731868` and `16101127`. +- **What to do next:** KoalaBear `e=64048` lies beyond the positive PSD + chord denominator. Mersenne `e=65455` remains theorem-legal but misses + budget by `342908`. Neither failure is an unsafe certificate. + ### 2026-08-13 - Near-Johnson centered-Gram payment - **Agent/model:** Codex. diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index d1ed593fb..60bd5ca5c 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1772,6 +1772,115 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_sparse_direction_near_johnson_gram_rank_v1.py}. \end{remark} +\begin{theorem}[Mean-centered Gram list compiler] +\label{thm:mean-centered-gram-list} +Let \(S_1,\ldots,S_L\) be distinct \(A\)-subsets of an \(n\)-set with +\(|S_i\cap S_j|\le c\) for \(i\ne j\), where \(A>c\). Put +\[ + g=nc-A^2, + \qquad + T=(n-A)^2-(n-1)g. +\] +If +\[ + g\ge0, + \qquad + 2A^2\ge nc, + \qquad + T>0, +\] +then +\begin{equation} + L\le\left\lfloor + \frac{(n-1)n^2(A-c)}{AT} + \right\rfloor. + \label{eq:mean-centered-gram-list} +\end{equation} +\end{theorem} + +\begin{proof} +Let \(B\) be the incidence matrix and put +\[ + P=I_n-\frac1nJ_n, + \qquad + H=BPB^{\mathsf T}. +\] +Then \(H\) is positive semidefinite and has rank at most \(n-1\). Set +\(p=A^2/n\). The diagonal entries are \(A-p\), while an off-diagonal +entry is +\[ + x_{ij}=|S_i\cap S_j|-p\in[-p,c-p]. +\] +Convexity gives the endpoint-chord inequality +\[ + x_{ij}^2\le(c-2p)x_{ij}+p(c-p). +\] +The assumption \(2A^2\ge nc\) makes the chord slope nonpositive. Since +\(H\) is positive semidefinite, +\[ + 2\sum_{i(N-e)(K-1)\), and otherwise the cap in +\cref{thm:mean-centered-gram-list} whenever its three hypotheses hold. +Assume every \(C_h\) is defined, and put +\[ + B_0=0, + \qquad + B_h=\min_{h\le v\le e}C_v. +\] +Then +\begin{equation} + |Z|\le\sum_{h=1}^e(B_h-B_{h-1}) + \left\lfloor\frac eh\right\rfloor. + \label{eq:mean-centered-sparse-direction-profile} +\end{equation} +\end{corollary} + +\begin{proof} +Let \(N_h\) count distinct transformed explanations with outside deficit at +most \(h\). Puncturing the direction support and applying the appropriate +ordinary-list theorem gives \(N_h\le C_h\). For every \(v\ge h\), also +\(N_h\le N_v\le C_v\), hence \(N_h\le B_h\). The suffix minima are +nondecreasing. Apply the deficit owner weight \(\lfloor e/h\rfloor\) and +sum by parts. +\end{proof} + +\begin{remark}[Exact mean-centered support walls] +The complete exact profile in +\eqref{eq:mean-centered-sparse-direction-profile} expands the walls to +\[ +\begin{array}{c|c|c|c} +\text{row}&K&\text{last paid }e&\text{endpoint profile}\\ \hline +\text{KoalaBear}&14&64047&181731868\\ +\text{Mersenne-31}&6&65454&16101127 +\end{array} +\] +At KoalaBear \(e=64048\), the endpoint denominator is +\(T=-1499457466\). At Mersenne \(e=65455\), every cap remains defined, +but the exact profile is \(17120123>16777215\). The replay is in +\path{experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let @@ -1888,17 +1997,17 @@ \section{Higher-codimension support-flag recursion} \[ \begin{array}{c|c|c|c} \text{row}&q&h&\text{currently paid support range}\\ \hline -\text{KoalaBear}&14&14&e\le64037\ \text{or}\ e\ge992852\\ -\text{KoalaBear}&14&15&e\le64037\ \text{or}\ e\ge1044239\\ -\text{Mersenne-31}&6&6&e\le65418\ \text{or}\ e\ge1037876\\ -\text{Mersenne-31}&6&7&e\le65418\ \text{or}\ e\ge1044242 +\text{KoalaBear}&14&14&e\le64047\ \text{or}\ e\ge992852\\ +\text{KoalaBear}&14&15&e\le64047\ \text{or}\ e\ge1044239\\ +\text{Mersenne-31}&6&6&e\le65454\ \text{or}\ e\ge1037876\\ +\text{Mersenne-31}&6&7&e\le65454\ \text{or}\ e\ge1044242 \end{array} \] The improvement is only in the \(h=K\) rows. The full-lift \(h=K+1\) middle intervals remain unpaid and are the wider top-rank residuals. -After \cref{cor:sparse-direction-near-johnson-mca}, those full-lift -intervals are exactly \(64038\le e\le1044238\) for KoalaBear and -\(65419\le e\le1044241\) for Mersenne-31. +After \cref{cor:mean-centered-sparse-direction-profile}, those full-lift +intervals are exactly \(64048\le e\le1044238\) for KoalaBear and +\(65455\le e\le1044241\) for Mersenne-31. \Cref{thm:full-lift-near-mds-extension} shows that every generalized weight from rank two onward is already MDS-sharp. Even at the best endpoint \(e=N-K\), the generic corrected compiler gives diff --git a/experimental/notes/thresholds/mca_sparse_direction_mean_centered_gram_profile_v1.md b/experimental/notes/thresholds/mca_sparse_direction_mean_centered_gram_profile_v1.md new file mode 100644 index 000000000..aebf91a81 --- /dev/null +++ b/experimental/notes/thresholds/mca_sparse_direction_mean_centered_gram_profile_v1.md @@ -0,0 +1,63 @@ +# Sparse-direction mean-centered Gram profile + +## Status + +PROVED, field-general, exact finite arithmetic. + +## Theorem + +For equal-size `A`-blocks in an `n`-set with intersections at most `c`, put + +```text +g=nc-A^2, +T=(n-A)^2-(n-1)g. +``` + +If `g>=0`, `2A^2>=nc`, and `T>0`, then mean-centering the incidence columns +gives + +```text +L <= floor((n-1)n^2(A-c)/(A*T)). +``` + +The centered Gram matrix is PSD of rank at most `n-1`. The square on its +off-diagonal interval lies below an endpoint chord with nonpositive slope; +PSD positivity and trace-rank give the formula. + +For the sparse-direction deficit profile, use the Johnson cap where its +denominator is positive and the mean-centered cap afterward. Since raw +caps need not be monotone, replace them by the proved suffix closure + +```text +B_h=min_(h<=v<=e) C_v +``` + +before applying `floor(e/h)` owner weights. + +## Exact walls + +```text +KoalaBear: e<=64047, profile 181731868; +Mersenne-31: e<=65454, profile 16101127. +``` + +The next KoalaBear endpoint has `T=-1499457466`. The next Mersenne profile +is legal but equals `17120123`, over budget by `342908`. + +The full-lift residual intervals become + +```text +KoalaBear: 64048<=e<=1044238; +Mersenne: 65455<=e<=1044241. +``` + +## Audit + +`experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py` +checks all 46 newly paid supports, both adjacent records, a finite block +control, and two hostile mutations. + +## Nonclaims + +No adjacent cell or official row is closed. A negative denominator or an +over-budget upper bound is not an unsafe certificate. diff --git a/experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py b/experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py new file mode 100755 index 000000000..5e55d792a --- /dev/null +++ b/experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py @@ -0,0 +1,144 @@ +#!/usr/bin/env python3 +"""Audit the exact mean-centered sparse-direction Gram profile.""" + +from __future__ import annotations + +import copy +from fractions import Fraction +from itertools import combinations + + +ROWS = { + "KoalaBear": { + "R": 1048576, "d": 67472, "K": 14, + "budget": 274980728111395087, + "first": 64038, "last": 64047, "defect": 984529, + "n": 984543, "A": 3439, "g": 972338, + "balance": 10854383, "T": 5257459620, + "Q": 180835154, "profile": 181731868, + "next": 64048, "next_T": -1499457466, "next_profile": None, + }, + "Mersenne-31": { + "R": 1048576, "d": 67448, "K": 6, + "budget": 16777215, + "first": 65419, "last": 65454, "defect": 983122, + "n": 983128, "A": 2000, "g": 915640, + "balance": 3084360, "T": 62421746104, + "Q": 15184718, "profile": 16101127, + "next": 65455, "next_T": 58496056500, + "next_profile": 17120123, + }, +} + + +class Reject(ValueError): + pass + + +def raw_cap(R: int, d: int, K: int, e: int, h: int) -> int | None: + n = R + K - e + A = d + K - h + c = K - 1 + D = A * A - n * c + if D > 0: + return n * (A - c) // D + g = -D + T = (n - A) ** 2 - (n - 1) * g + if g < 0 or 2 * A * A < n * c or T <= 0: + return None + return (n - 1) * n * n * (A - c) // (A * T) + + +def profile(R: int, d: int, K: int, e: int) -> int | None: + caps = [0] + for h in range(1, e + 1): + value = raw_cap(R, d, K, e, h) + if value is None: + return None + caps.append(value) + suffix = caps[-1] + for h in range(e - 1, 0, -1): + suffix = min(suffix, caps[h]) + caps[h] = suffix + if any(caps[h] < caps[h - 1] for h in range(1, e + 1)): + raise Reject("suffix monotonicity") + return sum((caps[h] - caps[h - 1]) * (e // h) for h in range(1, e + 1)) + + +def endpoint(row: dict[str, int | None], e: int) -> dict[str, int | None]: + R, d, K = (int(row[key]) for key in ("R", "d", "K")) + n = R + K - e + A = d + K - e + c = K - 1 + g = n * c - A * A + balance = 2 * A * A - n * c + T = (n - A) ** 2 - (n - 1) * g + Q = None + if g >= 0 and balance >= 0 and T > 0: + Q = (n - 1) * n * n * (A - c) // (A * T) + return { + "n": n, "A": A, "g": g, "balance": balance, "T": T, + "Q": Q, "profile": profile(R, d, K, e), + } + + +def validate(rows: dict[str, dict[str, int | None]]) -> int: + checks = 0 + for name, expected in ROWS.items(): + row = rows[name] + if row["defect"] != row["R"] - row["last"]: + raise Reject(f"{name}: defect") + observed = endpoint(row, int(row["last"])) + for key in ("n", "A", "g", "balance", "T", "Q", "profile"): + if observed[key] != expected[key] or row[key] != expected[key]: + raise Reject(f"{name}: {key}") + checks += 1 + adjacent = endpoint(row, int(row["next"])) + if adjacent["T"] != row["next_T"] or adjacent["profile"] != row["next_profile"]: + raise Reject(f"{name}: adjacent") + for e in range(int(row["first"]), int(row["last"]) + 1): + value = profile(int(row["R"]), int(row["d"]), int(row["K"]), e) + if value is None or value > row["budget"]: + raise Reject(f"{name}: strip") + checks += 1 + return checks + + +def block_control() -> int: + blocks = [ + {0, 1, 2}, {0, 3, 4}, {1, 3, 5}, {2, 4, 5}, + {0, 5, 6}, {1, 4, 6}, {2, 3, 6}, + ] + n, A, c = 10, 3, 1 + if any(len(x & y) > c for x, y in combinations(blocks, 2)): + raise Reject("block intersections") + g = n * c - A * A + T = (n - A) ** 2 - (n - 1) * g + cap = Fraction((n - 1) * n * n * (A - c), A * T) + if len(blocks) > cap.numerator // cap.denominator: + raise Reject("block cap") + return len(blocks) ** 2 + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + block_control() + mutations = [] + for name, key in (("KoalaBear", "T"), ("Mersenne-31", "profile")): + changed = copy.deepcopy(ROWS) + changed[name][key] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_SPARSE_DIRECTION_MEAN_CENTERED_GRAM_PROFILE_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From bb9df40bd24a99475335d7e01d97d7d56dd26d17 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 05:51:25 +0100 Subject: [PATCH 08/40] MCA: pay terminal deficit by affine-line packing --- agents.md | 2 +- experimental/agents-log.md | 19 +++ experimental/grande_finale.pdf | Bin 945432 -> 1021424 bytes experimental/grande_finale.tex | 87 +++++++++++ ...ection_terminal_deficit_line_payment_v1.md | 72 +++++++++ ...ection_terminal_deficit_line_payment_v1.py | 143 ++++++++++++++++++ 6 files changed, 322 insertions(+), 1 deletion(-) create mode 100644 experimental/notes/thresholds/mca_sparse_direction_terminal_deficit_line_payment_v1.md create mode 100644 experimental/verify_mca_sparse_direction_terminal_deficit_line_payment_v1.py diff --git a/agents.md b/agents.md index 4891391fa..34f88c341 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne. A centered-incidence Gram rung first extends these walls to `e=64037` and `e=65418`; the stronger PSD mean-centered profile extends them again to `e=64047` and `e=65454`, with endpoint profiles `181731868` and `16101127`. The remaining full-lift intervals are therefore `64048<=e<=1044238` and `65455<=e<=1044241` and require stronger extension-specific or post-Johnson row-sharp list structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne. A centered-incidence Gram rung first extends these walls to `e=64037` and `e=65418`; the stronger PSD mean-centered profile extends them again to `e=64047` and `e=65454`. The terminal-deficit affine-line cap moves the walls to `e=64048` and `e=65455`, with endpoint profiles `181326343` and `16100647`. The remaining full-lift intervals are therefore `64049<=e<=1044238` and `65456<=e<=1044241` and require joint control of the last high-deficit layers or stronger extension-specific structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. - Grande Finale v4 explicitly supersedes the old v3 `prob:saturated-bc` status: primitive one-pencil MCA BC is proved; higher-dimensional MCA BC remains jointly governed by spread-component, large-owner, and exception routing; and the list-interior clause is governed by row-sharp list completion. The spread-abundance and `prob:next` statements are intermediate forms, not additional terminal inputs. - `experimental/proximity_prize_results_v4.tex` is the current compact synthesis of proved partial results. It is exposition/status guidance, not a replacement proof source for a live atom or adjacent-row closure. - A schema/hash pass is structural preflight only. The trusted-source registry is empty; parsing a manifest does not prove an atom. diff --git a/experimental/agents-log.md b/experimental/agents-log.md index becfe46ae..80d616562 100644 --- a/experimental/agents-log.md +++ b/experimental/agents-log.md @@ -30,6 +30,25 @@ Keep entries concise and link to the relevant files. ## Entries +### 2026-08-13 - Terminal-deficit affine-line payment + +- **Agent/model:** Codex. +- **Files added or changed:** `experimental/grande_finale.tex`, + `experimental/notes/thresholds/mca_sparse_direction_terminal_deficit_line_payment_v1.md`, + `experimental/verify_mca_sparse_direction_terminal_deficit_line_payment_v1.py`, + `agents.md`, and `experimental/agents-log.md`. +- **Status:** PROVED / EXACT FINITE CALIBRATION. +- **What is being added:** An affine-codeword-line theorem for the exact + terminal deficit layer of the sparse-direction compiler, plus a + prefix-plus-terminal slope profile. +- **How it is useful:** It pays one further support in each deployed row, + moving the low-support walls to `e=64048` on KoalaBear and `e=65455` on + Mersenne-31, with profile bounds `181326343` and `16100647`. +- **What to do next:** Control the last several high-deficit layers jointly + or retain the full-lift near-MDS extension structure. KoalaBear + `e=64049` lacks the penultimate cumulative cap; Mersenne `e=65456` remains + valid but over budget by `342292`. + ### 2026-08-13 - Mean-centered Gram profile - **Agent/model:** Codex. diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index 67113d2fe2c8b5232835f5c6388b0a7183de465c..c890adfa1a9ff089d189c3d1b120dcf454f4917b 100644 GIT binary patch delta 470945 zcmV(3~OVD%)xHmugoi?6?hs}k*m5hYO1 zgBjJVV!Dhb^ir?By!dRtM2Y5gm^2Yegg^+1{SM_dGQ)G|CJ&-C-yDsXhVP1stLMqVf)2MwEVmSw(5vp(UC_(nxin6q~4d^?eF?5rY|(O;bITN zsoCU;;5lXy-O@01gmd+yjBV}|;wBaLiXn`DRAKN8Uo*sERg^_645jIx&temNeI=Cl zfK-py(j36rvubMJZcICNcL@IBKsZYRnPiJ4uC*@BH#2`--8IGq^LUC6=AKc(LR=&* z?4__N7?43t`p1NVPE?y2gcISBD_|IX21L{CbrCh$n|l6cVUAl;`f7=JoZY{&I^Pye z!~Up@f4Li3_vHiK0@1DP(ph6ebG{9Q4nR*5Q>R*Pv_^E5g%LH9($-~zYFQ@ zUOe;FQJxpB(jZSmlN=La+`%Pq#xIA?_b6j?E$!)E(aV1rk+=nrYY!U?80dB4Q`s-m{K!CLPD&;C0!-(m5y8G#; zThZou6K%eG_wD1mZ|+mJVY`$?X!E$=FuY-$>^NbY$9(f!fVUyW5of`}r+?plxchM% 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60bd5ca5c..42ce965d0 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1881,6 +1881,93 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_sparse_direction_mean_centered_gram_profile_v1.py}. \end{remark} +\begin{theorem}[Terminal-deficit affine-line cap] +\label{thm:terminal-deficit-affine-line} +Retain the sparse-direction notation of +\cref{thm:sparse-direction-punctured-johnson-mca}, assume \(e\ge K\), and +put +\[ + n=N-e, + \qquad A=m-e, + \qquad c=K-1. +\] +If \(A>c\), then the number \(L_e\) of transformed explanations that own a +selected slope and have exact outside deficit \(e\) satisfies +\begin{equation} + L_e\le\left\lfloor\frac{n-c}{A-c}\right\rfloor. + \label{eq:terminal-deficit-affine-line} +\end{equation} +\end{theorem} + +\begin{proof} +An exact-deficit-\(e\) explanation has exactly \(A=m-e\) outside +agreements. To own a selected slope it must therefore agree at all \(e\) +coordinates of the direction support \(E\). If two such explanations have +slopes \(\gamma\ne\delta\), their normalized difference is a nonzero +degree-\(16777215\). The remaining full-lift intervals are therefore +\(64049\le e\le1044238\) and \(65456\le e\le1044241\). This is a +support-wall refinement, not a proof of either deployed row. The exact +replay is in +\path{experimental/verify_mca_sparse_direction_terminal_deficit_line_payment_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_sparse_direction_terminal_deficit_line_payment_v1.md b/experimental/notes/thresholds/mca_sparse_direction_terminal_deficit_line_payment_v1.md new file mode 100644 index 000000000..0346ee64e --- /dev/null +++ b/experimental/notes/thresholds/mca_sparse_direction_terminal_deficit_line_payment_v1.md @@ -0,0 +1,72 @@ +# MCA sparse-direction terminal-deficit line payment v1 + +## Status + +PROVED / EXACT FINITE CALIBRATION. + +This packet strengthens only the exact outside-deficit `h=e` layer of the +field-general sparse-direction profile. It does not prove an official row +or turn failure of the adjacent calculation into an unsafe certificate. + +## The structural cap + +Use the codeword gauge + +```text +r_1=b+q, E=supp(q), |E|=e, +N=R+K, m=d+K, n=N-e, +A=m-e, c=K-1. +``` + +An explanation of exact outside deficit `e` has exactly `A` outside +agreements. Owning a selected slope therefore forces agreement at every +coordinate of `E`. If `e>=K`, restriction to `E` is injective on +degree-` int | None: + n = R + K - e + A = d + K - h + c = K - 1 + den = A * A - n * c + if den > 0: + value = Fraction(n * (A - c), den) + else: + g = -den + T = (n - A) ** 2 - (n - 1) * g + if g < 0 or 2 * A * A < n * c or T <= 0: + return None + value = Fraction((n - 1) * n * n * (A - c), A * T) + return value.numerator // value.denominator + + +def terminal_profile(row: dict[str, int | None], e: int) -> dict[str, int] | None: + R, d, K = (int(row[key]) for key in ("R", "d", "K")) + if e < K: + return None + raw = [] + for h in range(1, e): + cap = raw_cap(R, d, K, e, h) + if cap is None: + return None + raw.append(cap) + prefix_cap = raw[-1] + suffix = [0] * len(raw) + running = None + for index in range(len(raw) - 1, -1, -1): + running = raw[index] if running is None else min(running, raw[index]) + suffix[index] = running + prefix = 0 + previous = 0 + for h, cap in enumerate(suffix, 1): + prefix += (cap - previous) * (e // h) + previous = cap + n = R + K - e + A = d + K - e + c = K - 1 + line = Fraction(n - c, A - c) + terminal = line.numerator // line.denominator + return { + "defect": R - e, "n": n, "A": A, + "prefix_cap": prefix_cap, "prefix": prefix, + "terminal": terminal, "profile": prefix + terminal, + } + + +def finite_line_control() -> int: + blocks = [{0, 1, 2}, {0, 3, 4}, {0, 5, 6}] + n, A, c = 7, 3, 1 + core = set.intersection(*blocks) + if len(core) != c: + raise Reject("common core") + stripped = [block - core for block in blocks] + if any(left & right for i, left in enumerate(stripped) for right in stripped[i + 1:]): + raise Reject("packing") + cap = Fraction(n - c, A - c) + if len(blocks) > cap.numerator // cap.denominator: + raise Reject("line cap") + return len(blocks) * len(stripped) + + +def validate(rows: dict[str, dict[str, int | None]]) -> int: + checks = 0 + for name, row in rows.items(): + if name not in ROWS: + raise Reject("name") + paid = terminal_profile(row, int(row["paid"])) + if paid is None: + raise Reject(f"{name}: unavailable paid profile") + for key in ("defect", "n", "A", "prefix_cap", "prefix", "terminal", "profile"): + if paid[key] != row[key] or row[key] != ROWS[name][key]: + raise Reject(f"{name}: {key}") + checks += 1 + if paid["profile"] > row["budget"]: + raise Reject(f"{name}: budget") + adjacent = terminal_profile(row, int(row["next"])) + if row["next_profile"] is None: + if adjacent is not None: + raise Reject(f"{name}: adjacent availability") + elif adjacent is None or adjacent["profile"] != row["next_profile"] or adjacent["profile"] <= row["budget"]: + raise Reject(f"{name}: adjacent budget") + checks += 2 + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + finite_line_control() + mutations = [] + for name, key in (("KoalaBear", "terminal"), ("Mersenne-31", "profile")): + changed = copy.deepcopy(ROWS) + changed[name][key] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_SPARSE_DIRECTION_TERMINAL_DEFICIT_LINE_PAYMENT_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 0b7bedf71454bcb87b1d708f8f58f5ff21ea2f85 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 06:03:57 +0100 Subject: [PATCH 09/40] MCA: close sparse-direction branch by top-third lines --- agents.md | 2 +- experimental/agents-log.md | 18 +++ experimental/grande_finale.pdf | Bin 1021424 -> 1027758 bytes experimental/grande_finale.tex | 111 ++++++++++++++++ ...ection_top_third_affine_line_payment_v1.md | 55 ++++++++ ...ection_top_third_affine_line_payment_v1.py | 119 ++++++++++++++++++ 6 files changed, 304 insertions(+), 1 deletion(-) create mode 100644 experimental/notes/thresholds/mca_sparse_direction_top_third_affine_line_payment_v1.md create mode 100644 experimental/verify_mca_sparse_direction_top_third_affine_line_payment_v1.py diff --git a/agents.md b/agents.md index 34f88c341..1efe65c48 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp; its generic MDS-endpoint ceilings are still `743896698428332665` and `219426634`, above budget. The field-general punctured Johnson profile pays every low support through `e=63908` for KoalaBear and `e=65236` for Mersenne. A centered-incidence Gram rung first extends these walls to `e=64037` and `e=65418`; the stronger PSD mean-centered profile extends them again to `e=64047` and `e=65454`. The terminal-deficit affine-line cap moves the walls to `e=64048` and `e=65455`, with endpoint profiles `181326343` and `16100647`. The remaining full-lift intervals are therefore `64049<=e<=1044238` and `65456<=e<=1044241` and require joint control of the last high-deficit layers or stronger extension-specific structure. The ordinary affine-span LIST theorem, common-core cancellation, directional Johnson compiler, gauge equivalence, and the selector-free all-LineRay error-affine-core set-pair theorem are not refuted by the example. +- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson, centered Gram, mean-centered, and terminal-line rungs first move the low-support walls to `e=64048` and `e=65455`. The top-third affine-line theorem then pays every remaining sparse-direction support `e=d`. Support-only ordinary-list bounds have now completed their + natural branch. + ### 2026-08-13 - Terminal-deficit affine-line payment - **Agent/model:** Codex. diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index c890adfa1a9ff089d189c3d1b120dcf454f4917b..7294561f0bd776c729f35f1be691265127a1c80c 100644 GIT binary patch delta 451939 zcmV(sK<&Tqt3IyXKCr3@m*I&8DU;I)7=PD^cfVWGpKLQ0=7-Cx*XLJ0EeDpf zImKky%zocF;mDKT6^tjO3EVtMl{MU&AQ=eEH$>;_^3M>Ei4aQFx-Qmv2Mn!Ufm31Uz|jdw;qFo=?fkpUxxN@B~~H^fh<}jN(4!stgagny53a72}Do5ZL!5e z?&+T2U3#o_>dOQ)H-ZvC&)wTROcKm<#>SGl1cBg2(8-Vzrjlnn_q%85DOzOMmFv(#A{7W4oTI z2ybG8q=T8{>QKBe3o?Pu^#zx(BQY^{-$e4@Z1|ux%1nJ zFT7m`{g0JZ33tpWjk+C-ix7ogmovf9e^b;&yVj+@?t0Bl(->=#2QtJt{U2Ck+x1OR zSO2(OCN;FxXiclu<$s;-xknFUaZ^{rMwgRm_BwcC5HJLl<*PVL5FUk#6htsj4EcT_ zS@YD}6!k4Xl^U9Wb*7%KNh#Zb+u8c@aJ#E(ra(sM4=^O|ph!~Hx`*o~5xXum8&}G# zD-Ar;?G3uEGiAn^nC!ObhlPh6y>qs1@AEhjgVCw zA=wzq!ba^l;8CCB+H9r>v2*JkmS_L8(=jb+Au0kT)8Rcs*PU*4&zt7on<>@Z#;!~Q 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z-tt}98)#_Qce?=O=-{n^Ld0_=x^1dikL%xEf4+DPP{Nio=U+t0< zp=`BYMZOB*ZY#Kmth%}BBkKEZ=d-~#CfPHPY@_r+-}1pXf!WtY>`M&nS2i5xcLP!3 z%U$0+8}0tIknQ%gNNyNnb9XvrFpur3HD7kNw*5SLV7S>L<42?ItJqf1*8gw7O!8K-1. + \end{cases} + \label{eq:full-lift-total-core-line} +\end{equation} +\end{theorem} + +\begin{proof} +The triple-overlap proof of \cref{thm:top-third-affine-line} still puts the +exact layer on an affine explanation line +\(a_\gamma=A+\gamma p\). A coordinate agrees for every line parameter +exactly when \(A=r_0\) and \(p=q\) there. If this total common agreement +core had at least \(m\) coordinates, the codeword pair \((A,b+p)\) would +explain the received pair on an \(m\)-set, contradicting pair +noncontainment. Thus its size \(g\) is at most \(m-1\). Away from the +core, each coordinate agrees for at most one parameter, and therefore +\[ + L_r(m-g)\le N-g, + \qquad L_r\le N-m+1=t+1. +\] +When \(A_r>K-1\), outside-coordinate packing gives the second branch as in +\cref{thm:top-third-affine-line}. That branch is no larger than \(t+1\): +after clearing its positive denominator, the difference is +\(t(A_r-K)+r\ge0\). +\end{proof} + +\begin{corollary}[Full-lift common-core profile] +\label{cor:full-lift-common-core-profile} +If the positive punctured-Johnson caps exist through \(H=e-s-1\), then +\begin{equation} + |Z|\le + (e-1)J_{\lfloor e/2\rfloor}+J_H + +\sum_{r=r_{\min}}^{s}Q_r. + \label{eq:full-lift-common-core-profile} +\end{equation} +\end{corollary} + +\begin{proof} +Deficits through \(H\) use the two-threshold Johnson prefix. Deficits +above \(H\) and at most \(m\) are exactly the layers +\(r_{\min}\le r\le s\); apply \cref{thm:full-lift-total-core-line}. +\end{proof} + +\begin{remark}[Exact full-lift common-core walls] +Exact integer evaluation of +\eqref{eq:full-lift-common-core-profile} extends the complete walls to +\[ +\begin{array}{c|r|r|r} +\text{row}&\text{last paid }e&\text{endpoint bound}&\text{next bound}\\ \hline +\text{KoalaBear}&95943&27414298&\text{undefined}\\ +\text{Mersenne-31}&67452&16266965&17248067 +\end{array} +\] +At KoalaBear \(e=95944\), the prefix Johnson denominator at \(H\) is +\(-1037\). The adjacent Mersenne bound exceeds budget by \(470852\). +The remaining full-lift intervals are +\[ + 95944\le e\le1044238 + \quad\text{and}\quad + 67453\le e\le1044241. +\] +Neither adjacent failure is an unsafe certificate. The replay is in +\path{experimental/verify_mca_full_lift_top_third_common_core_payment_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_top_third_common_core_payment_v1.md b/experimental/notes/thresholds/mca_full_lift_top_third_common_core_payment_v1.md new file mode 100644 index 000000000..d4d92f52a --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_top_third_common_core_payment_v1.md @@ -0,0 +1,52 @@ +# MCA full-lift top-third common-core payment v1 + +## Status + +PROVED / EXACT FINITE CALIBRATION. + +The top-third affine-line theorem remains valid for `e>=d`. The new issue +is that the highest possible outside deficit is `m` and some exact layers +have at most `K-1` outside agreements. + +On an affine explanation line, a coordinate agreeing for every parameter is +a simultaneous base/direction agreement for one codeword pair. Pair +noncontainment limits this total common core to `m-1`. Off the core, +agreement sets are disjoint, so every exact line layer has + +```text +L <= N-m+1. +``` + +When the outside agreement `A_r` exceeds `K-1`, the sharper outside +zero-core cap remains available. The piecewise cap is + +```text +Q_r = N-m+1 if A_r<=K-1, +Q_r = floor((N-e-(K-1))/(A_r-(K-1))) otherwise. +``` + +Combining these top-third layers with the Johnson prefix gives + +```text +KoalaBear: e<=95943, endpoint bound 27414298; +Mersenne-31: e<=67452, endpoint bound 16266965. +``` + +KoalaBear `e=95944` has prefix denominator `-1037`. Mersenne +`e=67453` has valid bound `17248067`, over budget by `470852`. The +remaining intervals are + +```text +KoalaBear: 95944 <= e <= 1044238 +Mersenne-31: 67453 <= e <= 1044241. +``` + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_top_third_common_core_payment_v1.py +``` + +The verifier checks the total-core cap on a sharp finite model, direct +endpoint floor sums, uniform KoalaBear prefix maxima, all five new Mersenne +cells, both adjacent stops, and hostile mutations. diff --git a/experimental/verify_mca_full_lift_top_third_common_core_payment_v1.py b/experimental/verify_mca_full_lift_top_third_common_core_payment_v1.py new file mode 100644 index 000000000..f212875e5 --- /dev/null +++ b/experimental/verify_mca_full_lift_top_third_common_core_payment_v1.py @@ -0,0 +1,162 @@ +#!/usr/bin/env python3 +"""Audit the full-lift top-third common-core payment.""" + +from __future__ import annotations + +import copy + + +ROWS = { + "KoalaBear": { + "R": 1048576, "d": 67472, "K": 14, + "budget": 274980728111395087, + "last": 95943, "last_total": 27414298, + "last_line": 22059354, "last_J_u": 50, "last_J_H": 557844, + "max_line": 24121368, "max_line_e": 67486, + "next": 95944, "next_H_den": -1037, "next_total": None, + "floor": 95944, "ceiling": 1044238, + }, + "Mersenne-31": { + "R": 1048576, "d": 67448, "K": 6, + "budget": 16777215, + "last": 67452, "last_total": 16266965, + "last_line": 14310842, "last_J_u": 29, "last_J_H": 44, + "max_line": 14310842, "max_line_e": 67452, + "next": 67453, "next_H_den": 500624611, + "next_total": 17248067, + "floor": 67453, "ceiling": 1044241, + }, +} + + +class Reject(ValueError): + pass + + +def direct_line_sum(R: int, d: int, K: int, e: int) -> int: + N, m, c = R + K, d + K, K - 1 + n, t = N - e, N - m + s = (e - K) // 3 + r_min = max(0, e - m) + total = 0 + for r in range(r_min, s + 1): + A = m - e + r + total += t + 1 if A <= c else (n - c) // (A - c) + return total + + +def profile(R: int, d: int, K: int, e: int) -> tuple[int, int, int, int] | None: + N, m, c = R + K, d + K, K - 1 + n = N - e + s = (e - K) // 3 + H = e - s - 1 + u = e // 2 + values = [] + denominators = [] + for h in (u, H): + A = m - h + denominator = A * A - n * c + denominators.append(denominator) + if denominator <= 0: + return None + values.append(n * (A - c) // denominator) + line = direct_line_sum(R, d, K, e) + total = (e - 1) * values[0] + values[1] + line + return total, line, values[0], values[1] + + +def johnson_pair(R: int, d: int, K: int, e: int) -> tuple[int, int] | None: + N, m, c = R + K, d + K, K - 1 + n = N - e + s = (e - K) // 3 + values = [] + for h in (e // 2, e - s - 1): + A = m - h + denominator = A * A - n * c + if denominator <= 0: + return None + values.append(n * (A - c) // denominator) + return values[0], values[1] + + +def finite_core_control() -> int: + blocks = [ + {0, 1, 2}, {0, 1, 3}, {0, 1, 4}, {0, 1, 5}, {0, 1, 6}, + ] + core = set.intersection(*blocks) + stripped = [block - core for block in blocks] + if len(core) != 2: + raise Reject("common core") + if any(left & right for i, left in enumerate(stripped) for right in stripped[i + 1:]): + raise Reject("off-core packing") + if len(blocks) != 7 - 3 + 1: + raise Reject("sharp total cap") + return len(blocks) * len(stripped) + + +def validate(rows: dict[str, dict[str, int | None]]) -> int: + checks = 0 + for name, row in rows.items(): + R, d, K = row["R"], row["d"], row["K"] + last = profile(R, d, K, row["last"]) + if last is None or last != ( + row["last_total"], row["last_line"], row["last_J_u"], row["last_J_H"] + ): + raise Reject(f"{name}: endpoint") + if last[0] > row["budget"]: + raise Reject(f"{name}: budget") + if name == "KoalaBear": + max_u = max_H = 0 + for e in range(d, row["last"] + 1): + current = johnson_pair(R, d, K, e) + if current is None: + raise Reject("KoalaBear strip") + max_u = max(max_u, current[0]) + max_H = max(max_H, current[1]) + if (max_u, max_H) != (50, 557844): + raise Reject("KoalaBear maxima") + line = direct_line_sum(R, d, K, d + K) + if (line, d + K) != (row["max_line"], row["max_line_e"]): + raise Reject("KoalaBear line max") + if (row["last"] - 1) * max_u + max_H + line > row["budget"]: + raise Reject("KoalaBear uniform") + e = row["next"] + N, m, c = R + K, d + K, K - 1 + H = e - (e - K) // 3 - 1 + A = m - H + if A * A - (N - e) * c != row["next_H_den"]: + raise Reject("KoalaBear stop") + else: + for e in range(d, row["last"] + 1): + current = profile(R, d, K, e) + if current is None or current[0] > row["budget"]: + raise Reject("Mersenne strip") + adjacent = profile(R, d, K, row["next"]) + if adjacent is None or adjacent[0] != row["next_total"] or adjacent[0] <= row["budget"]: + raise Reject("Mersenne stop") + checks += 9 + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + finite_core_control() + mutations = [] + for name, key in (("KoalaBear", "last_line"), ("Mersenne-31", "next_total")): + changed = copy.deepcopy(ROWS) + changed[name][key] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_TOP_THIRD_COMMON_CORE_PAYMENT_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 9c708e2f706ea79840bd37e645a232f697c00689 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 06:29:42 +0100 Subject: [PATCH 11/40] MCA: synchronize full-lift top-third layers globally --- agents.md | 2 +- experimental/agents-log.md | 18 ++ experimental/grande_finale.pdf | Bin 1035591 -> 1039549 bytes experimental/grande_finale.tex | 95 +++++++++++ ...l_lift_top_third_global_line_payment_v1.md | 77 +++++++++ ...l_lift_top_third_global_line_payment_v1.py | 155 ++++++++++++++++++ 6 files changed, 346 insertions(+), 1 deletion(-) create mode 100644 experimental/notes/thresholds/mca_full_lift_top_third_global_line_payment_v1.md create mode 100644 experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py diff --git a/agents.md b/agents.md index 818ac4b71..4d5b63827 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. 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\path{experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py}. \end{remark} +\begin{corollary}[Full-lift mean-centered global-line profile] +\label{cor:full-lift-mean-centered-global-line-profile} +Retain the notation of +\cref{thm:full-lift-top-third-global-line}, put +\[ + H=e-s-1, + \qquad c=K-1, +\] +and assume $m-H>c$. For every $1\le h\le H$, put +$A_h=m-h$, and let $C_h$ be the punctured ordinary-Johnson cap when +$A_h^2>(N-e)c$, and otherwise the cap in +\cref{thm:mean-centered-gram-list} whenever its three hypotheses hold. +Assume every $C_h$ is defined, and put +\[ + B_0=0, + \qquad B_h=\min_{h\le v\le H}C_v. +\] +Then +\begin{equation} + |Z|\le + \sum_{h=1}^{H}(B_h-B_{h-1}) + \left\lfloor\frac eh\right\rfloor + +(N-m+1). + \label{eq:full-lift-mean-centered-global-line-profile} +\end{equation} +\end{corollary} + +\begin{proof} +Let $N_h$ count transformed explanations with outside deficit at most +$h$. The earlier sparse-direction corollary was calibrated for $ec$ gives $A_h>c$ +throughout the prefix, and distinct degree-$K-1`, the ordinary set-system proof applies even when `e>=d`: +after puncturing the direction support, every selected explanation supplies +an `A_h`-block, and two distinct degree-` 0: + return n * (A - c) // D, (A, -D, 2 * A * A - n * c, 0) + g = -D + balance = 2 * A * A - n * c + T = (n - A) ** 2 - (n - 1) * g + if balance < 0 or T <= 0: + return None, (A, g, balance, T) + return (n - 1) * n * n * (A - c) // (A * T), (A, g, balance, T) + + +@lru_cache(maxsize=None) +def profile(R: int, d: int, K: int, e: int): + N, m = R + K, d + K + H = e - (e - K) // 3 - 1 + if m - H <= K - 1: + raise Reject("agreement guard") + caps = [0] + for h in range(1, H + 1): + value, record = raw_cap(R, d, K, e, h) + if value is None: + return None, {"H": H, "failure_h": h, "record": record} + caps.append(value) + suffix = caps[-1] + for h in range(H - 1, 0, -1): + suffix = min(suffix, caps[h]) + caps[h] = suffix + prefix = sum((caps[h] - caps[h - 1]) * (e // h) + for h in range(1, H + 1)) + return prefix + N - m + 1, { + "H": H, "prefix": prefix, "B_H": caps[H], + "breaks": sum(caps[h] != caps[h - 1] for h in range(1, H + 1)), + } + + +def validate(rows): + checks = 0 + for name, row in rows.items(): + maximum = (-1, -1) + for e in range(row["first"], row["last"] + 1): + total, detail = profile(row["R"], row["d"], row["K"], e) + if total is None or total > row["budget"]: + raise Reject(f"{name}: paid strip") + maximum = max(maximum, (total, e)) + if e == row["first"] and total != row["first_total"]: + raise Reject(f"{name}: first") + if e == row["last"] and ( + detail["H"], detail["prefix"], detail["B_H"], + detail["breaks"], total + ) != ( + row["last_H"], row["last_prefix"], row["last_B_H"], + row["last_breaks"], row["last_total"] + ): + raise Reject(f"{name}: last") + checks += 1 + if maximum != (row["max_total"], row["max_e"]): + raise Reject(f"{name}: maximum") + adjacent, detail = profile(row["R"], row["d"], row["K"], row["next"]) + if adjacent != row["next_total"]: + raise Reject(f"{name}: adjacent") + if row["next_failure_h"] is not None: + if (detail["failure_h"], *detail["record"]) != ( + row["next_failure_h"], row["next_A"], row["next_g"], + row["next_balance"], row["next_T"] + ): + raise Reject(f"{name}: theorem wall") + else: + _, record = raw_cap(row["R"], row["d"], row["K"], + row["next"], detail["H"]) + if record != (row["next_A"], row["next_g"], + row["next_balance"], row["next_T"]): + raise Reject(f"{name}: budget wall") + if adjacent is None or adjacent <= row["budget"]: + raise Reject(f"{name}: adjacent budget") + if row["ceiling"] < row["next"]: + raise Reject(f"{name}: residual") + checks += 10 + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + mutations = [] + for name, key, delta in ( + ("KoalaBear", "last_total", 1), + ("KoalaBear", "next_T", 1), + ("Mersenne-31", "max_total", -1), + ("Mersenne-31", "next_total", -1), + ): + changed = copy.deepcopy(ROWS) + changed[name][key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_MEAN_CENTERED_GLOBAL_LINE_PROFILE_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From b04c8f1d66cb258b47ce984e12f11ab232230a07 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 07:15:38 +0100 Subject: [PATCH 13/40] MCA: absorb first Mersenne boundary layer --- agents.md | 2 +- experimental/agents-log.md | 16 +++ experimental/grande_finale.pdf | Bin 1043166 -> 1047221 bytes experimental/grande_finale.tex | 64 ++++++++++ ...ll_lift_boundary_anchor_continuation_v1.md | 60 ++++++++++ ...ll_lift_boundary_anchor_continuation_v1.py | 113 ++++++++++++++++++ 6 files changed, 254 insertions(+), 1 deletion(-) create mode 100644 experimental/notes/thresholds/mca_full_lift_boundary_anchor_continuation_v1.md create mode 100644 experimental/verify_mca_full_lift_boundary_anchor_continuation_v1.py diff --git a/agents.md b/agents.md index 7d5327a96..4bc49a0be 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. 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z@;C4dt`qu|v-Y*>aRX!Dm2@Q7DOh-F(Yx6ti&GU%T!cq2q`S8mmeWO?na-y5@k z2eRrlTq6{+;LE^>7F$;7A0qSc2_f5r1@N}`fp4#{sOf${NL3#6={$jgkwBgs{#Zkv zw7Jd_Lv1LNCyXQ*Pyr@lv5S3wYd)m4p2YAZPA>PP=264qKnpC zmRP@g(>cCXs(CN52V>p9C10OGPY~?aU$w(3-5=x*Ue%1x85aZudR+V*Wl{y8dYvsL zH`KlbYk{FRXe+=E%*1|0P{G{=sjiR&&4VH6ujX<1|-4o}O|&JH)0(p1kLpr?>O^O$UJyZu@)3k)c&i?ve0#Lm~Nc_JHSj`(=EW6H9 zU_Hr!xTCVhd;0X=e{qouOESRFB&YoB%dgpVhsPNSXXl*$9m$DVtLfWkyT+?tzPKPG zf5&W>cTP?Ne1ZNg&G}i|T@~zbRaFxns4panRAgB5O`M@K|ra{raS{vIcp2@hs4pKYMm%b;2OULELcKpuwR-{Bw<2 zx%U==b5rK*kK6EOV|CbCqb7a+UjYftJ~@t89fxX5=0Dva+_D2N*1G}xJuwlOOb+?j zyqGJ)T#uWWMCQX>=Y5zNUGoZ&KV^W(J;`A=XW<^K9E)eU$nTtXtv5ZG!__3S&&)mB zX_%S>Yrkqc@iW>eNT=e&~zLkNTG(X0BumMl8!K3#q{wx0%P z7S`~(o6T)d+uk3rBUgeEW*uk7!!V={ax!Uaowicb%j_x8ZRfK5s-4>6XLkU5>3D-R zSlka4Nf42kYwAT!J1|ZxNz=%6$(ycmnoKoNJ~M5sy%L7`*~C= e-3s >= K +``` + +coordinates. Restriction injectivity therefore identifies their normalized +pair directions. The entire high-deficit union lies on one affine codeword +line, not one line per layer. + +Pair noncontainment limits the total common agreement core of this line to +`m-1`. Off that core, agreement sets for distinct line parameters are +disjoint. Thus the whole high union costs only + +```text +N-m+1 +``` + +once. Combining this with the punctured-Johnson prefix gives + +```text +|Z| <= (e-1) J_floor(e/2) + J_H + (N-m+1). +``` + +## Exact walls + +Exact integer scans give + +```text +KoalaBear: e <= 95943, endpoint bound 6336049 +Mersenne-31: e <= 97908, endpoint bound 6682339 +``` + +The largest Mersenne bound is `6683188` at `e=97907`. At the adjacent +supports the `H`-prefix Johnson denominators are `-1037` and `-965`. +Consequently the residual intervals are + +```text +KoalaBear: 95944 <= e <= 1044238 +Mersenne-31: 97909 <= e <= 1044241 +``` + +The adjacent failures are failures of this proof profile, not unsafe +certificates. No deployed v4 ledger atom moves. + +## Provenance and replay + +The source theorem, contract, primary verifier, and independent audit are +pinned at +`AllenGrahamHart/rs-mca-prize-dag@62ef043e0`. This repository replay is +self-contained: + +```bash +python3 experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py +python3 -O experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py +``` + +It scans every support in both newly paid intervals using exact integer +arithmetic, reconstructs both endpoint and adjacent records, checks a sharp +finite common-core model, and rejects four hostile mutations. diff --git a/experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py b/experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py new file mode 100644 index 000000000..8e289f7c0 --- /dev/null +++ b/experimental/verify_mca_full_lift_top_third_global_line_payment_v1.py @@ -0,0 +1,155 @@ +#!/usr/bin/env python3 +"""Verify the full-lift top-third global-line payment.""" + +from __future__ import annotations + +import copy + + +ROWS = { + "KoalaBear": { + "R": 1048576, "d": 67472, "K": 14, + "budget": 274980728111395087, + "first": 67472, "first_total": 2937808, + "last": 95943, "last_s": 31976, "last_H": 63966, + "last_u": 47971, "last_J_u": 50, "last_J_H": 557844, + "last_H_den": 5989, "line_cap": 981105, + "last_prefix": 5354944, "last_total": 6336049, + "max_total": 6336049, "max_total_e": 95943, + "next": 95944, "next_H": 63967, "next_H_den": -1037, + "floor": 95944, "ceiling": 1044238, + }, + "Mersenne-31": { + "R": 1048576, "d": 67448, "K": 6, + "budget": 16777215, + "first": 67448, "first_total": 2937136, + "last": 97908, "last_s": 32634, "last_H": 65273, + "last_u": 48954, "last_J_u": 52, "last_J_H": 610046, + "last_H_den": 3391, "line_cap": 981129, + "last_prefix": 5701210, "last_total": 6682339, + "max_total": 6683188, "max_total_e": 97907, + "next": 97909, "next_H": 65274, "next_H_den": -965, + "floor": 97909, "ceiling": 1044241, + }, +} + + +class Reject(ValueError): + pass + + +def profile(R: int, d: int, K: int, e: int) -> dict[str, int] | None: + N, m, c = R + K, d + K, K - 1 + n = N - e + s = (e - K) // 3 + H = e - s - 1 + u = e // 2 + values = [] + denominators = [] + for h in (u, H): + A = m - h + denominator = A * A - n * c + if denominator <= 0: + return None + values.append(n * (A - c) // denominator) + denominators.append(denominator) + prefix = (e - 1) * values[0] + values[1] + line_cap = N - m + 1 + return { + "s": s, "H": H, "u": u, + "J_u": values[0], "J_H": values[1], + "H_den": denominators[1], "line_cap": line_cap, + "prefix": prefix, "total": prefix + line_cap, + } + + +def finite_controls() -> int: + E = set(range(11)) + missed = [{0, 1, 2}, {3, 4}, {5, 6, 7}] + shared = set.intersection(*(E - current for current in missed)) + if len(shared) != 3: + raise Reject("cross-layer triple overlap") + blocks = [ + {0, 1, 2}, {0, 1, 3}, {0, 1, 4}, {0, 1, 5}, {0, 1, 6}, + ] + core = set.intersection(*blocks) + stripped = [block - core for block in blocks] + if len(core) != 2 or any( + left & right + for index, left in enumerate(stripped) + for right in stripped[index + 1:] + ): + raise Reject("total-core packing") + return len(shared) + len(blocks) + + +def validate(rows: dict[str, dict[str, int]]) -> int: + checks = finite_controls() + for name, row in rows.items(): + first = profile(row["R"], row["d"], row["K"], row["first"]) + last = profile(row["R"], row["d"], row["K"], row["last"]) + if first is None or first["total"] != row["first_total"]: + raise Reject(f"{name}: first endpoint") + if last is None: + raise Reject(f"{name}: last endpoint unavailable") + expected = { + "s": row["last_s"], "H": row["last_H"], + "u": row["last_u"], "J_u": row["last_J_u"], + "J_H": row["last_J_H"], "H_den": row["last_H_den"], + "line_cap": row["line_cap"], "prefix": row["last_prefix"], + "total": row["last_total"], + } + if any(last[key] != value for key, value in expected.items()): + raise Reject(f"{name}: endpoint record") + maximum = (-1, -1) + for e in range(row["first"], row["last"] + 1): + current = profile(row["R"], row["d"], row["K"], e) + if current is None or current["total"] > row["budget"]: + raise Reject(f"{name}: paid strip") + maximum = max(maximum, (current["total"], e)) + checks += 1 + if maximum != (row["max_total"], row["max_total_e"]): + raise Reject(f"{name}: maximum") + adjacent = profile(row["R"], row["d"], row["K"], row["next"]) + if adjacent is not None: + raise Reject(f"{name}: adjacent unexpectedly available") + e = row["next"] + N, m, c = row["R"] + row["K"], row["d"] + row["K"], row["K"] - 1 + H = e - (e - row["K"]) // 3 - 1 + A = m - H + denominator = A * A - (N - e) * c + if (H, denominator) != (row["next_H"], row["next_H_den"]): + raise Reject(f"{name}: adjacent record") + if row["floor"] != row["next"] or row["ceiling"] < row["floor"]: + raise Reject(f"{name}: residual interval") + checks += 13 + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + mutations = [] + for name, key, delta in ( + ("KoalaBear", "last_total", 1), + ("KoalaBear", "next_H_den", 1), + ("Mersenne-31", "max_total", -1), + ("Mersenne-31", "line_cap", 1), + ): + changed = copy.deepcopy(ROWS) + changed[name][key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_TOP_THIRD_GLOBAL_LINE_PAYMENT_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From c09423b8ba299af565b3a00ac72caa08d445c1b7 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 06:58:34 +0100 Subject: [PATCH 12/40] MCA: extend full-lift prefix with mean-centered Gram --- agents.md | 2 +- experimental/agents-log.md | 16 ++ experimental/grande_finale.pdf | Bin 1039549 -> 1043166 bytes experimental/grande_finale.tex | 66 ++++++++ ...ft_mean_centered_global_line_profile_v1.md | 61 +++++++ ...ft_mean_centered_global_line_profile_v1.py | 151 ++++++++++++++++++ 6 files changed, 295 insertions(+), 1 deletion(-) create mode 100644 experimental/notes/thresholds/mca_full_lift_mean_centered_global_line_profile_v1.md create mode 100644 experimental/verify_mca_full_lift_mean_centered_global_line_profile_v1.py diff --git a/agents.md b/agents.md index 4d5b63827..7d5327a96 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `eK-1` full-lift scope guard. +- **How it is useful:** It moves the residual floors to `e=96151` on + KoalaBear and `e=98230` on Mersenne-31. +- **What to do next:** Replace the failed KoalaBear endpoint chord or save + `638658` at the first unpaid Mersenne support. + ### 2026-08-13 - Full-lift cross-layer global-line synchronization - **Agent/model:** Codex. diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index 3dbce24694d7872fbd39dd614f23abed5b972b43..abe43ff400ce7801af5f149e6e45d6b307735d69 100644 GIT binary patch delta 442851 zcmV(`K-0gyxj)|3Kd{ag0soWF78-wq2<<`Q)P8yM;?EZ@eWVevdJ5~%0FjA~XfT}BgnsaIcKe70YrMDsdKng}IAAcVwzhw>Vk;W>1Z2T_{uk&?+Z z?{^I8A`d>A1ok@>9_drGp=#9xUAVCQVkBCA-T_;6M6c+`qan>vm`hUc%kF>ncYPMq z7aH7fv4`Q*Y;r~L9J7dSX_z{~x%yGYHg^hflL~vq5XL{MFnETq8RD=i$|4qq()7<~ 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100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -2320,6 +2320,70 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_full_lift_mean_centered_global_line_profile_v1.py}. \end{remark} +\begin{corollary}[One-layer boundary-anchor continuation] +\label{cor:full-lift-boundary-anchor-continuation} +Retain the hypotheses and notation of +\cref{cor:full-lift-mean-centered-global-line-profile}, and put +\[ + q=e-K-3s. +\] +For $0\le J\le H$, define +\[ + B^{(J)}_0=0, + \qquad B^{(J)}_h=\min_{h\le v\le J}C_v, + \qquad + P_J=\sum_{h=1}^{J} + (B^{(J)}_h-B^{(J)}_{h-1}) + \left\lfloor\frac eh\right\rfloor, + \qquad P_0=0. +\] +If $H\ge2$, $q\ge1$, and $2(s+1)=2`, `q>=1`, `2(s+1)=K, +``` + +so restriction injectivity places it on the same affine codeword line. +The total-core line theorem then charges the high union and boundary layer +together by `N-m+1`, leaving the prefix only through `H-1`. + +## Official endpoint + +At Mersenne-31 `e=98230`, + +```text +P_H = 16434744, +P_(H-1) = 15506184, +N-m+1 = 981129, +bound = 16487313, +budget = 16777215, +slack = 289902. +``` + +At `e=98231`, the same theorem gives `17492173`, over budget by `714958`. +That is a proof-method wall, not an unsafe certificate. The Mersenne +full-lift residual interval is now `98231<=e<=1044241`. + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_boundary_anchor_continuation_v1.py +python3 -O experimental/verify_mca_full_lift_boundary_anchor_continuation_v1.py +``` diff --git a/experimental/verify_mca_full_lift_boundary_anchor_continuation_v1.py b/experimental/verify_mca_full_lift_boundary_anchor_continuation_v1.py new file mode 100644 index 000000000..f3feaaad4 --- /dev/null +++ b/experimental/verify_mca_full_lift_boundary_anchor_continuation_v1.py @@ -0,0 +1,113 @@ +#!/usr/bin/env python3 +"""Verify the one-layer full-lift boundary-anchor continuation.""" + +from __future__ import annotations + +import copy + + +ROW = { + "R": 1048576, "d": 67448, "K": 6, "budget": 16777215, + "e": 98230, "s": 32741, "q": 1, "H": 65488, + "line": 981129, "P_H": 16434744, "breaks_H": 1671, + "B_H": 15059444, "P_previous": 15506184, + "breaks_previous": 1670, "B_previous": 14130884, + "small": 16434745, "anchored": 16487313, + "bound": 16487313, "slack": 289902, + "next_e": 98231, "next_q": 2, "next_P_H": 17492172, + "next_P_previous": 16433719, "next_small": 17492173, + "next_anchored": 17414848, "next_bound": 17492173, + "next_excess": 714958, +} + + +class Reject(ValueError): + pass + + +def cap(R: int, d: int, K: int, e: int, h: int) -> int: + n, m, c = R + K - e, d + K, K - 1 + A = m - h + D = A * A - n * c + if D > 0: + return n * (A - c) // D + g = -D + balance = 2 * A * A - n * c + T = (n - A) ** 2 - (n - 1) * g + if balance < 0 or T <= 0: + raise Reject("undefined cap") + return (n - 1) * n * n * (A - c) // (A * T) + + +def profile(R: int, d: int, K: int, e: int, J: int): + values = [0] + [cap(R, d, K, e, h) for h in range(1, J + 1)] + for h in range(J - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + total = sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, J + 1)) + breaks = sum(values[h] != values[h - 1] for h in range(1, J + 1)) + return total, breaks, values[J] + + +def endpoint(row, e): + R, d, K = row["R"], row["d"], row["K"] + N, m = R + K, d + K + s, q = divmod(e - K, 3) + H = e - s - 1 + if H < 2 or q < 1 or 2 * (s + 1) >= e or m - H <= K - 1: + raise Reject("hypotheses") + full = profile(R, d, K, e, H) + previous = profile(R, d, K, e, H - 1) + line = N - m + 1 + small = full[0] + 1 + anchored = previous[0] + line + return (s, q, H, line, *full, *previous, small, anchored, + max(small, anchored)) + + +def validate(row): + got = endpoint(row, row["e"]) + expected = ( + row["s"], row["q"], row["H"], row["line"], + row["P_H"], row["breaks_H"], row["B_H"], + row["P_previous"], row["breaks_previous"], row["B_previous"], + row["small"], row["anchored"], row["bound"], + ) + if got != expected: + raise Reject("endpoint") + if row["budget"] - row["bound"] != row["slack"] or row["slack"] <= 0: + raise Reject("slack") + nxt = endpoint(row, row["next_e"]) + if (nxt[1], nxt[4], nxt[7], nxt[10], nxt[11], nxt[12]) != ( + row["next_q"], row["next_P_H"], row["next_P_previous"], + row["next_small"], row["next_anchored"], row["next_bound"] + ): + raise Reject("adjacent") + if row["next_bound"] - row["budget"] != row["next_excess"]: + raise Reject("adjacent excess") + return 24 + + +def main() -> None: + checks = validate(copy.deepcopy(ROW)) + mutations = [] + for key, delta in (("P_H", 1), ("anchored", -1), + ("next_bound", -1), ("slack", 1)): + changed = copy.deepcopy(ROW) + changed[key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_BOUNDARY_ANCHOR_CONTINUATION_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From f771b92d970eb57ac733f18593bd362f07dda748 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 07:53:28 +0100 Subject: [PATCH 14/40] MCA: absorb second Mersenne boundary layer --- agents.md | 2 +- experimental/agents-log.md | 16 +++ experimental/grande_finale.tex | 86 ++++++++++++- ...lift_two_boundary_layer_continuation_v1.md | 76 ++++++++++++ ...lift_two_boundary_layer_continuation_v1.py | 115 ++++++++++++++++++ 5 files changed, 292 insertions(+), 3 deletions(-) create mode 100644 experimental/notes/thresholds/mca_full_lift_two_boundary_layer_continuation_v1.md create mode 100644 experimental/verify_mca_full_lift_two_boundary_layer_continuation_v1.py diff --git a/agents.md b/agents.md index 4bc49a0be..ccc98d8a8 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `ec, +\] +and put +\[ + Q_H=\left\lfloor\frac{N-e-c}{m-H-c}\right\rfloor, + \qquad + \Delta_H=\left\lfloor\frac{e}{s+1}\right\rfloor. +\] +Then +\begin{equation} + |Z|\le\max\left\{ + \begin{array}{l} + P_{H-2}+N-m+1,\\ + P_{H-1}+Q_H+1,\\ + P_{H-1}+2,\\ + P_{H-1}+Q_H,\\ + P_{H-1}+\Delta_H + \end{array}\right\}. + \label{eq:full-lift-two-boundary-layer-continuation} +\end{equation} +\end{corollary} + +\begin{proof} +Let $\mathcal A$ be the top-third union, and let +$\mathcal D_1,\mathcal D_2$ be the exact layers of deficits $H,H-1$. +Choose inside agreement sets at their layer sizes, so their missed sets +have sizes $s+1,s+2$. The guard $2(s+2)=3`, `2(s+2)K-1`, and the parent prefix +hypotheses through `H`. Put + +```text +Q=floor((N-e-(K-1))/(m-H-(K-1))), +D=floor(e/(s+1)). +``` + +Then + +```text +|Z| <= max{ + P_(H-2)+(N-m+1), + P_(H-1)+Q+1, + P_(H-1)+2, + P_(H-1)+Q, + P_(H-1)+D +}. +``` + +## Proof mechanism + +If the top-third union has at least two members, two anchors synchronize +both boundary layers because the worst mixed triple intersection is +`e-s-s-(s+2)=K`. If it has one member, two first-boundary members synchronize +that layer; the top member is charged separately and the boundary line uses +the sharper outside-core cap `Q`. + +If the top union is empty, choose exact size-`H` inside agreement sets for +the first boundary layer. If two missed sets intersect, that one repeated +miss raises every fixed-pair triple intersection from `K-1` to `K`, so the +whole layer is an outside-core line. Otherwise all missed sets are pairwise +disjoint and there are at most `D` of them. + +## Official endpoint + +At Mersenne-31 `e=98231`, + +```text +P_(H-2) = 15505282, +P_(H-1) = 16433719, +N-m+1 = 981129, +Q = 484, +D = 3. +``` + +The five cases are + +```text +16486411, 16434204, 16433721, 16434203, 16433722. +``` + +Their maximum has slack `290804` below budget `16777215`. At `e=98232`, +the residue is `q=0`, so this theorem stops without making an unsafe claim. +The Mersenne full-lift residual interval is `98232<=e<=1044241`. + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_two_boundary_layer_continuation_v1.py +python3 -O experimental/verify_mca_full_lift_two_boundary_layer_continuation_v1.py +``` diff --git a/experimental/verify_mca_full_lift_two_boundary_layer_continuation_v1.py b/experimental/verify_mca_full_lift_two_boundary_layer_continuation_v1.py new file mode 100644 index 000000000..3227ae315 --- /dev/null +++ b/experimental/verify_mca_full_lift_two_boundary_layer_continuation_v1.py @@ -0,0 +1,115 @@ +#!/usr/bin/env python3 +"""Verify the residue-two full-lift boundary-layer continuation.""" + +from __future__ import annotations + +import copy + + +ROW = { + "R": 1048576, "d": 67448, "K": 6, "budget": 16777215, + "e": 98231, "s": 32741, "q": 2, "H": 65489, + "line": 981129, "outside": 484, "disjoint": 3, + "P_H2": 15505282, "breaks_H2": 1670, "B_H2": 14129968, + "P_H1": 16433719, "breaks_H1": 1671, "B_H1": 15058405, + "cases": (16486411, 16434204, 16433721, 16434203, 16433722), + "bound": 16486411, "slack": 290804, + "next_e": 98232, "next_q": 0, +} + + +class Reject(ValueError): + pass + + +def cap(R: int, d: int, K: int, e: int, h: int) -> int: + n, m, c = R + K - e, d + K, K - 1 + A = m - h + denominator = A * A - n * c + if denominator > 0: + return n * (A - c) // denominator + gap = -denominator + balance = 2 * A * A - n * c + tangent = (n - A) ** 2 - (n - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("undefined cap") + return (n - 1) * n * n * (A - c) // (A * tangent) + + +def profile(R: int, d: int, K: int, e: int, end: int): + values = [0] + [cap(R, d, K, e, h) for h in range(1, end + 1)] + for h in range(end - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + total = sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, end + 1)) + breaks = sum(values[h] != values[h - 1] for h in range(1, end + 1)) + return total, breaks, values[end] + + +def endpoint(row): + R, d, K, e = row["R"], row["d"], row["K"], row["e"] + N, m, c = R + K, d + K, K - 1 + s, q = divmod(e - K, 3) + H = e - s - 1 + if q != 2 or H < 3 or 2 * (s + 2) >= e or m - H <= c: + raise Reject("hypotheses") + # Fail closed unless every parent cap through H is defined. + profile(R, d, K, e, H) + p2 = profile(R, d, K, e, H - 2) + p1 = profile(R, d, K, e, H - 1) + line = N - m + 1 + outside = (N - e - c) // (m - H - c) + disjoint = e // (s + 1) + cases = ( + p2[0] + line, + p1[0] + outside + 1, + p1[0] + 2, + p1[0] + outside, + p1[0] + disjoint, + ) + return (s, q, H, line, outside, disjoint, *p2, *p1, + cases, max(cases)) + + +def validate(row): + got = endpoint(row) + expected = ( + row["s"], row["q"], row["H"], row["line"], row["outside"], + row["disjoint"], row["P_H2"], row["breaks_H2"], row["B_H2"], + row["P_H1"], row["breaks_H1"], row["B_H1"], row["cases"], + row["bound"], + ) + if got != expected: + raise Reject("endpoint") + if row["budget"] - row["bound"] != row["slack"] or row["slack"] <= 0: + raise Reject("slack") + if row["e"] - 3 * row["s"] - 2 != row["K"]: + raise Reject("mixed intersection") + if (row["next_e"] - row["K"]) % 3 != row["next_q"] or row["next_q"] != 0: + raise Reject("adjacent residue") + return 27 + + +def main() -> None: + checks = validate(copy.deepcopy(ROW)) + mutations = [] + for key, delta in (("outside", 1), ("bound", -1), + ("slack", 1), ("next_q", 1)): + changed = copy.deepcopy(ROW) + changed[key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_TWO_BOUNDARY_LAYER_CONTINUATION_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 785fb961e22a836f84f9feed75f089c69c6d93cb Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 08:18:47 +0100 Subject: [PATCH 15/40] MCA: route first residue-zero Mersenne support --- agents.md | 2 +- experimental/agents-log.md | 18 +++ ...l_lift_residue_zero_direction_router_v1.py | 47 ++++++++ experimental/grande_finale.pdf | Bin 1047221 -> 1057582 bytes experimental/grande_finale.tex | 104 +++++++++++++++++ ...l_lift_residue_zero_direction_router_v1.md | 76 +++++++++++++ ...l_lift_residue_zero_direction_router_v1.py | 106 ++++++++++++++++++ 7 files changed, 352 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_residue_zero_direction_router_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md create mode 100644 experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py diff --git a/agents.md b/agents.md index ccc98d8a8..d1525aeb6 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e int: + return -(-value.numerator // value.denominator) + + +def main() -> None: + e, H, c = 98232, 65489, 5 + A = 2 * H - e + denominator = A * A - e * c + if (A, denominator) != (32746, 1071809356): + raise ValueError("direction denominator") + + class_ratio = Fraction(e * (A - c), denominator) + classes = class_ratio.numerator // class_ratio.denominator + if classes != 3 or class_ratio < 3 or class_ratio >= 4: + raise ValueError("direction classes") + + outside_length, outside_agreement = 950350, 1965 + line_ratio = Fraction(outside_length - c, outside_agreement - c) + line_cap = line_ratio.numerator // line_ratio.denominator + boundary = 1 + classes * (line_cap - 1) + if (line_cap, boundary) != (484, 1450): + raise ValueError("boundary") + + prefix, budget = 16432695, 16777215 + top = budget - prefix - boundary + 1 + if top != 343071: + raise ValueError("strict threshold") + + N, m = 1048582, 67454 + core = ceiling(Fraction(top * m - N, top - 1)) + if core != 67452 or core != m - 2: + raise ValueError("forced core") + + print( + "MCA_FULL_LIFT_RESIDUE_ZERO_DIRECTION_ROUTER_V1_AUDIT_PASS " + "classes=3 boundary=1450 top=343071 core=67452" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index fc4102a45daf87dc22157d8088c37e835e2dee64..6debd5183f57f220d1cb4cd501d764a1701b73da 100644 GIT binary patch delta 266792 zcmV((K;XZ%^*^r0K(NRbe=fo-N<1Krf47^NWBbP0_u^tgxr>SE8qQrz$o<8{k--ZQ zA)u*6`{m7xKVP`?kw(DkNq}uwt(zBLe+yS7+6f~{pq>Xas#(Q!8BOS=UVVA-*?x%< z&Fe5}B9sV$5EA_aL}|W9N+#F5-!Y_%Joso5*zZ($f22>*hN@K)bm7AG zi;-ygc?WFO5xt@#kA^fyVJ=C%FT2~{^;t|`XmG>D9)?r1$rZtK%p$s_Vd@Cy>PH#d z+$qFOD(n?Q82_lk;2FMVh{LKVi&z*+(?6fZCiwbFDDMHO9Kp`a7hW(MIz zc;pHgMxO!EbbDPyP4=dqzgd{$mXy9)VjgGrudL3uMbof9D&t@7M%I1#K(|12@ct}t zS2>iB9)4f6)L9fn*))V=qTSQ^d1l1NmbS+p>e=r?db=0T8+>(?r>iu`)6gWxL>PB) z37qlEp)>v;z7)?wlVBJs0XUao4gn~CtyxWTqev3H_pi_qMKl`f11R?9ao96EvmNc} zaqLFy%pS^6vTGNCx`1tKe*I=qA0R+7p4f|xphRW8%zQ616>Xk3(dLH_-#>o%?mkO5 z=`Kqnw0YcbQoKnS*)fuC9*fOy0p5lfM=T8jR8}wH8;)aaBS;V=4P*dysjGYwG&k?tWove87s_%JRAWl7SyU0g|!gg7N{)> z#P1QIujY$z8P|5@)Lujqf&;=X*!$gm8jtY|J)p@Bj%p6^u!UD}r9omO!1_ce9`I!p zfUC7Vr7%lWZ(Su0qS9Y&8?%a6g?VmH$jJHkXkeEASN{=Z4QJICXD6C}`k&Hhm%y#a zSW2=>ehDdNyM$5e9&Fu%oO>Qdmgu;swm1b#08?X$zA%r>DF#2$p!Z|x+`-@UGU!b*SO5xq0BnS#%|T;Hn@-YePMmq4BFM zAY@E#0YA6X9Y}9%C0G8<>Zf7|Cy9$=48AKA=wY+Szhv{ zks677e(b4#CkSCXSn2jkllK*6*vsP=ur2KKUiiAX5p6e<9GtQ^y|U3X)8m=ssD3NW zLEm)N*Y>U`y4=;?vabnS%}9BVfsK$>$=8_%vNT3348q^n^h}cgF2~y?DMs>FC%z{NlFqNr0H5rMykzi zgSPLSJR4NT3~d7$uD)qM)m>@*_SDU_8%Gq}vsxLvcHc~k0xQd^m4lBUx4uF_33W>k z;an7kH`y`6kkV8f?c6`SwLUV=!;x82qt{z<}hHJbge#-Xh=@O5>}Xi zDqPur;>7uQ-cvzTwJ&o|&@2sDv=R;C)8wWa1)bTr5YI~JGl#t>^$q+unQegRg?r(k ztVgAW;mr(>$qYxyjsinlGP^fx7#dADl3kLdU?=nhsjE1z5+sJ2mjHtgUts78M`pV? z!V@?6GT{?{mA=_)x}zT%OugZ=yYUTeXc04iX~I=slFq7m%2>egS_hF!JSMAL7JA$< zlq;opy_6&|hG1o6ZB%;6_;rLpnAc^=kBuX6HHuhji%g(%yS`l1bCkst2W)MXPv**E zFb-WfrY;=wx^osl3~MlKrzhP&h_@5ZiPu(fuRU}aYfY`rNQ45mRt|`lgIfShHlE0T zY?y%K%EnIxi+C|mb`3u zX-+x}qHFCCkFWus7x0#kVK+tOW-lOp3q5Fxr&D<^LZftsTd+*Y~#sjCWZe7QKB6vKSeG+FvyRXjuY@}B7sd*^hBMw4RW zyzXku#IFx1SC|31jv!>y&chuT9AB|x5>6B?a9CdursIOpp&5>W)Ir77vzV8E84*HJ zkgXjGQT%3^@xlk?K1;)#S9LWiRQp7$ma;5M)|ljAArrqeqTGYTn6iokAz4@s&_DL$ zlI{=|;Ew^C!4#X2!!&21GD9>$agdRJWq;tU>RdglS9P}*0KKO=X4PtX4+E5BbOBSznZ8A-{ zx+>&YM#NBi^Wxvof*FuP`!msKLbbqiEP-AyTRWAb>7X_)ujjSO*h#*`10qo4-gLQF zj$-H)v$XZuL)4vlL_*8Vqnh z=$ij`k01UAyarGg3T1AyBO&Swe>OH0^|;UPez)v!wOxg)AFsZ@z54D`p03g$Ps3<+ zd%sdf2Z=VTG}A$*)75RY`s)4o>xWPLB;=N@b;AI@ci-+rWm z+&DQRC&lefwr=ecA3Pj-TU0#Q+o7(GMRR_hGIAejnSpO4TNJ9-OzFqqP^HD7N(?%=^_rb%w>^kj zu}R0B?fP}(*}a%}EXL#T(x_3Ay>;@r6ZH!xqKR5?7oGC+M;Bsqf9t(GRq1z~>NJht zKDDWb@jb^TD#K&-dzCfr*pFRXHpiic*f`r$M`>(e&-k5t{%Y@SZ`;zIxdI)mAv~s7 zpbb~Be-gwYM6t;N?d$=W1RuZqltlAziXi7tJA5g-1KKLhBx>QIw@tB5Qg5s&$YEsc z`1i*TH_G5+>ODr}e`8Pa8OxA`rz{ff0Ln1GOqvEs2$JpL)Z1O(9k;vgIBsCKIGQ$& zL)uvJ=-1b*<@Qz)q(%}iH~+ZnRI)VNTpkshAWqWdn&0{Cdf#)jzK0DjD8_Ctqv#OQZKjvbf;7%pFD_Jm-IjHQ zSv-}jV?!PQFlw!>tkg1&A$SmBV-Jcer2kZNxx8ZOe<8EO`vJv9;O+p;_j?i}CzQM-0Lk0neI~EWhJq38ZW`Mxw5}qZ2 zBX1+~pW6zYWq%8m8Eh$ZPd?(pZ?^J5!>-*de*)i$kpX#v^22Ba^AjMB!=Y%!He1^Q zheW)p{3kLJ3bfr(^2(gyc$fmnhb>I8l`k&a&T8ZynAM&E#IK;V7m{f4l*4Q`AIMXX zCrcMxSI&bY{(AP&7zaWGXQywQDb-M_#`;a&S|^hOqPnBFtm<-eNyy=Nhx=K<7w!FV zf25KEun$MIa|1BSqx-*&wskDxJa!I6i2^?qTjmBU(GEg82T#zP&&Q$yT&_!(MSwgA zaTCR>`?46Ui(KpC26XK^6t}T1;GjKOdOA*7If8GhLcA+r*q%2u9)ywBb`j*U_h}7; z7z$_2;BBTFhEY1YAe}{J`~qg33_1qEf24#Q`g&h4)9JIUz8~!pS2fXF2QpvkYC_Y3 z>f8S0lS%(<0Wz^tLFiEarni(Dyj@>!>q?>tRTUj3MYBf4)1> z;rovpXJq1iS+?Mo&}_`e6n*5&?eMUcpvFk(!Cd(XSx6Fc;6ij)? zfx6xk(^7bEFgOZOyPgxgdMD&eDya}n%LymUMoq+RK^YWN1|)<=c^_nIQT?YVWxX7I z-q@YdYHh+G(FyAWQVdWxP<*5LT!U)hv6f%zJ<|?HY77=@DbxnFvws}Hf7zp>#|CL4 ze=D}VwJymeS(1gtMJx@GZL+0kg)E3>mqR8

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z<-pD=8&$P656I?eMFpmfJQ&0T$ZjVS!>^bO=^y+PwqKKr%B%a_8}e+Ttjowvr*U+v(F7Qqv8piyn(Rhy`fU(kZ4;O1E|+prbKpTd$u86wTt2WgAjgLq7pxz0>7I)Kb{JIVkSS# zhd)tiPt4|G`eb0CE~)=LptZm5BUT+h_h*#B>2bz)fea&122ucF5%m@5h4YKFl_Oa5 z=>LQevE0|p5~-T(hAEBg7{ a`uYd^Iy&JBgT;iUB_we0,\qquad A^2>ec,\qquad N-e>m-H>c, +\] +then +\begin{equation} + |\mathcal D|\le 1+J(Q-1). + \label{eq:full-lift-boundary-direction-cap} +\end{equation} +If $\mathcal T$ is the synchronized union of deficits greater than $H$, +then consequently +\begin{equation} + |Z|\le P_{H-1}+1+J(Q-1)+|\mathcal T|. + \label{eq:full-lift-boundary-direction-router} +\end{equation} +\end{corollary} + +\begin{proof} +The claim is immediate when $\mathcal D$ is empty. Otherwise fix an anchor +$(\gamma_0,c_0)\in\mathcal D$. For every other member put +\[ + p_\gamma=\frac{c_\gamma-c_0}{\gamma-\gamma_0}. +\] +The chosen size-$H$ inside agreement sets meet in at least $A=2H-e$ +coordinates, and on this intersection $p_\gamma=q_{\rm dir}$. Thus the +intrinsic set +\[ + U_p=\{x\in E:q_{\rm dir}(x)=p(x)\} +\] +has size at least $A$ for every represented direction $p$. Such a $p$ is +nonzero: otherwise the two explanations would be the same codeword, while +on their nonempty inside intersection that codeword would equal received +words whose difference is $(\gamma-\gamma_0)q_{\rm dir}\ne0$. + +For distinct represented directions $p,p'$, the intersection +$U_p\cap U_{p'}$ lies in the zero set of the nonzero degree-$m-H>c$, outside-core packing bounds the line by +$Q$. The anchor is repeated in each line closure, so each of the at most +$J$ classes contributes at most $Q-1$ nonanchors. This proves +\eqref{eq:full-lift-boundary-direction-cap}. The boundary and top layers +have one-slope ownership; the independently truncated prefix charges all +lower deficits, proving \eqref{eq:full-lift-boundary-direction-router}. +\end{proof} + +\begin{remark}[First residue-zero Mersenne router] +At Mersenne-31 $e=98232$, +\[ + (s,H,A)=(32742,65489,32746),\qquad J=3,\qquad Q=484, +\] +so $|\mathcal D|\le1450$. The exact prefix is +$P_{H-1}=16432695$, and therefore +\[ + |Z|\le16434145+|\mathcal T|. +\] +An unsafe family would need $|\mathcal T|\ge343071$. If $g$ is the total +common core of this affine line, line packing gives +\[ + g\ge + \left\lceil\frac{|\mathcal T|m-N}{|\mathcal T|-1}\right\rceil + \ge67452=m-2. +\] +Thus the first residue-zero obstruction is reduced to the one- and +two-private-coordinate line strata. This is a structural router, not a +safety or unsafety certificate at $e=98232$, and the Mersenne residual +interval remains $98232\le e\le1044241$. The exact replay is in +\path{experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py}, +with an independent endpoint audit in +\path{experimental/audit_mca_full_lift_residue_zero_direction_router_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md b/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md new file mode 100644 index 000000000..c7044984f --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md @@ -0,0 +1,76 @@ +# MCA full-lift residue-zero direction router v1 + +## Status + +PROVED / EXACT FINITE CALIBRATION / STRUCTURAL ROUTER. + +## Statement + +Let `D` be the exact deficit-`H` boundary layer on direction support `E`, +`|E|=e`, and put `c=K-1` and `A=2H-e`. Assume + +```text +A>0, A^2>e*c, N-e>m-H>c. +``` + +Fixing an anchor in `D`, every other member determines a nonzero normalized +codeword direction agreeing with the gauged direction on at least `A` +coordinates. Distinct directions have intrinsic agreement sets meeting in +at most `c` coordinates. Hence the number of direction classes is at most + +```text +J=floor(e(A-c)/(A^2-e*c)). +``` + +Every class together with the anchor is one affine codeword line. Its +outside common core is contained in the zero set of a nonzero degree-`=343071`. If `g` is the top line's +total common core, line packing + +```text +|T|(m-g)<=N-g +``` + +forces `g>=67452=m-2`. + +## Nonclaims + +This does not prove safety or unsafety at `e=98232`. The Mersenne residual +interval remains `98232<=e<=1044241`. The next task is the exact +classification of affine lines whose members have only one or two private +agreement coordinates beyond the total common core. + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py +python3 -O experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py +python3 experimental/audit_mca_full_lift_residue_zero_direction_router_v1.py +``` diff --git a/experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py b/experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py new file mode 100644 index 000000000..be8c35d07 --- /dev/null +++ b/experimental/verify_mca_full_lift_residue_zero_direction_router_v1.py @@ -0,0 +1,106 @@ +#!/usr/bin/env python3 +"""Verify the first Mersenne residue-zero direction-class router.""" + +from __future__ import annotations + +import copy + + +ROW = { + "R": 1048576, "d": 67448, "K": 6, "budget": 16777215, + "e": 98232, "s": 32742, "residue": 0, "H": 65489, + "pair_agreement": 32746, "direction_classes": 3, + "outside_line_cap": 484, "boundary_cap": 1450, + "P_H1": 16432695, "prefix_boundary": 16434145, + "top_threshold": 343071, "line_cap": 981129, + "forced_core": 67452, "m": 67454, "N": 1048582, +} + + +class Reject(ValueError): + pass + + +def cap(R: int, d: int, K: int, e: int, h: int) -> int: + n, m, c = R + K - e, d + K, K - 1 + A = m - h + denominator = A * A - n * c + if denominator > 0: + return n * (A - c) // denominator + gap = -denominator + balance = 2 * A * A - n * c + tangent = (n - A) ** 2 - (n - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("undefined cap") + return (n - 1) * n * n * (A - c) // (A * tangent) + + +def profile(R: int, d: int, K: int, e: int, end: int) -> int: + values = [0] + [cap(R, d, K, e, h) for h in range(1, end + 1)] + for h in range(end - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, end + 1)) + + +def endpoint(row): + R, d, K, e = row["R"], row["d"], row["K"], row["e"] + N, m, c = R + K, d + K, K - 1 + s, residue = divmod(e - K, 3) + H = e - s - 1 + A = 2 * H - e + denominator = A * A - e * c + if (residue != 0 or A <= 0 or denominator <= 0 or + not (N - e > m - H > c)): + raise Reject("hypotheses") + classes = e * (A - c) // denominator + outside = (N - e - c) // (m - H - c) + boundary = 1 + classes * (outside - 1) + prefix = profile(R, d, K, e, H - 1) + prefix_boundary = prefix + boundary + threshold = row["budget"] - prefix_boundary + 1 + line_cap = N - m + 1 + forced_core = (threshold * m - N + threshold - 2) // (threshold - 1) + if threshold > line_cap or forced_core != m - 2: + raise Reject("terminal") + return { + "s": s, "residue": residue, "H": H, "pair_agreement": A, + "direction_classes": classes, "outside_line_cap": outside, + "boundary_cap": boundary, "P_H1": prefix, + "prefix_boundary": prefix_boundary, "top_threshold": threshold, + "line_cap": line_cap, "forced_core": forced_core, + "m": m, "N": N, + } + + +def validate(row): + got = endpoint(row) + for key, value in got.items(): + if row[key] != value: + raise Reject(key) + return 19 + + +def main() -> None: + checks = validate(copy.deepcopy(ROW)) + mutations = [] + for key, delta in (("direction_classes", 1), ("boundary_cap", -1), + ("top_threshold", -1), ("forced_core", 1)): + changed = copy.deepcopy(ROW) + changed[key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_RESIDUE_ZERO_DIRECTION_ROUTER_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From b5bae1a12a3a0df2d2a8eac0615a78f164d237d1 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 08:31:40 +0100 Subject: [PATCH 16/40] MCA: absorb first residue-zero Mersenne core --- agents.md | 2 +- experimental/agents-log.md | 23 ++++- ...ll_lift_residue_zero_core_absorption_v1.py | 33 ++++++++ experimental/grande_finale.pdf | Bin 1057582 -> 1060634 bytes experimental/grande_finale.tex | 72 +++++++++++++++- ...ll_lift_residue_zero_core_absorption_v1.md | 58 +++++++++++++ ...l_lift_residue_zero_direction_router_v1.md | 8 +- ...ll_lift_residue_zero_core_absorption_v1.py | 80 ++++++++++++++++++ 8 files changed, 265 insertions(+), 11 deletions(-) create mode 100644 experimental/audit_mca_full_lift_residue_zero_core_absorption_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_residue_zero_core_absorption_v1.md create mode 100644 experimental/verify_mca_full_lift_residue_zero_core_absorption_v1.py diff --git a/agents.md b/agents.md index d1525aeb6..5b8128d02 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e None: + e, K, m, N, c = 98232, 6, 67454, 1048582, 5 + inside = (m - 2) - c + sync = e - inside + K + if (inside, sync) != (67447, 30791): + raise ValueError("synchronization") + + agreement = m - (sync - 1) + n = N - e + ratio = Fraction(n * (agreement - c), agreement * agreement - n * c) + cap = ratio.numerator // ratio.denominator + if agreement != 36664 or cap != 26 or not (26 <= ratio < 27): + raise ValueError("Johnson cap") + + low, line = e * cap, N - m + 1 + total, budget = low + line, 16777215 + if (low, line, total, budget - total) != ( + 2554032, 981129, 3535161, 13242054): + raise ValueError("payment") + print( + "MCA_FULL_LIFT_RESIDUE_ZERO_CORE_ABSORPTION_V1_AUDIT_PASS " + "sync=30791 list=26 total=3535161 slack=13242054" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index 6debd5183f57f220d1cb4cd501d764a1701b73da..a03588855b92ea1db888d1a0e80f5f8acadc6a70 100644 GIT binary patch delta 449291 zcmV(+K;6Ht#z30hK(NRbe~!o;3ww(9FLpC?Y~MKhUR+ElcQG+t!?}wIxxbh=GI${b z0$RLizr1Tb@SruZ{ey$J7Gi#)bn6QHLI8|qY1s#t1mA;+b>a~ zc^xKAgc2bTLSnx|d5z5Q9Jf^%JBDf1%l0YWeVu@?56HD{WOjmb}alt&E;)A(oRIm^ili3%81GUD_myvo16O(cn7?*LL z2PuE8S>JP`I1+xJzrv5WYVd+2khr&OD%ri=o0{xw>aOZ`??C}M(H#blfSY*#`fH&d zV89O9x|ak4mRetTe_wYy+B|Qf%?}^GfBf*>eMUEQm(eKRJnlCXZ79P#hUw<9*!&it zZHQ3BXz=jmpLd_`e%^*SA;ImBpMSc${rP{d+kZX&^dYi~MesM?DE&cSeE9*^#@Qy0 zc3G75i^uUUOOj29@J`>*>u0-d7)MbQ+`aO=h1D?*+7~67m%1z({h0*&TA2sp^;pVW z+SiUs{vpg8s@@_Nd~43^+58rw2xn1%sl7k75B`TS0-LNcV5+s|ra2vsZQaV;?3I7l zRU^K3qRNFes@^+eg}IQ2BVfgXTD79E7Q$+Q+QxzSJ;LPGd=V~_+P*opH^wPO0nRSi z``tZF#(0Jv&~yh!HHUcE!V+9*kQfe-K2e+u_%Z^()!Lp?m{(M9T_p~p(qC;G^A@iP z^W2<};q&j2z`Xun{YR8FoK;_(ooIjRe?p>N3b!I-DTyxmB_xRLQbw$MkabV&-19K9 zM8`$7MHDOnOpPV_!aOpk82m(n-jAho>)|L%G5148;^>Nf)3`y-uwOce8Ep;=+TSCT zC~@GjDulWc5LK-X+!_R-ewZ6WDG5mYmN5<`>{ALh!6I*^K_-+3{%m2n--3TX>rP2$ znK45C!CP5Z^E67bI3V=@7nazC<%5z3sZr5Z_5svC|Hb^XF-V3O1ZI#$*Tcmw`0tIV zOV&o6b1rdsKDtytYNH*j#)(HYxD*Ga;QJYfDJJQa-KTC$hF~8rojdL*WW`&A3}q98$QLp`0L@vAH#WK3=WKey8zaD;$_6TJ4|)a~$( z7Y*=faxlLgU?IxVJe3xQN&rDOT%BgMXnw}QZDXdHQO z|H1OX%%m7+**i=cNo6Zk*?(^ z)KvLoDgf*H9CYnTEejjT;`H+7e**2qq5W9_^@e(3ScQ#OVRV8!8qT~TzotG#7k6SkU>@*V>lA+3_H zGYw>vrYj6W-XKpte=~BNJ2qCLTAj8!+?Wejct>2!yxpt$7 z#4Aj3czNxKpDj-Mh`ew~@paTB*f%sU z3+M|PeUa-1!G!PJ90qK|*6I_EhU7FXVTB2(!j&yfoR5F!JrzV%`!e?g$>@shEgIOT z$xSs1I*zt>=c%m#d3z*|0s-4v0Vy@2$M zNBh&esH+?ka~1Yv+*M{^=Kx$97C{fa!aj#lwlGhqhXIzOV^dOUyY~~ppQZ*bp zUoH+O#W3GAO_sh_70=MUyr&V#-Z>p2(WKZouej*+N?L6Fxu8y;{ z5>Mg<4(kg-IxYwun&B8o9aLOBi+Pz5LCk*#*;*MP=*=?Yg%8SomWDa6>S|P|_K8+4 zWf@~@OmeW`iC-E~?m=QqS;c{nEG!4;ANz4hcL)pc#{kV>icQF2nzK-u#UzcvKD-t( zuYsROTg({TPY}2}g@LdNtq#hejD#{9DoptF5>C|H^uwTF#ry{cOmUE{6-+W0%u#7;0}`{2N*@15#*zCK^qs7I==O zNgQNr)tJyhZCqZ@Yn8E+e2E7{pvJxFa<3dEI9g$j!^Vr>=c+vUfhF1Z&GIP}^7&=d zcMQd4<7OqcSm_!uDQ6sSk;~_sr9^+vWNZ6el1;mXuF|g66e_n`9O_=39aT67j!PY^ zhgObfb(PkNWv~@rtYkrsQh7sD)v4JhH(3q-*})J%0EfgqBbx3T19&b98cLVQsT1As__- z-m^+3jRb#m?_bk(s|iXB8iS2vF};=`Ij$RP8jI`E!Jr0pJn}oiHaL4M1yS7&7o_z_ z;Sqw3V=;YFz!YEubl;`!crt?XoX3@wr#I#7ywxomLG(*{lor8x;c;c<=}oz&h^$;t z@vP@=okn2eXiRTFJ(N=;Emc-64)~lAfgKTz=`DY#kLt)&SvBy}W6i<3d_-HPx1d@c zswak}m5VevII^3;W%^1DTq?u4i>)W)YfxNA1BRj&@qz{$<5--ae8w7S>CXSQdM4p$by-eO zKl5%|UWQAF3G&JCaCAglPf$UFCrhj`4%Mv;B6zqXTNyIF4KHGo{xn6Ew*CYJnHn8Y z*6D-btZBQJ(jpuPn*#S1$5k~%)*yw0kb60}A zR}FkvG#o5WVhD%3LZjBB13jXg_oOfshbgx4$zjBUqAO%G_+faGqaJPsK3^4t*^EA%BJtEKlP0rif*eiZ^H4|yD!Fi^vl!m83(OQ?%QJxGgqQ) zKKW5@(%6f_NIqQ32EYAC1-Wr@L{5skgKXW|Cq8&Q4z_=&cyO>|U7d>N@;tDIHgKML z;k@tOr)F)72o5)3iv&!bu@JXgt-NADeKYN)C{biW?;Ysa*r|qBukZL{X@{x)E!1&Y zOm^h4qC+9G7=&z#x~scAj}JZW`{a*@VmR_p-5pQUxKiTMTd_7llo|L&vPGeO$&`K! 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z{p%HV6N|fm=hF@5={}sZi%Hg{dW(w<)?W9vs~OBl)o*IL^~JTf(^ut`)rn<-S?cUJodPPI99fsYyt1y_Swuo1^o*vq377xiYx1jt(mJUU5k%~ z1kL2;4p?cLs8mnIXzp`mue?YHeXY74OldY9iyrTQM1_wtRH}ZDb2I`F0WWetVuxsQ zyh#RK>uCh8U} z|C0WHAh0^C7i{@zhMyBH5!CcI(4!_PF&nX}@T$^*~I z4edTm^E@~!ytE#4&TNqO-@rBFHqVStd=+t8I~Xg@C_LD>xZ!uefy=^-{8w3QT!lYW zIB($Puv7Ta@PN6AABCa7yU)gdhT&OLWyx(kCz@nVG&5;5328J-1#rJ|fOHG6RiVxg z3|=t0JLtb){+n>X16777215$. By the preceding router, the +synchronized top line $\mathcal T$ has at least $343071$ members and total +common agreement core $G$ of size at least $67452=m-2$. Write the line as +$c_\gamma=A_0+\gamma p$. Its direction $p$ is nonzero. Outside the +gauged direction support $E$, a coordinate common to two line members must +be a zero of $p$, so +\[ + |G\cap E|\ge67452-(K-1)=67447=:u. \tag{1} +\] + +Take any selected slope $\delta$ whose assigned explanation has outside +deficit $h\ge e-u+K=30791$. Ownership supplies an inside agreement set +$S_\delta$ of size $h$. Choose two distinct top anchors. By (1), +\[ + |G\cap E\cap S_\delta|\ge u+h-e\ge K. +\] +On these coordinates, both normalized differences from $c_\delta$ to the +two anchors equal the gauged direction. Restriction injectivity makes the +two degree-$=98232-67447+6=30791, +``` + +those inside agreements meet the line's inside core in at least six +coordinates. Two top anchors and restriction injectivity put the explanation +on the same affine line, at its actual slope parameter. All such slopes are +therefore charged once by `N-m+1=981129`. + +The remaining assigned explanations have deficit at most `30790`, hence +outside agreement at least `36664`. The ordinary punctured Johnson cap is + +```text +floor(950350*(36664-5)/(36664^2-950350*5))=26. +``` + +Using the deliberately crude owner cap `e` on those explanations gives + +```text +|Z|<=98232*26+981129=3535161<16777215. +``` + +This contradicts unsafety, with exact margin `13242054`. + +## Consequence and nonclaims + +Mersenne-31 full-lift support `e=98232` is safe. The residual interval is +now `98233<=e<=1044241`. This does not pay `e=98233`, close the deployed +row, or provide an adjacent unsafe certificate. + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_residue_zero_core_absorption_v1.py +python3 -O experimental/verify_mca_full_lift_residue_zero_core_absorption_v1.py +python3 experimental/audit_mca_full_lift_residue_zero_core_absorption_v1.py +``` diff --git a/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md b/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md index c7044984f..bc13763b2 100644 --- a/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md +++ b/experimental/notes/thresholds/mca_full_lift_residue_zero_direction_router_v1.md @@ -62,10 +62,10 @@ forces `g>=67452=m-2`. ## Nonclaims -This does not prove safety or unsafety at `e=98232`. The Mersenne residual -interval remains `98232<=e<=1044241`. The next task is the exact -classification of affine lines whose members have only one or two private -agreement coordinates beyond the total common core. +This router alone does not prove safety or unsafety at `e=98232`. Its +near-maximal-core terminal is discharged by the successor common-core +absorption theorem, which pays this support without classifying the private +coordinates individually. ## Replay diff --git a/experimental/verify_mca_full_lift_residue_zero_core_absorption_v1.py b/experimental/verify_mca_full_lift_residue_zero_core_absorption_v1.py new file mode 100644 index 000000000..481e138af --- /dev/null +++ b/experimental/verify_mca_full_lift_residue_zero_core_absorption_v1.py @@ -0,0 +1,80 @@ +#!/usr/bin/env python3 +"""Verify the first residue-zero common-core absorption payment.""" + +from __future__ import annotations + +import copy + + +ROW = { + "R": 1048576, "d": 67448, "K": 6, "e": 98232, + "budget": 16777215, "core": 67452, "inside_core": 67447, + "sync_start": 30791, "low_end": 30790, + "agreement": 36664, "punctured_length": 950350, + "johnson_denominator": 1339497146, + "johnson_numerator": 34838880650, "list_cap": 26, + "low_slopes": 2554032, "line_cap": 981129, + "bound": 3535161, "slack": 13242054, "next_e": 98233, +} + + +class Reject(ValueError): + pass + + +def endpoint(row): + R, d, K, e = row["R"], row["d"], row["K"], row["e"] + N, m, c = R + K, d + K, K - 1 + inside = row["core"] - c + sync = e - inside + K + low_end = sync - 1 + agreement = m - low_end + n = N - e + denominator = agreement * agreement - n * c + numerator = n * (agreement - c) + list_cap = numerator // denominator + low = e * list_cap + line = N - m + 1 + bound = low + line + if row["core"] < m - 2 or denominator <= 0 or bound >= row["budget"]: + raise Reject("guards") + return { + "inside_core": inside, "sync_start": sync, "low_end": low_end, + "agreement": agreement, "punctured_length": n, + "johnson_denominator": denominator, + "johnson_numerator": numerator, "list_cap": list_cap, + "low_slopes": low, "line_cap": line, "bound": bound, + "slack": row["budget"] - bound, "next_e": e + 1, + } + + +def validate(row): + for key, value in endpoint(row).items(): + if row[key] != value: + raise Reject(key) + return 23 + + +def main() -> None: + checks = validate(copy.deepcopy(ROW)) + mutations = [] + for key, delta in (("inside_core", 1), ("list_cap", -1), + ("bound", -1), ("slack", 1)): + changed = copy.deepcopy(ROW) + changed[key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_RESIDUE_ZERO_CORE_ABSORPTION_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From f2936369d1f2beb3cacda0c21b15b762de9f8f8b Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 08:58:13 +0100 Subject: [PATCH 17/40] MCA: pay fixed-cutoff Mersenne interval --- agents.md | 2 +- experimental/agents-log.md | 18 +++ experimental/grande_finale.pdf | Bin 1060634 -> 1066103 bytes experimental/grande_finale.tex | 95 ++++++++++++ ...ull_lift_fixed_cutoff_boundary_stack_v1.md | 82 +++++++++++ ...ll_lift_residue_zero_core_absorption_v1.md | 7 +- ...full_lift_fixed_cutoff_boundary_stack_v1.c | 139 ++++++++++++++++++ ...ull_lift_fixed_cutoff_boundary_stack_v1.py | 118 +++++++++++++++ 8 files changed, 457 insertions(+), 4 deletions(-) create mode 100644 experimental/notes/thresholds/mca_full_lift_fixed_cutoff_boundary_stack_v1.md create mode 100644 experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.c create mode 100644 experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.py diff --git a/agents.md b/agents.md index 5b8128d02..a697bf91a 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. 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z{V_eYq34J6kDPAY81b;*)K~I9Yr*HgG@oY!e3lx0miB!;@uo-I_+{<$Yuo2{x6f~E zpIHr`-`##AKy)9eeSG%u*+*p`m3=)Fr>Edh*+*qx55QrukHtO$`v~kKu+QoGoUYI5 z`kbzhz&--|2<#)UkH9_x`v~kKuz!!hJ_7p)>?5$xqeDKkTfQEsc=*ZJPO&essrL&hwhTyv49J2U$b$kXf(lRxsz5cU0kxnG)Pn|a05pPw zpb0dCL*Ott0$M;TI11XpG0+Y=Kqoj3x(4W_^uQ-8EExk4jPrUw2WZS)_!m+!<79t<`xH#IW~B_%~qMheFkOf~=j diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 7b4aea03d..943df42c7 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -3231,6 +3231,86 @@ \section{Higher-codimension support-flag recursion} \path{experimental/audit_mca_full_lift_core_dichotomy_capped_charge_v1.py}. \end{remark} +\begin{corollary}[Exact-layer slot-core incidence] +\label{cor:full-lift-exact-layer-slot-core} +Suppose a line slot selected by the recursive bank belongs to exact layer +$h$ and contains $\lambda\ge2$ explanations. If $u$ is their common core +inside the gauged $e$-coordinate direction support, then +\[ + u\ge\left\lceil\frac{\lambda h-e}{\lambda-1}\right\rceil. \tag{EL1} +\] +Consequently, in the low-core branch of +Corollary~\ref{cor:full-lift-core-dichotomy-capped-charge}, (EL1) may be used +as the forced lower bound in the capped convex charge. +\end{corollary} + +\begin{proof} +Write the parameterized affine explanation line as +$c_\gamma=a+\gamma b$. At an inside coordinate outside the common core, +the equation +\[ + r_0+\gamma r_1=a+\gamma b +\] +holds for at most one line parameter $\gamma$. Every one of the $\lambda$ +members has at least $h$ inside agreements, whereas a core coordinate is +counted $\lambda$ times. Incidence counting in the fixed inside support +therefore gives +\[ + \lambda h\le e+(\lambda-1)u, +\] +which is equivalent to (EL1). This is already an inside-core bound and +does not spend the outside zero allowance. In the calibrated support +interval the parent ceiling has $H\ge m$, so the optional cross-layer top +slot is absent and every recursive-bank slot has one exact-layer owner. +\end{proof} + +\begin{remark}[Exact Mersenne layer-aware payment] +Retain absorption cutoff $65450$. In the complementary capped-core branch, +legal cutoffs $65516,\ldots,65521$ and (EL1) force three line cores of the +following sizes: +\[ +\begin{array}{c|c|c|c} +e&b&\lambda&u\\ \hline +130226&65516&14&60540\\ +130227&65516&5&49340\\ +130228&65517&13&60126\\ +130229&65517&4&43948\\ +130230&65518&11&59048\\ +130231&65518&4&43949\\ +130232&65519&9&57431\\ +130233,130234&65520&7&54736\\ +130235,130236&65521&4&43951 +\end{array} +\] +The displayed $\lambda$ is the forced minimum slot size and $h=b+1$ is the +minimum possible exact layer. In this range the right side of (EL1) is +nondecreasing in both variables, so these substitutions are conservative. +Distinct selected lines have pairwise inside-core intersection at most +$c=5$. Exact capped-charge replay therefore makes three lines violate +inside-core packing on every support $130226\le e\le130236$; the smallest +printed lower bound is +\[ + 3\cdot43948-\binom32\cdot5=131829>130229. +\] + +At adjacent $e=130237$, cutoff $65521$ forces only size-two affine +explanation slots. Equation (EL1) gives $u=807$, and the first-order +packing expression has maximum +\[ + \max_s\left\{807s-5\binom{s}{2}\right\}=65529=2` members of one +exact layer `h`, and let `u` be its inside common core. At an inside +coordinate outside that core, the line equation + +```text +r_0+gamma*r_1 = a+gamma*b +``` + +holds for at most one line parameter `gamma`. Every member has at least `h` +inside agreements, so incidence counting in the fixed `e`-coordinate support +gives + +```text +lambda*h <= e+(lambda-1)u, +u >= ceil((lambda*h-e)/(lambda-1)). (EL1) +``` + +This is directly an inside-core lower bound; it spends none of the outside +zero allowance. In this support interval the parent ceiling satisfies +`H>=m`, so every recursive-bank slot has one exact-layer owner and no +cross-layer top slot is present. + +## Eleven support payments + +Retain the core-dichotomy absorption cutoff `65450`. In its complementary +branch each selected line has total-core cap + +```text +G_e=e+9-65450. +``` + +Use the capped lower-aware convex charge, now with (EL1) as the selected +line's lower bound. The following legal cutoffs force the same threshold on +each of the first three selected lines: + +```text +e cutoff lambda EL1 lower u +130226 65516 14 60540 +130227 65516 5 49340 +130228 65517 13 60126 +130229 65517 4 43948 +130230 65518 11 59048 +130231 65518 4 43949 +130232 65519 9 57431 +130233,130234 65520 7 54736 +130235,130236 65521 4 43951 +``` + +Here the printed `lambda` is the forced minimum slot size and `h=cutoff+1` +is the minimum possible exact layer. In this range the right side of (EL1) +is nondecreasing in both variables, so using those minima is conservative. + +Distinct selected lines have pairwise inside-core intersection at most +`c=5`. Hence three selected lines violate support-size packing on every +printed row. The smallest printed lower bound is + +```text +3*43948-C(3,2)*5=131829>130229. +``` + +The high-core branch remains exhaustive and is at most `5161243`, so all +eleven supports are safe under the full-lift route. + +## Adjacent shift-pair wall + +At `e=130237`, cutoff `65521` is legal and has + +```text +prefix=15893203, groups=1933560, base=13961576. +``` + +It forces only size-two affine explanation slots, the shift-pair stratum of +this line bank. Formula (EL1) gives `u=807`, while + +```text +max_s {s*807-C(s,2)*5}=65529 int: + shortened, agreement = N - e, M - h + if agreement <= C: + raise Reject("prefix agreement") + johnson = agreement * agreement - shortened * C + if johnson > 0: + return shortened * (agreement - C) // johnson + gap = -johnson + balance = 2 * agreement * agreement - shortened * C + tangent = (shortened - agreement) ** 2 - (shortened - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("prefix cap") + return ((shortened - 1) * shortened * shortened * (agreement - C) + // (agreement * tangent)) + + +def prefix(e: int, cutoff: int) -> int: + values = [0] + [raw_cap(e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def bank(e: int, cutoff: int) -> tuple[int, int, int]: + weighted = prefix(e, cutoff) + groups = 0 + for h in range(cutoff + 1, M + 1): + overlap = 2 * h - e + denominator = overlap * overlap - e * C + if 2 * h <= e or overlap <= C or denominator <= 0: + raise Reject("bank guard") + groups += e * (overlap - C) // denominator + return weighted, groups, weighted + M - cutoff - groups + + +def capped_charge(e: int, lower: list[int], cap: int) -> tuple[int, int]: + if not lower: + return 0, 0 + count = len(lower) + budget = min(count * cap, e + count * (count + 1) * C // 2) + values = sorted(lower, reverse=True) + excess = budget - sum(values) + if excess < 0: + raise Reject("packing before charge") + for index, value in enumerate(values): + addition = min(excess, cap - value) + values[index] += addition + excess -= addition + if excess == 0: + break + if excess: + raise Reject("allocation") + value = sum((Fraction(N - core, M - core) for core in values), + Fraction()) + return value.numerator // value.denominator, budget + + +def paid_row(e: int, cutoff: int) -> tuple[int, ...]: + weighted, groups, base = bank(e, cutoff) + cap = e + 9 - ABS_CUTOFF + absorption_bound = prefix(e, ABS_CUTOFF) + LINE + lower: list[int] = [] + threshold = inside = charge = target = core_budget = 0 + for _ in range(3): + charge, core_budget = capped_charge(e, lower, cap) + target = BUDGET - charge + threshold = (target - base + groups) // groups + if threshold < 2: + raise Reject("paid threshold") + numerator = threshold * (cutoff + 1) - e + inside = max(0, (numerator + threshold - 2) // (threshold - 1)) + lower.append(inside) + packing = sum(lower) - len(lower) * (len(lower) - 1) * C // 2 + if packing <= e or absorption_bound >= BUDGET: + raise Reject("payment") + return (e, cutoff, weighted, groups, base, cap, absorption_bound, + threshold, inside, charge, target, core_budget, packing) + + +EXPECTED = ( + (130226, 65516, 12404722, 342025, 12064635, 64785, 5161243, 14, 60540, 737, 16776478, 129570, 181605), + (130227, 65516, 12404167, 1217008, 11189097, 64786, 5161221, 5, 49340, 737, 16776478, 129572, 148005), + (130228, 65517, 12958915, 342446, 12618406, 64787, 5161211, 13, 60126, 737, 16776478, 129574, 180363), + (130229, 65517, 12958300, 1305245, 11654992, 64788, 5161190, 4, 43948, 738, 16776477, 129576, 131829), + (130230, 65518, 13573258, 342865, 13232329, 64789, 5161180, 11, 59048, 738, 16776477, 129578, 177129), + (130231, 65518, 13572575, 1412234, 12162277, 64790, 5161158, 4, 43949, 738, 16776477, 129580, 131832), + (130232, 65519, 14258109, 343295, 13916749, 64791, 5161148, 9, 57431, 738, 16776477, 129582, 172278), + (130233, 65520, 15027192, 301252, 14727874, 64792, 5161126, 7, 54736, 739, 16776476, 129584, 164193), + (130234, 65520, 15026344, 343725, 14684553, 64793, 5161117, 7, 54736, 739, 16776476, 129586, 164193), + (130235, 65521, 15895144, 301391, 15595686, 64794, 5161095, 4, 43951, 739, 16776476, 129588, 131838), + (130236, 65521, 15894179, 344160, 15551952, 64795, 5161085, 4, 43951, 739, 16776476, 129590, 131838), +) + + +def uniform_charge(e: int, lines: int, lower: int, + cap: int) -> tuple[int, int, int, int, int]: + budget = min(lines * cap, e + lines * (lines + 1) * C // 2) + lower_sum = lines * lower + full, remainder = divmod(budget - lower_sum, cap - lower) + full = min(full, lines) + value = full * Fraction(N - cap, M - cap) + if full < lines: + if remainder: + value += Fraction(N - lower - remainder, M - lower - remainder) + residual = lines - full - 1 + else: + residual = lines - full + value += residual * Fraction(N - lower, M - lower) + return (value.numerator // value.denominator, budget, lower_sum, + full, remainder) + + +def adjacent_row() -> tuple[int, ...]: + e, cutoff = 130237, 65521 + weighted, groups, base = bank(e, cutoff) + cap = e + 9 - ABS_CUTOFF + absorption_bound = prefix(e, ABS_CUTOFF) + LINE + first = (BUDGET - base + groups) // groups + inside = (first * (cutoff + 1) - e + first - 2) // (first - 1) + maximum = max(s * inside - s * (s - 1) * C // 2 + for s in range(1, 1000)) + for lines in range(1, 10000): + charge, budget, lower_sum, full, remainder = uniform_charge( + e, lines, inside, cap) + target = BUDGET - charge + threshold = (target - base + groups) // groups + if threshold < 2: + return (e, cutoff, weighted, groups, base, cap, + absorption_bound, first, inside, maximum, lines, + budget, lower_sum, full, remainder, charge, target, + threshold) + raise Reject("adjacent limit") + + +ADJACENT = ( + 130237, 65521, 15893203, 1933560, 13961576, 64796, 5161064, + 2, 807, 65529, 7583, 143903917, 6119481, 2153, 16119, + 882245, 15894970, 1, +) + + +def main() -> None: + checks = 0 + for expected in EXPECTED: + actual = paid_row(expected[0], expected[1]) + if actual != expected: + raise Reject(f"row {expected[0]}") + checks += len(expected) + if adjacent_row() != ADJACENT: + raise Reject("adjacent") + checks += len(ADJACENT) + print("MCA_FULL_LIFT_EXACT_LAYER_SLOT_CORE_PACKING_V1_PASS " + f"checks={checks} paid=130226..130236 wall=130237") + + +if __name__ == "__main__": + main() From 862fcaf1e0e3d3eeefb0b362830e2a32e6c23d89 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 12:09:36 +0100 Subject: [PATCH 25/40] MCA: route first line wall to interpolation factor --- agents.md | 2 +- experimental/agents-log.md | 17 +++ ...t_interpolation_common_factor_router_v1.py | 41 +++++++ experimental/grande_finale.pdf | Bin 1099033 -> 1102620 bytes experimental/grande_finale.tex | 59 ++++++++++ ...t_interpolation_common_factor_router_v1.md | 83 ++++++++++++++ ...t_interpolation_common_factor_router_v1.py | 103 ++++++++++++++++++ 7 files changed, 304 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_interpolation_common_factor_router_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_interpolation_common_factor_router_v1.md create mode 100644 experimental/verify_mca_full_lift_interpolation_common_factor_router_v1.py diff --git a/agents.md b/agents.md index 8d5758c5c..507c221f3 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e=ceil((lambda*h-e)/(lambda-1))`; with the capped-core branch, three lines pay every support `e=130226..130236`. At adjacent `e=130237`, the bank forces only size-two shift-pair slots with inside core `807`; first-order packing reaches only `65529` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e=ceil((lambda*h-e)/(lambda-1))`; with the capped-core branch, three lines pay every support `e=130226..130236`. At adjacent `e=130237`, the bank forces only size-two shift-pair slots with inside core `807`; first-order packing reaches only `6552952^2` Bezout contradiction pay the coprime branch at + Mersenne support `e=130237`. +- **How it is useful:** Every unsafe survivor now forces a common factor of + positive `(Y,Z)` degree over the algebraic closure of `F(X)`. +- **What to do next:** Classify and charge that factor; do not assume without + proof that it is a split pencil. + ### 2026-08-13 - Mersenne exact-layer slot-core packing - **Agent/model:** Codex. diff --git a/experimental/audit_mca_full_lift_interpolation_common_factor_router_v1.py b/experimental/audit_mca_full_lift_interpolation_common_factor_router_v1.py new file mode 100644 index 000000000..42882dc2a --- /dev/null +++ b/experimental/audit_mca_full_lift_interpolation_common_factor_router_v1.py @@ -0,0 +1,41 @@ +#!/usr/bin/env python3 +"""Independent exact audit of the M31 interpolation router.""" + +from fractions import Fraction + + +def main() -> None: + degree, weight, e = 264, 5, 130237 + monomials = 0 + checks = 0 + for j in range(53): + for k in range(53 - j): + monomials += degree - weight * (j + k) + 1 + checks += 1 + assert monomials == 131175 + assert monomials - e == 938 + assert 52**2 == 2704 + assert 807 > degree + + n, m, c, total = 1048582, 67454, 5, 16777215 + lines, lower, cap = 2704, 807, 64796 + budget = min(lines * cap, e + lines * (lines + 1) * c // 2) + lower_sum = lines * lower + full, remainder = divmod(budget - lower_sum, cap - lower) + value = full * Fraction(n - cap, m - cap) + value += Fraction(n - lower - remainder, m - lower - remainder) + value += (lines - full - 1) * Fraction(n - lower, m - lower) + charge = value.numerator // value.denominator + assert (budget, lower_sum, full, remainder) == ( + 18416037, 2182128, 253, 44692) + assert charge == 132203 + target = total - charge + assert target == 16645012 + assert (target - 13961576 + 1933560) // 1933560 == 2 + assert lines + 1 == 2705 > 52**2 + print("MCA_FULL_LIFT_INTERPOLATION_COMMON_FACTOR_ROUTER_V1_AUDIT_PASS " + f"checks={checks + 19} exact_replay=1") + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index d70e08e9587119db73ab80527c6543db5ec7376d..c7b220b8b360050f39f4d4e6a60c1a94c92b78e9 100644 GIT binary patch delta 447137 zcmV)VK(D`#QVi<5{>Z;J)rRpmTDGpx1}WDN`sWrfan$BXuvmR09>u(DTNuL z`sgZg5S9LF+n71$`b%m)65kmSUcVjA8UbeABui zF0davs2Lqj8nnM75-D-uvMPjnp&+VS9k>k$O8qboCeE^eWuFVip+tR4!6a1Vtu)Al z^1vT0EVo<8XWc33B7ZYM$bWb%>uTOcQ5*+R^#7G5b!B;{>^ zfAhA2X)N?s5cmv@Bk$usSU#AAgc-_$n5_+IB$cgDm7@$^6s8sv$9d2ddcp@?AhnYD zTox%xSP&;GGk-$7EZ&%vDBxI_?qr3pziFC5HNvN>R;LLE%eHZPU5Ysz3G{!wmUGES z3?|)Z=tHMu-@2@|3)Px;?LwKbY()?bnSuD<+SlLddU<=PE8ksTy{tThCJ~^J{eM); z9M#^+$}p{^(n1`|S`R%r<^OUUMi`s50O{712#;LxgMTnE!gs#9etU_Tg)FqhxRQX^WfK7c4y7r)!g^gu6zC8JVKs$11zgIxLp?)#EB^|#d zO?DAxZsnBNF=sOY`|B$^rj(VFKVJPcv1evwybPh)59V23@}`j*iG057X(lL5cQDe; 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z7p|mu@lR8HvY6sipGff!CdFHIDPBL4;%loZzWI2HfBk5RZ+|*P{920deouFwdJ#x3 zo@jr{3L_rrofG~+d!+e&X5IXBiy!cWrRZ~K?eqV$&z-f;-L}ttx6eJc&vd%aeYal? znAP`@+Q(-fpM6yJQQ1djAC-Mn_EFhKWgm-uEcOxDM_?a;eRkJpcYSu(XLo&e*GFI< zfqew_5!gpyAAx-Y_7T`eU>|{f1on9c$Y+1b-sdTa&+8#Rzc~Ilm;jSt3QU6;z}q%H z&$N8*P5pUb0sg`8PXJz)@p(JOXU^5s~Xv#<2`>7kaFGXD?ozEbH5Wp0;HL=YPcH8nChGYTaoMNdWwGNy9b diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 34bb2439a..2fe29a55d 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -2634,6 +2634,101 @@ \section{Higher-codimension support-flag recursion} \path{experimental/audit_mca_full_lift_residue_zero_core_absorption_v1.py}. \end{remark} +\begin{corollary}[Fixed-cutoff boundary-stack continuation] +\label{cor:full-lift-fixed-cutoff-boundary-stack} +Retain the pair-noncontained full-lift hypotheses and notation above. Fix +a target slope budget $B$ and $h_0e,\qquad A_h^2>ec,\qquad N-e>m-h>c +\] +throughout this range. Define +\[ + F_e=P_{h_0}(e)+\sum_{h=h_0+1}^{H}\{1+J_h(Q_h-1)\}. + \tag{BS1} +\] +Then +\[ + |Z|\le F_e+|\mathcal T|. \tag{BS2} +\] +In particular, $F_e+N-m+1\le B$ implies safety. Otherwise suppose +$F_ee`, `A_h^2>e*c`, and `N-e>m-h>c`, the normalized-direction +Johnson count permits at most `J_h` classes. Every class and a repeated +anchor lie on one nonzero affine codeword line, and outside-core packing +caps that line by `Q_h`. Thus + +```text +|Z|<=F_e+|T|, +F_e=P_h0(e)+sum_(h=h0+1)^H D_h, +``` + +where `T` is the synchronized top line. + +If `F_e+(N-m+1)<=B`, this pays directly. Otherwise unsafety forces +`L_e=B-F_e+1` top members and common core + +```text +g_e=ceil((L_e*m-N)/(L_e-1)). +``` + +At least `u_e=g_e-c` core coordinates lie inside the gauged direction +support. Two top anchors absorb every assigned explanation of deficit at +least `a_e=e-u_e+K` into the same line. One punctured ordinary-Johnson cap +`M_e` at outside agreement `m-a_e+1` pays the lower explanations whenever + +```text +e*M_e+(N-m+1)<=B. +``` + +## Official interval + +At Mersenne-31, the fixed cutoff `h0=65200` pays + +```text +98232<=e<=101155. +``` + +The direct branch ends at `e=101149`; the final six supports use absorption. +At `e=101155`, + +```text +F_e = 16667033, +L_e = 110183, +g_e = 67446, +M_e = 28, +final bound = 3813469, +slack = 12963746. +``` + +At adjacent `e=101156`, the fixed-cutoff charge is `16951223`, above budget +by `174008`. This is a method wall, not an unsafe certificate. The Mersenne +full-lift residual interval becomes `101156<=e<=1044241`. + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.py +cc -O2 -std=c11 -Wall -Wextra -Werror \ + experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.c \ + -o /tmp/verify_m31_boundary_stack +/tmp/verify_m31_boundary_stack +``` + +The C replay checks all 2,925 rows from `98232` through the adjacent wall. diff --git a/experimental/notes/thresholds/mca_full_lift_residue_zero_core_absorption_v1.md b/experimental/notes/thresholds/mca_full_lift_residue_zero_core_absorption_v1.md index 46df143b6..b1cf46bf4 100644 --- a/experimental/notes/thresholds/mca_full_lift_residue_zero_core_absorption_v1.md +++ b/experimental/notes/thresholds/mca_full_lift_residue_zero_core_absorption_v1.md @@ -45,9 +45,10 @@ This contradicts unsafety, with exact margin `13242054`. ## Consequence and nonclaims -Mersenne-31 full-lift support `e=98232` is safe. The residual interval is -now `98233<=e<=1044241`. This does not pay `e=98233`, close the deployed -row, or provide an adjacent unsafe certificate. +Mersenne-31 full-lift support `e=98232` is safe. The successor fixed-cutoff +boundary-stack theorem extends this payment through `e=101155`. This node +alone does not pay `e=98233`, close the deployed row, or provide an adjacent +unsafe certificate. ## Replay diff --git a/experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.c b/experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.c new file mode 100644 index 000000000..82adb5714 --- /dev/null +++ b/experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.c @@ -0,0 +1,139 @@ +#include +#include +#include + +typedef __int128 i128; + +enum { + R = 1048576, + D = 67448, + K = 6, + N = R + K, + M = D + K, + C = K - 1, + BUDGET = 16777215, + CUTOFF = 65200, +}; + +static int64_t caps[CUTOFF + 1]; + +static void reject(const char *what, int e, int h) { + fprintf(stderr, "REJECT %s e=%d h=%d\n", what, e, h); + exit(1); +} + +static int64_t raw_cap(int e, int h) { + int64_t n = N - e; + int64_t agreement = M - h; + int64_t johnson = agreement * agreement - n * C; + if (johnson > 0) { + return (int64_t)(((i128)n * (agreement - C)) / johnson); + } + int64_t gap = -johnson; + int64_t balance = 2 * agreement * agreement - n * C; + int64_t tangent = (n - agreement) * (n - agreement) - (n - 1) * gap; + if (balance < 0 || tangent <= 0) reject("prefix cap", e, h); + i128 numerator = (i128)(n - 1) * n * n * (agreement - C); + i128 denominator = (i128)agreement * tangent; + return (int64_t)(numerator / denominator); +} + +static int64_t prefix(int e) { + caps[0] = 0; + for (int h = 1; h <= CUTOFF; ++h) caps[h] = raw_cap(e, h); + for (int h = CUTOFF - 1; h >= 1; --h) { + if (caps[h + 1] < caps[h]) caps[h] = caps[h + 1]; + } + int64_t total = 0; + for (int h = 1; h <= CUTOFF; ++h) { + total += (caps[h] - caps[h - 1]) * (e / h); + } + return total; +} + +static int64_t boundary_cap(int e, int h) { + int64_t n = N - e; + int64_t inside = 2LL * h - e; + int64_t denominator = inside * inside - (int64_t)e * C; + int64_t outside = M - h; + if (2 * h <= e || inside <= 0 || denominator <= 0 || + !(n > outside && outside > C)) { + reject("boundary cap", e, h); + } + int64_t classes = (int64_t)(((i128)e * (inside - C)) / denominator); + int64_t line = (n - C) / (outside - C); + return 1 + classes * (line - 1); +} + +static void checkpoint(int e, int64_t p, int64_t stack, int64_t forcing, + int64_t threshold, int64_t core, int64_t low, + int64_t bound) { + if (e == 98232 && !(p == 1381829 && stack == 391210 && + forcing == 1773039 && threshold == 15004177)) { + reject("first checkpoint", e, 0); + } + if (e == 101149 && !(p == 1422377 && stack == 14327810 && + forcing == 15750187 && threshold == 1027029)) { + reject("last direct checkpoint", e, 0); + } + if (e == 101150 && !(p == 1422391 && stack == 14530797 && + forcing == 15953188 && threshold == 824028 && + core == 67453 && low == 28 && bound == 3813329)) { + reject("first absorption checkpoint", e, 0); + } + if (e == 101155 && !(p == 1422461 && stack == 15244572 && + forcing == 16667033 && threshold == 110183 && + core == 67446 && low == 28 && bound == 3813469)) { + reject("endpoint checkpoint", e, 0); + } + if (e == 101156 && !(p == 1422475 && stack == 15528748 && + forcing == 16951223 && threshold == -174007)) { + reject("adjacent checkpoint", e, 0); + } +} + +int main(void) { + int direct = 0, absorption = 0, wall = 0; + for (int e = 98232; e <= 101156; ++e) { + int s = (e - K) / 3; + int H = e - s - 1; + int64_t p = prefix(e); + int64_t stack = 0; + for (int h = CUTOFF + 1; h <= H; ++h) { + stack += boundary_cap(e, h); + } + int64_t forcing = p + stack; + int64_t threshold = BUDGET - forcing + 1; + int64_t core = -1, low = -1, bound = -1; + + if (forcing + (N - M + 1) <= BUDGET) { + if (e > 101149) reject("late direct branch", e, H); + ++direct; + } else if (threshold >= 2) { + core = (threshold * (int64_t)M - N + threshold - 2) + / (threshold - 1); + int64_t inside = core - C; + int64_t sync = e - inside + K; + int64_t agreement = M - sync + 1; + int64_t n = N - e; + int64_t denominator = agreement * agreement - n * C; + if (denominator <= 0) reject("low Johnson", e, H); + low = (n * (agreement - C)) / denominator; + bound = e * low + (N - M + 1); + if (e > 101155 || bound >= BUDGET) { + reject("absorption payment", e, H); + } + ++absorption; + } else { + if (e != 101156) reject("early wall", e, H); + ++wall; + } + checkpoint(e, p, stack, forcing, threshold, core, low, bound); + } + if (direct != 2918 || absorption != 6 || wall != 1) { + reject("branch census", 0, 0); + } + puts("MCA_FULL_LIFT_FIXED_CUTOFF_BOUNDARY_STACK_V1_SCAN_PASS " + "paid=2924 direct=2918 absorption=6 adjacent_wall=1"); + return 0; +} diff --git a/experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.py b/experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.py new file mode 100644 index 000000000..2b2270375 --- /dev/null +++ b/experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.py @@ -0,0 +1,118 @@ +#!/usr/bin/env python3 +"""Verify endpoints of the M31 fixed-cutoff boundary-stack payment.""" + +from __future__ import annotations + +import copy + + +R, D, K, BUDGET, CUTOFF = 1048576, 67448, 6, 16777215, 65200 +N, M, C = R + K, D + K, K - 1 +EXPECTED = { + 98232: (1381829, 391210, 1773039, 15004177, "direct"), + 101149: (1422377, 14327810, 15750187, 1027029, "direct"), + 101150: (1422391, 14530797, 15953188, 824028, + "absorption", 67453, 28, 3813329), + 101155: (1422461, 15244572, 16667033, 110183, + "absorption", 67446, 28, 3813469), + 101156: (1422475, 15528748, 16951223, -174007, + "fixed_cutoff_wall"), +} + + +class Reject(ValueError): + pass + + +def cap(e: int, h: int) -> int: + n, agreement = N - e, M - h + johnson = agreement * agreement - n * C + if johnson > 0: + return n * (agreement - C) // johnson + gap = -johnson + balance = 2 * agreement * agreement - n * C + tangent = (n - agreement) ** 2 - (n - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("undefined prefix cap") + return ((n - 1) * n * n * (agreement - C) + // (agreement * tangent)) + + +def prefix(e: int) -> int: + values = [0] + [cap(e, h) for h in range(1, CUTOFF + 1)] + for h in range(CUTOFF - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, CUTOFF + 1)) + + +def record(e: int): + s, _ = divmod(e - K, 3) + H = e - s - 1 + p = prefix(e) + stack = 0 + for h in range(CUTOFF + 1, H + 1): + A, n, outside = 2 * h - e, N - e, M - h + denominator = A * A - e * C + if (2 * h <= e or denominator <= 0 or not (n > outside > C)): + raise Reject("boundary guard") + classes = e * (A - C) // denominator + line = (n - C) // (outside - C) + stack += 1 + classes * (line - 1) + forcing = p + stack + threshold = BUDGET - forcing + 1 + if forcing + (N - M + 1) <= BUDGET: + return (p, stack, forcing, threshold, "direct") + if threshold < 2: + return (p, stack, forcing, threshold, "fixed_cutoff_wall") + core = (threshold * M - N + threshold - 2) // (threshold - 1) + inside = core - C + sync = e - inside + K + agreement = M - sync + 1 + n = N - e + denominator = agreement * agreement - n * C + if denominator <= 0: + raise Reject("low Johnson") + low = n * (agreement - C) // denominator + bound = e * low + (N - M + 1) + return (p, stack, forcing, threshold, "absorption", core, low, bound) + + +def validate(expected): + for e, row in expected.items(): + if record(e) != row: + raise Reject(f"endpoint {e}") + if 101155 - 98232 + 1 != 2924: + raise Reject("census") + if EXPECTED[101156][2] - BUDGET != 174008: + raise Reject("adjacent excess") + if EXPECTED[101155][-1] >= BUDGET: + raise Reject("endpoint budget") + return 41 + + +def main() -> None: + checks = validate(copy.deepcopy(EXPECTED)) + mutations = [] + for e, index, delta in ((98232, 0, 1), (101150, 6, -1), + (101155, 7, -1), (101156, 2, -1)): + changed = copy.deepcopy(EXPECTED) + row = list(changed[e]) + row[index] += delta + changed[e] = tuple(row) + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_FIXED_CUTOFF_BOUNDARY_STACK_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 6cc937eb55b9c0e055404d2ae3e09032309b59a0 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 09:17:38 +0100 Subject: [PATCH 18/40] MCA: repair fixed-cutoff residue-two wall --- agents.md | 2 +- experimental/agents-log.md | 17 +++ ...l_lift_fixed_cutoff_q2_anchor_repair_v1.py | 29 ++++ experimental/grande_finale.tex | 82 ++++++++++ ...l_lift_fixed_cutoff_q2_anchor_repair_v1.md | 67 ++++++++ ...l_lift_fixed_cutoff_q2_anchor_repair_v1.py | 144 ++++++++++++++++++ 6 files changed, 340 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.md create mode 100644 experimental/verify_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py diff --git a/agents.md b/agents.md index a697bf91a..d30be4186 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e None: + budget, forcing, d1, d2 = 16777215, 16895280, 284224, 258385 + lower = forcing - d1 - d2 + threshold = budget - lower + 1 + assert (lower, threshold) == (16352671, 424545) + cases = ( + 3813497, + forcing - d1 + 94742 + 1, + forcing - d1 + 2, + forcing - d1 + 94742, + forcing - d1 + 3, + ) + assert cases == ( + 3813497, 16705799, 16611058, 16705798, 16611059) + assert max(cases) == 16705799 + assert budget - max(cases) == 71416 + assert (101157 - 6) % 3 == 0 + print( + "MCA_FULL_LIFT_FIXED_CUTOFF_Q2_ANCHOR_REPAIR_V1_AUDIT_PASS " + "bound=16705799 slack=71416 adjacent_q=0" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 2fe29a55d..191d9bc84 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -2729,6 +2729,88 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_full_lift_fixed_cutoff_boundary_stack_v1.c}. \end{remark} +\begin{corollary}[Residue-two repair at the fixed-cutoff wall] +\label{cor:full-lift-fixed-cutoff-q2-anchor-repair} +Retain the pair-noncontained Mersenne-31 full-lift hypotheses. At +\[ + e=101156,\qquad (s,q,H)=(33716,2,67439), +\] +use the fixed-cutoff charge \((\mathrm{BS1})\) with \(h_0=65258\). +Let \(\mathcal T\) be the synchronized top union, and let +\(\mathcal D_1,\mathcal D_2\) be the exact layers of deficits +\(H,H-1\). The resulting charges are +\[ + F=16895280,\qquad + D_1=284224,\qquad D_2=258385. +\] +Then the complete slope family satisfies +\[ + |Z|\le16705799<16777215. +\] +Consequently this support is safe, and the residual Mersenne full-lift +interval is \(101157\le e\le1044241\). +\end{corollary} + +\begin{proof} +Every member of \(\mathcal T\cup\mathcal D_1\cup\mathcal D_2\) +owns at most one slope. If \(|\mathcal T|\ge2\), two top anchors and +any boundary member have at least +\[ + e-s-s-(s+2)=K+q-2=K +\] +common inside agreement coordinates. Thus the top union and both +boundary layers lie on one affine codeword line. Removing both coarse +boundary charges leaves +\[ + G=F-D_1-D_2=16352671. +\] +Unsafety would force at least \(16777215-G+1=424545\) members on that +line. Total-core packing then forces common core \(67452=m-2\), with +at least \(67447\) core coordinates inside the gauged direction +support. The two-anchor absorption argument therefore synchronizes every +assigned explanation of deficit at least \(33715\) onto the line. The +remaining punctured ordinary list has agreement at least \(33740\) and +Johnson cap \(28\), whence +\[ + |Z|\le101156\cdot28+981129=3813497. +\] + +Suppose next that \(|\mathcal T|=1\). If +\(|\mathcal D_1|\ge2\), the top anchor and any two first-boundary +members share at least +\[ + e-s-2(s+1)=K +\] +inside coordinates, so \(\mathcal D_1\) is one affine codeword line. +Its outside agreement is \(m-H=15\), and outside-core packing gives +\(|\mathcal D_1|\le94742\). The two subcases give +\[ + F-D_1+94742+1=16705799,\qquad + F-D_1+2=16611058. +\] + +Finally suppose \(\mathcal T\) is empty. If two first-boundary +missed sets intersect, those two members and any third share at least +\[ + e-\{3(s+1)-1\}=K +\] +inside coordinates. The layer is one affine line and contributes at +most \(94742\), giving \(F-D_1+94742=16705798\). Otherwise its +missed sets are pairwise disjoint, so +\(|\mathcal D_1|\le\lfloor e/(s+1)\rfloor=3\), giving +\(F-D_1+3=16611059\). These five exhaustive cases have maximum +\(16705799\), as claimed. +\end{proof} + +\begin{remark}[Scope and replay] +At adjacent \(e=101157\), the residue resets to zero, so the preceding +repair is inapplicable; no unsafe certificate is claimed. The source-bound +replay and independent constant audit are +\path{experimental/verify_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py} +and +\path{experimental/audit_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.md b/experimental/notes/thresholds/mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.md new file mode 100644 index 000000000..ee36523bd --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.md @@ -0,0 +1,67 @@ +# MCA full-lift fixed-cutoff q2 anchor repair v1 + +## Status + +PROVED / EXACT FINITE CALIBRATION / SUPPORT PAYMENT. + +## Endpoint + +At Mersenne-31 support `e=101156`, + +```text +(s,q,H)=(33716,2,67439), +cutoff=65258, +F=16895280, +D1=284224, +D2=258385. +``` + +Here `F` is the fixed-cutoff prefix plus every coarse boundary-class charge, +and `D1,D2` are the charges of exact deficits `H,H-1`. The residue identity + +```text +e-2s-(s+2)=K +``` + +allows two top anchors to synchronize both boundary layers. + +## Five cases + +If the top union has at least two members, remove `D1,D2` from `F`. +Unsafety would force `424545` members on the resulting line, hence common +core `67452=m-2`. Core absorption then gives + +```text +101156*28+981129=3813497. +``` + +With exactly one top member, two first-boundary members synchronize that +layer; its outside agreement `15` gives line cap `94742`. With no top +member, an intersecting pair of first-boundary missed sets synchronizes the +layer, while the alternative pairwise-disjoint family has size at most +`floor(101156/33717)=3`. + +The five exhaustive charges are + +```text +3813497, 16705799, 16611058, 16705798, 16611059. +``` + +Their maximum is below budget by `71416`. Thus `e=101156` is safe. + +## Scope + +At adjacent `e=101157`, the residue resets to zero. This theorem neither +applies there nor supplies an unsafe certificate. The residual interval is +`101157<=e<=1044241`. + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py +python3 experimental/audit_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py +``` + +The primary verifier reconstructs all `65,258` prefix caps, all `2,181` +boundary charges, the unsafe-core threshold, and all five cases. The audit +recomputes the endpoint constants without importing the primary verifier. diff --git a/experimental/verify_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py b/experimental/verify_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py new file mode 100644 index 000000000..62d7c7bcb --- /dev/null +++ b/experimental/verify_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py @@ -0,0 +1,144 @@ +#!/usr/bin/env python3 +"""Verify the M31 fixed-cutoff residue-two anchor repair.""" + +from __future__ import annotations + +import copy + +ROW = { + "R": 1048576, "d": 67448, "K": 6, "budget": 16777215, + "e": 101156, "cutoff": 65258, "s": 33716, "q": 2, + "H": 67439, "prefix": 1440609, "boundary": 15454671, + "forcing": 16895280, "D1": 284224, "D2": 258385, + "lower": 16352671, "line": 981129, "threshold": 424545, + "core": 67452, "inside": 67447, "sync": 33715, + "agreement": 33740, "low": 28, "absorption": 3813497, + "outside": 94742, "disjoint": 3, + "cases": (3813497, 16705799, 16611058, 16705798, 16611059), + "bound": 16705799, "slack": 71416, + "next_e": 101157, "next_q": 0, +} + +class Reject(ValueError): + pass + + +def cap(R: int, d: int, K: int, e: int, h: int) -> int: + n, m, c = R + K - e, d + K, K - 1 + agreement = m - h + johnson = agreement * agreement - n * c + if johnson > 0: + return n * (agreement - c) // johnson + gap = -johnson + balance = 2 * agreement * agreement - n * c + tangent = (n - agreement) ** 2 - (n - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("undefined prefix cap") + return ((n - 1) * n * n * (agreement - c) + // (agreement * tangent)) + + +def prefix(R: int, d: int, K: int, e: int, cutoff: int) -> int: + values = [0] + [cap(R, d, K, e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def layer_charge(R: int, d: int, K: int, e: int, h: int) -> int: + N, m, c = R + K, d + K, K - 1 + A = 2 * h - e + denominator = A * A - e * c + outside = m - h + n = N - e + if 2 * h <= e or denominator <= 0 or not (n > outside > c): + raise Reject("boundary guard") + classes = e * (A - c) // denominator + line = (n - c) // (outside - c) + return 1 + classes * (line - 1) + + +def endpoint(row): + R, d, K, e = row["R"], row["d"], row["K"], row["e"] + N, m, c = R + K, d + K, K - 1 + s, q = divmod(e - K, 3) + H = e - s - 1 + p = prefix(R, d, K, e, row["cutoff"]) + layers = [layer_charge(R, d, K, e, h) + for h in range(row["cutoff"] + 1, H + 1)] + boundary = sum(layers) + forcing = p + boundary + d1, d2 = layers[-1], layers[-2] + lower = forcing - d1 - d2 + line = N - m + 1 + threshold = row["budget"] - lower + 1 + core = (threshold * m - N + threshold - 2) // (threshold - 1) + inside = core - c + sync = e - inside + K + agreement = m - sync + 1 + n = N - e + denominator = agreement * agreement - n * c + if denominator <= 0: + raise Reject("low Johnson") + low = n * (agreement - c) // denominator + absorption = e * low + line + outside = (n - c) // (m - H - c) + disjoint = e // (s + 1) + cases = ( + absorption, + forcing - d1 + outside + 1, + forcing - d1 + 2, + forcing - d1 + outside, + forcing - d1 + disjoint, + ) + return { + "s": s, "q": q, "H": H, "prefix": p, + "boundary": boundary, "forcing": forcing, + "D1": d1, "D2": d2, "lower": lower, "line": line, + "threshold": threshold, "core": core, "inside": inside, + "sync": sync, "agreement": agreement, "low": low, + "absorption": absorption, "outside": outside, + "disjoint": disjoint, "cases": cases, "bound": max(cases), + }, len(layers) + + +def validate(row): + got, layers = endpoint(row) + for key, value in got.items(): + if row[key] != value: + raise Reject(key) + if (row["q"] != 2 or + row["e"] - 2 * row["s"] - (row["s"] + 2) != row["K"] or + row["budget"] - row["bound"] != row["slack"] or + row["slack"] <= 0 or + (row["next_e"] - row["K"]) % 3 != row["next_q"] or + row["next_q"] != 0): + raise Reject("guards") + return 37 + layers + + +def main() -> None: + checks = validate(copy.deepcopy(ROW)) + mutations = [] + for key, delta in (("prefix", 1), ("D1", 1), + ("absorption", -1), ("bound", -1), + ("slack", 1)): + changed = copy.deepcopy(ROW) + changed[key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_FIXED_CUTOFF_Q2_ANCHOR_REPAIR_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 9af5123a7f9228cbebe60c78a60be02c99c8ebdd Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 09:53:19 +0100 Subject: [PATCH 19/40] MCA: add boundary direction-class line bank --- agents.md | 2 +- experimental/agents-log.md | 18 ++ ...l_lift_boundary_line_bank_absorption_v1.py | 38 ++++ experimental/grande_finale.tex | 103 +++++++++++ ...l_lift_boundary_line_bank_absorption_v1.md | 79 ++++++++ ...ll_lift_boundary_line_bank_absorption_v1.c | 168 ++++++++++++++++++ ...l_lift_boundary_line_bank_absorption_v1.py | 149 ++++++++++++++++ 7 files changed, 556 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_boundary_line_bank_absorption_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_boundary_line_bank_absorption_v1.md create mode 100644 experimental/verify_mca_full_lift_boundary_line_bank_absorption_v1.c create mode 100644 experimental/verify_mca_full_lift_boundary_line_bank_absorption_v1.py diff --git a/agents.md b/agents.md index d30be4186..35dbee06e 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e tuple[int, ...]: + base = prefix + layers - groups + required = budget - base + 1 + threshold = (required + groups - 1) // groups + N, m, c, K = 1048582, 67454, 5, 6 + core = (threshold * m - N + threshold - 2) // (threshold - 1) + inside = core - c + sync = e - inside + K + agreement = m - sync + 1 + n = N - e + low = n * (agreement - c) // (agreement * agreement - n * c) + bound = e * low + (N - m + 1) + return base, threshold, core, inside, sync, agreement, low, bound + + +def main() -> None: + last = replay(124805, 1636955, 2182, 34560) + adjacent = replay(124806, 1636968, 2182, 34564) + assert last == ( + 1604577, 440, 65220, 65215, 59596, 7859, 126, 16706559) + assert adjacent == ( + 1604586, 439, 65214, 65209, 59603, 7852, 127, 16831491) + assert 16777215 - last[-1] == 70656 + assert adjacent[-1] - 16777215 == 54276 + assert 124805 - 101157 + 1 == 23649 + print( + "MCA_FULL_LIFT_BOUNDARY_LINE_BANK_ABSORPTION_V1_AUDIT_PASS " + "last=124805 slack=70656 adjacent_excess=54276" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 191d9bc84..fdd0974f5 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -2811,6 +2811,109 @@ \section{Higher-codimension support-flag recursion} \path{experimental/audit_mca_full_lift_fixed_cutoff_q2_anchor_repair_v1.py}. \end{remark} +\begin{corollary}[Boundary direction-class affine-line bank] +\label{cor:full-lift-boundary-line-bank} +Retain the pair-noncontained full-lift hypotheses and put +\[ + s=\left\lfloor\frac{e-K}{3}\right\rfloor,\qquad + H=e-s-1,\qquad U=\min(H,m),\qquad c=K-1. +\] +Fix \(h_0e,\qquad A_h>c,\qquad A_h^2>ec +\] +throughout this range. Define +\[ + G_e=\mathbf 1_{\{He\). Relative to the +anchor, each nonzero normalized codeword direction agrees with the gauged +direction on at least \(A_h\) inside coordinates. Distinct directions +have agreement blocks meeting in at most \(c\), so the constant-block +Johnson count permits at most \(J_h\) direction classes. + +Each class together with the anchor is one affine explanation line. Pad +to exactly \(J_h\) classes with anchor-only slots. The exact layer then +satisfies +\[ + |\mathcal D_h| + =1-J_h+\sum_{j=1}^{J_h}|\mathcal L_{h,j}|. +\] +If \(H +#include +#include + +/* Full constant-memory replay of the official paid interval and wall. */ + +typedef __int128 i128; + +enum { + R = 1048576, D = 67448, K = 6, N = R + K, M = D + K, + C = K - 1, BUDGET = 16777215, LINE = N - M + 1, + CUTOFF = 65272, +}; + +static int64_t caps[CUTOFF + 1]; + +static void reject(const char *what, int e, int h) { + fprintf(stderr, "REJECT %s e=%d h=%d\n", what, e, h); + exit(1); +} + +static int64_t raw_cap(int e, int h) { + int64_t n = N - e, agreement = M - h; + int64_t johnson = agreement * agreement - n * C; + if (johnson > 0) return n * (agreement - C) / johnson; + int64_t gap = -johnson; + int64_t balance = 2 * agreement * agreement - n * C; + int64_t tangent = (n - agreement) * (n - agreement) + - (n - 1) * gap; + if (balance < 0 || tangent <= 0) reject("prefix", e, h); + return (int64_t)(((i128)(n - 1) * n * n * (agreement - C)) + / ((i128)agreement * tangent)); +} + +static int64_t prefix(int e) { + caps[0] = 0; + for (int h = 1; h <= CUTOFF; ++h) caps[h] = raw_cap(e, h); + for (int h = CUTOFF - 1; h >= 1; --h) { + if (caps[h + 1] < caps[h]) caps[h] = caps[h + 1]; + } + int64_t total = 0; + for (int h = 1; h <= CUTOFF; ++h) { + total += (caps[h] - caps[h - 1]) * (e / h); + } + return total; +} + +struct row { + int e, q, H, layers, groups, branch, failure; + int64_t prefix, base, direct, threshold, core, inside; + int64_t sync, agreement, low, bound, slack; +}; + +static struct row record(int e) { + struct row x = {0}; + x.e = e; + x.q = (e - K) % 3; + int s = (e - K) / 3; + x.H = e - s - 1; + int upper = x.H < M ? x.H : M; + if (CUTOFF >= upper) reject("empty line bank", e, upper); + x.prefix = prefix(e); + x.base = x.prefix; + x.groups = x.H < M; + for (int h = CUTOFF + 1; h <= upper; ++h) { + int64_t inside = 2LL * h - e; + int64_t denominator = inside * inside - (int64_t)e * C; + if (2 * h <= e || inside <= C || denominator <= 0) { + x.branch = 2; + x.failure = 1; + return x; + } + int64_t classes = + (int64_t)(((i128)e * (inside - C)) / denominator); + if (classes < 1) reject("class count", e, h); + x.base += 1 - classes; + x.groups += (int)classes; + ++x.layers; + } + x.direct = x.base + (int64_t)x.groups * LINE; + if (x.direct <= BUDGET) { + x.branch = 0; + x.bound = x.direct; + x.slack = BUDGET - x.bound; + return x; + } + int64_t required = BUDGET - x.base + 1; + if (required <= 0 || x.groups == 0) { + x.branch = 2; + x.failure = 2; + return x; + } + x.threshold = (required + x.groups - 1) / x.groups; + if (x.threshold < 2) { + x.branch = 2; + x.failure = 3; + return x; + } + x.core = (x.threshold * M - N + x.threshold - 2) + / (x.threshold - 1); + x.inside = x.core - C; + x.sync = e - x.inside + K; + x.agreement = M - x.sync + 1; + int64_t n = N - e; + int64_t denominator = x.agreement * x.agreement - n * C; + if (denominator <= 0) { + x.branch = 2; + x.failure = 4; + return x; + } + x.low = n * (x.agreement - C) / denominator; + x.bound = (int64_t)e * x.low + LINE; + x.slack = BUDGET - x.bound; + x.branch = x.bound <= BUDGET ? 1 : 2; + x.failure = x.branch == 1 ? 0 : 5; + return x; +} + +static void print_row(const char *name, const struct row *x) { + printf("%s e=%d q=%d H=%d branch=%d failure=%d prefix=%lld " + "layers=%d groups=%d base=%lld direct=%lld threshold=%lld " + "core=%lld inside=%lld sync=%lld agreement=%lld low=%lld " + "bound=%lld slack=%lld\n", + name, x->e, x->q, x->H, x->branch, x->failure, + (long long)x->prefix, x->layers, x->groups, + (long long)x->base, (long long)x->direct, + (long long)x->threshold, (long long)x->core, + (long long)x->inside, (long long)x->sync, + (long long)x->agreement, (long long)x->low, + (long long)x->bound, (long long)x->slack); +} + +int main(void) { + struct row first = {0}, last = {0}, wall = {0}; + int paid = 0, direct = 0, absorption = 0; + for (int e = 101157; e <= 140000; ++e) { + struct row x = record(e); + if (x.branch == 2) { + wall = x; + break; + } + if (!paid) first = x; + last = x; + ++paid; + if (x.branch == 0) ++direct; else ++absorption; + } + if (!(paid == 23649 && direct == 0 && absorption == 23649 && + first.e == 101157 && first.prefix == 1502226 && + first.groups == 6502 && first.base == 1497892 && + first.threshold == 2350 && first.bound == 3813525 && + last.e == 124805 && last.prefix == 1636955 && + last.groups == 34560 && last.base == 1604577 && + last.threshold == 440 && last.core == 65220 && + last.low == 126 && last.bound == 16706559 && + wall.e == 124806 && wall.failure == 5 && + wall.groups == 34564 && wall.base == 1604586 && + wall.threshold == 439 && wall.low == 127 && + wall.bound == 16831491)) { + reject("interval census", wall.e, wall.H); + } + print_row("FIRST", &first); + print_row("LAST_PAID", &last); + print_row("FIRST_WALL", &wall); + printf("MCA_FULL_LIFT_BOUNDARY_LINE_BANK_ABSORPTION_V1_SCAN_PASS " + "paid=%d direct=%d absorption=%d\n", + paid, direct, absorption); + return 0; +} diff --git a/experimental/verify_mca_full_lift_boundary_line_bank_absorption_v1.py b/experimental/verify_mca_full_lift_boundary_line_bank_absorption_v1.py new file mode 100644 index 000000000..c9b162fca --- /dev/null +++ b/experimental/verify_mca_full_lift_boundary_line_bank_absorption_v1.py @@ -0,0 +1,149 @@ +#!/usr/bin/env python3 +"""Verify endpoints of the M31 boundary-line-bank payment.""" + +from __future__ import annotations + +import copy + + +ROW = { + "R": 1048576, "d": 67448, "K": 6, "budget": 16777215, + "cutoff": 65272, "paid_count": 23649, + "first": { + "e": 101157, "q": 0, "H": 67439, "prefix": 1502226, + "layers": 2167, "groups": 6502, "base": 1497892, + "direct": 6380798650, "threshold": 2350, "core": 67037, + "inside": 67032, "sync": 34131, "agreement": 33324, + "low": 28, "bound": 3813525, "slack": 12963690, + }, + "last": { + "e": 124805, "q": 2, "H": 83205, "prefix": 1636955, + "layers": 2182, "groups": 34560, "base": 1604577, + "direct": 33909422817, "threshold": 440, "core": 65220, + "inside": 65215, "sync": 59596, "agreement": 7859, + "low": 126, "bound": 16706559, "slack": 70656, + }, + "adjacent": { + "e": 124806, "q": 0, "H": 83205, "prefix": 1636968, + "layers": 2182, "groups": 34564, "base": 1604586, + "direct": 33913347342, "threshold": 439, "core": 65214, + "inside": 65209, "sync": 59603, "agreement": 7852, + "low": 127, "bound": 16831491, "slack": -54276, + }, +} + + +class Reject(ValueError): + pass + + +def raw_cap(R: int, d: int, K: int, e: int, h: int) -> int: + n, m, c = R + K - e, d + K, K - 1 + agreement = m - h + johnson = agreement * agreement - n * c + if johnson > 0: + return n * (agreement - c) // johnson + gap = -johnson + balance = 2 * agreement * agreement - n * c + tangent = (n - agreement) ** 2 - (n - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("undefined prefix cap") + return ((n - 1) * n * n * (agreement - c) + // (agreement * tangent)) + + +def prefix(R: int, d: int, K: int, e: int, cutoff: int) -> int: + values = [0] + [ + raw_cap(R, d, K, e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def endpoint(row, expected): + R, d, K, e = row["R"], row["d"], row["K"], expected["e"] + N, m, c = R + K, d + K, K - 1 + s, q = divmod(e - K, 3) + H = e - s - 1 + upper = min(H, m) + p = prefix(R, d, K, e, row["cutoff"]) + class_sum = 0 + layers = 0 + for h in range(row["cutoff"] + 1, upper + 1): + A = 2 * h - e + denominator = A * A - e * c + if 2 * h <= e or A <= c or denominator <= 0: + raise Reject("line-bank guard") + classes = e * (A - c) // denominator + if classes < 1: + raise Reject("class count") + class_sum += classes + layers += 1 + groups = int(H < m) + class_sum + base = p + layers - class_sum + direct = base + groups * (N - m + 1) + required = row["budget"] - base + 1 + threshold = (required + groups - 1) // groups + if required <= 0 or threshold < 2: + raise Reject("pigeonhole") + core = (threshold * m - N + threshold - 2) // (threshold - 1) + inside = core - c + sync = e - inside + K + agreement = m - sync + 1 + n = N - e + denominator = agreement * agreement - n * c + if denominator <= 0: + raise Reject("low Johnson") + low = n * (agreement - c) // denominator + bound = e * low + (N - m + 1) + return { + "e": e, "q": q, "H": H, "prefix": p, "layers": layers, + "groups": groups, "base": base, "direct": direct, + "threshold": threshold, "core": core, "inside": inside, + "sync": sync, "agreement": agreement, "low": low, + "bound": bound, "slack": row["budget"] - bound, + }, layers + + +def validate(row): + checks = 47 + for name in ("first", "last", "adjacent"): + got, layers = endpoint(row, row[name]) + if got != row[name]: + raise Reject(name) + checks += layers + if (row["paid_count"] != row["last"]["e"] - row["first"]["e"] + 1 or + row["adjacent"]["e"] != row["last"]["e"] + 1 or + row["last"]["bound"] > row["budget"] or + row["adjacent"]["bound"] <= row["budget"]): + raise Reject("summary") + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROW)) + mutations = [] + for section, key, delta in ( + ("last", "prefix", 1), ("last", "groups", 1), + ("last", "bound", -1), ("adjacent", "bound", -1), + (None, "paid_count", 1)): + changed = copy.deepcopy(ROW) + target = changed if section is None else changed[section] + target[key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_BOUNDARY_LINE_BANK_ABSORPTION_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From bc6f8286b2f0d6265170494becd8f78c48f7cb9c Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 10:33:53 +0100 Subject: [PATCH 20/40] MCA: recursively peel full-lift boundary lines --- agents.md | 2 +- experimental/agents-log.md | 20 ++ ..._recursive_line_peeling_core_packing_v1.py | 114 ++++++++++ experimental/grande_finale.tex | 98 ++++++++ ..._recursive_line_peeling_core_packing_v1.md | 119 ++++++++++ ...t_recursive_line_peeling_core_packing_v1.c | 211 ++++++++++++++++++ ..._recursive_line_peeling_core_packing_v1.py | 206 +++++++++++++++++ 7 files changed, 769 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_recursive_line_peeling_core_packing_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_recursive_line_peeling_core_packing_v1.md create mode 100644 experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c create mode 100644 experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.py diff --git a/agents.md b/agents.md index 35dbee06e..170294fe2 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e int: + shortened = N - e + agreement = M - h + assert agreement > C + delta = agreement * agreement - shortened * C + if delta > 0: + return shortened * (agreement - C) // delta + gap = -delta + assert 2 * agreement * agreement >= shortened * C + tangent = ((shortened - agreement) ** 2 + - (shortened - 1) * gap) + assert tangent > 0 + return ((shortened - 1) * shortened * shortened * (agreement - C) + // (agreement * tangent)) + + +def weighted_prefix(e: int, cutoff: int) -> int: + suffix = [0] + [gram_cap(e, h) for h in range(1, cutoff + 1)] + running = suffix[cutoff] + for h in range(cutoff - 1, 0, -1): + running = min(running, suffix[h]) + suffix[h] = running + return sum((suffix[h] - suffix[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def bank(e: int, cutoff: int, upper: int) -> tuple[int, int]: + groups = 0 + for h in range(cutoff + 1, upper + 1): + overlap = 2 * h - e + denominator = overlap * overlap - e * C + assert 2 * h > e and overlap > C and denominator > 0 + groups += e * (overlap - C) // denominator + base = weighted_prefix(e, cutoff) + upper - cutoff - groups + return groups, base + + +def forced_inside(threshold: int) -> int: + numerator = threshold * M - N + core = 0 if numerator <= 0 else ( + numerator + threshold - 2) // (threshold - 1) + return max(core - C, 0) + + +def thresholds(groups: int, base: int, count: int) -> list[int]: + answer = [] + for r in range(count): + required = BUDGET - r * LINE - base + 1 + assert required > 0 + answer.append((required + groups - 1) // groups) + return answer + + +def main() -> None: + groups, base = bank(124806, 65304, M) + threshold = thresholds(groups, base, 1)[0] + inside = forced_inside(threshold) + sync = 124806 - inside + K + low = weighted_prefix(124806, sync - 1) + assert (groups, base, threshold, inside, sync, low + LINE) == ( + 33873, 2138482, 433, 65178, 59634, 2603990) + + groups, base = bank(128340, 65304, M) + threshold1 = thresholds(groups, base, 1)[0] + inside1 = forced_inside(threshold1) + sync1 = 128340 - inside1 + K + groups2, base2 = bank(128340, 65304, sync1 - 1) + threshold2 = thresholds(groups2, base2, 2)[1] + inside2 = forced_inside(threshold2) + assert (threshold1, inside1, sync1, groups2, base2, + threshold2, inside2) == ( + 213, 62822, 65524, 12523, 1749442, 1122, 66574) + assert inside1 + inside2 - C == 129391 > 128340 + + groups, base = bank(130198, 65504, M) + last_thresholds = thresholds(groups, base, 5) + last_inside = [forced_inside(value) for value in last_thresholds] + last_lhs = sum(last_inside) - 10 * C + assert (groups, base, last_thresholds, last_inside, last_lhs) == ( + 260297, 8163020, [34, 30, 26, 22, 19], + [37718, 33617, 28204, 20729, 12942], 133160) + assert last_lhs > 130198 + + groups, base = bank(130199, 65504, M) + wall_thresholds = thresholds(groups, base, 9) + wall_inside = [forced_inside(value) for value in wall_thresholds] + wall_lhs = sum(wall_inside) - 36 * C + next_target = BUDGET - 9 * LINE + assert (groups, base, wall_thresholds, wall_inside, + wall_lhs, next_target) == ( + 269019, 8154082, [33, 29, 25, 22, 18, 14, 11, 7, 3], + [36789, 32409, 26569, 20729, 9736, 0, 0, 0, 0], + 126052, 7947054) + assert wall_lhs <= 130199 and next_target - base + 1 <= 0 + + print( + "MCA_FULL_LIFT_RECURSIVE_LINE_PEELING_CORE_PACKING_V1_AUDIT_PASS " + "first=124806 packing=128340 last=130198 wall=130199" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index fdd0974f5..5ad50de79 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -2914,6 +2914,104 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_full_lift_boundary_line_bank_absorption_v1.c}. \end{remark} +\begin{corollary}[Recursive affine-line peeling and inside-core packing] +\label{cor:full-lift-recursive-line-peeling} +Assume the hypotheses of +Corollary~\ref{cor:full-lift-boundary-line-bank}, suppose \(H\ge m\), and +put \(Q=N-m+1\). Fix a cutoff \(b0\) and +\(T_r-C_r+1>0\), an unsafe residual forces an affine line of size at least +\[ + \lambda_r=\left\lceil\frac{T_r-C_r+1}{G_r}\right\rceil. \tag{RP1} +\] +For \(\lambda_r\ge2\), define +\[ + g_r=\max\left(0, + \left\lceil\frac{\lambda_rm-N}{\lambda_r-1}\right\rceil\right), + \qquad u_r=\max(g_r-c,0), + \qquad U_{r+1}=\min(U_r,e-u_r+K-1). \tag{RP2} +\] +Remove the whole selected line and repeat. The selected lines are distinct, +and their inside-core sets \(I_i\) satisfy +\[ + |I_i|\ge u_i,\qquad |I_i\cap I_j|\le c. +\] +Consequently, if at any stage +\[ + \sum_{i=0}^{r}u_i-\binom{r+1}{2}c>e, \tag{RP3} +\] +the assumed unsafe family is impossible. Thus termination either at the +weighted prefix or at (RP3) certifies safety. +\end{corollary} + +\begin{proof} +Apply the exact-layer line-bank identity to the residual assigned family. +Empty layers only improve the resulting inequality, so +\[ + |\mathcal Z_r|\le C_r+\sum_{i=1}^{G_r}|\mathcal L_i|. +\] +If the original family is unsafe after \(r\) line charges, then +\(|\mathcal Z_r|>T_r\), and pigeonhole gives (RP1). The guard +\(\lambda_r\ge2\) makes the selected slot an actual affine line rather than +a padded singleton. Total-core line packing gives \(g_r\). At most \(c\) +core coordinates lie outside the gauged direction support, and the +common-core absorption argument places every residual explanation of +deficit at least \(e-u_r+K\) on the same line. Removing the whole line costs +at most \(Q\) and gives (RP2). Even when (RP2) does not lower the deficit +ceiling, the selected line is absent from the next residual. + +Write a selected line as \(c_\gamma=a_i+\gamma b_i\). Its total common +core consists of coordinates where +\((r_0,r_1)=(a_i,b_i)\). Distinct selected lines have distinct codeword +pairs. At least one of \(a_i-a_j\) and \(b_i-b_j\) is therefore a nonzero +degree-\(128340\). At the last support, the five inside-core +lower bounds give +\[ + 37718+33617+28204+20729+12942-10\cdot5 + =133160>130198. +\] +At adjacent \(e=130199\), nine legal peels give packing lower bound +\(126052\le130199\); the next residual target is \(7947054\), while the +certified base charge is \(8154082\), so (RP1) no longer forces a line. This +is a method wall, not an unsafe certificate. The Mersenne residual interval +is now \(130199\le e\le1044241\). + +The endpoint replay, independent audit, and full constant-memory interval +replay are +\path{experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.py}, +\path{experimental/audit_mca_full_lift_recursive_line_peeling_core_packing_v1.py}, +and +\path{experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_recursive_line_peeling_core_packing_v1.md b/experimental/notes/thresholds/mca_full_lift_recursive_line_peeling_core_packing_v1.md new file mode 100644 index 000000000..c94810350 --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_recursive_line_peeling_core_packing_v1.md @@ -0,0 +1,119 @@ +# Full-lift recursive line peeling and core packing + +## Scope + +This packet continues the Mersenne-31 full-lift support payment from the +boundary-line-bank wall `e=124806`. It does not claim that the adjacent +failure is unsafe and does not close the full Mersenne row. + +Use + +```text +N=1048582, m=67454, K=6, c=5, +B=16777215, Q=N-m+1=981129. +``` + +## Residual recursion + +After `r` affine lines have been removed and charged, let `Z_r` be the +remaining assigned slopes, let `U_r` be their deficit ceiling, and set + +```text +T_r = B-rQ. +``` + +For a legal prefix cutoff `b`, the exact-layer direction-class bank gives + +```text +|Z_r| <= C_r + sum_(i=1)^G_r |L_i|, +G_r = sum_(h=b+1)^U_r J_h, +C_r = P_b(e)+(U_r-b)-G_r. +``` + +If the residual is still unsafe, it forces one line of size + +```text +lambda_r = ceil((T_r-C_r+1)/G_r). +``` + +For `lambda_r>=2`, total-core packing and the inside-support correction give + +```text +g_r = max(0,ceil((lambda_r*m-N)/(lambda_r-1))), +u_r = max(g_r-c,0), +U_(r+1) = min(U_r,e-u_r+K-1). +``` + +Remove every assigned slope on that affine line and repeat. The next +selected line is distinct even when `U_(r+1)=U_r`. + +## Core-packing invariant + +Write a peeled line as `c_gamma=a_i+gamma*b_i`. Its common core is the set +where `(r_0,r_1)=(a_i,b_i)`. For two distinct lines, at least one codeword +difference `a_i-a_j` and `b_i-b_j` is nonzero, so the two cores meet in at +most `K-1=5` coordinates. Their inside cores all lie in the same +`e`-coordinate direction support. Hence + +```text +sum_i u_i-C(r,2)*5 <= e. +``` + +A strict violation contradicts unsafety. Otherwise, if the recursion +lowers the residual ceiling into a defined suffix-minimum weighted prefix, +that prefix plus the peeled-line charges proves safety. + +## Exact Mersenne interval + +Start at cutoff `65304`. If its first boundary layer fails a guard, use +the least legal cutoff plus two guard layers. Exact replay checks every +prefix and line-bank guard and proves + +```text +124806<=e<=130198. +``` + +The branch census is: + +```text +paid supports: 5393 +weighted prefix: 3837 +core packing: 1556 +direct line bank: 0 +maximum peels: 5 +line counts 1..5: 3534,397,1397,59,6 +``` + +The first packing termination is `e=128340`: + +```text +62822+66574-5 = 129391 > 128340. +``` + +At the last paid support `e=130198`, five forced lines give + +```text +37718+33617+28204+20729+12942-10*5 + = 133160 > 130198. +``` + +At adjacent `e=130199`, nine legal peels only give packing lower bound +`126052`. The next residual target is `7947054`, below base charge +`8154082`, so the pigeonhole numerator is nonpositive and cannot force +another line. This is a method wall, not an unsafe certificate. The +Mersenne residual becomes + +```text +130199<=e<=1044241. +``` + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.py +python3 experimental/audit_mca_full_lift_recursive_line_peeling_core_packing_v1.py +cc -O2 -std=c11 -Wall -Wextra -Werror \ + experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c \ + -o /tmp/verify_m31_recursive_line_peeling +/tmp/verify_m31_recursive_line_peeling +``` diff --git a/experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c b/experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c new file mode 100644 index 000000000..2751658a1 --- /dev/null +++ b/experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c @@ -0,0 +1,211 @@ +#include +#include +#include + +/* Full constant-memory replay of recursive M31 affine-line peeling. */ + +typedef __int128 i128; + +enum { + R = 1048576, D = 67448, K = 6, N = R + K, M = D + K, + C = K - 1, BUDGET = 16777215, LINE = N - M + 1, + INITIAL_CUTOFF = 65304, ADAPTIVE_MARGIN = 2, MAX_LINES = 32, +}; + +static int64_t caps[M + 1]; + +static void reject(const char *what, int e, int value) { + fprintf(stderr, "REJECT %s e=%d value=%d\n", what, e, value); + exit(1); +} + +static int raw_cap(int e, int h, int64_t *out) { + int64_t shortened = N - e, agreement = M - h; + if (agreement <= C) return 0; + int64_t johnson = agreement * agreement - shortened * C; + if (johnson > 0) { + *out = shortened * (agreement - C) / johnson; + return 1; + } + int64_t gap = -johnson; + int64_t balance = 2 * agreement * agreement - shortened * C; + int64_t tangent = (shortened - agreement) * (shortened - agreement) + - (shortened - 1) * gap; + if (balance < 0 || tangent <= 0) return 0; + *out = (int64_t)(((i128)(shortened - 1) * shortened * shortened + * (agreement - C)) + / ((i128)agreement * tangent)); + return 1; +} + +static int prefix(int e, int cutoff, int64_t *out) { + if (cutoff < 1 || cutoff > M) return 0; + caps[0] = 0; + for (int h = 1; h <= cutoff; ++h) { + if (!raw_cap(e, h, &caps[h])) return 0; + } + for (int h = cutoff - 1; h >= 1; --h) { + if (caps[h + 1] < caps[h]) caps[h] = caps[h + 1]; + } + int64_t total = 0; + for (int h = 1; h <= cutoff; ++h) { + total += (caps[h] - caps[h - 1]) * (e / h); + } + *out = total; + return 1; +} + +static int choose_cutoff(int e) { + int cutoff = INITIAL_CUTOFF; + while (cutoff < M) { + int h = cutoff + 1; + int64_t agreement = 2LL * h - e; + if (2 * h > e && agreement > C && + agreement * agreement > (int64_t)e * C) break; + ++cutoff; + } + if (cutoff > INITIAL_CUTOFF && cutoff < M) { + cutoff += ADAPTIVE_MARGIN; + if (cutoff > M) cutoff = M; + } + return cutoff; +} + +struct row { + int e, failure, finish, cutoff, lines, upper; + int first_threshold, last_threshold; + int64_t piece, bound, slack, inside_sum, packing; + int64_t groups, base, target; +}; + +static struct row record(int e) { + struct row x = {0}; + x.e = e; + int H = e - (e - K) / 3 - 1; + if (H < M) { x.failure = 1; return x; } + x.cutoff = choose_cutoff(e); + int64_t bank_prefix; + if (x.cutoff >= M || !prefix(e, x.cutoff, &bank_prefix)) { + x.failure = 2; + return x; + } + int upper = M; + while (x.lines < MAX_LINES) { + int64_t target = BUDGET - (int64_t)x.lines * LINE; + if (target < 0) { x.failure = 3; return x; } + int64_t whole_prefix; + if (prefix(e, upper, &whole_prefix) && whole_prefix <= target) { + x.finish = 1; + x.upper = upper; + x.piece = whole_prefix; + x.bound = (int64_t)x.lines * LINE + whole_prefix; + x.slack = BUDGET - x.bound; + return x; + } + + int64_t groups = 0; + for (int h = x.cutoff + 1; h <= upper; ++h) { + int64_t agreement = 2LL * h - e; + int64_t denominator = agreement * agreement - (int64_t)e * C; + if (2 * h <= e || agreement <= C || denominator <= 0) { + x.failure = 4; + return x; + } + groups += (int64_t)(((i128)e * (agreement - C)) / denominator); + } + int64_t base = bank_prefix + upper - x.cutoff - groups; + int64_t required = target - base + 1; + if (required <= 0) { + x.failure = 5; + x.upper = upper; + x.groups = groups; + x.base = base; + x.target = target; + return x; + } + if (groups < 1) { x.failure = 6; return x; } + int64_t threshold = (required + groups - 1) / groups; + if (threshold < 2 || threshold > LINE) { + x.failure = 7; + return x; + } + if (!x.lines) x.first_threshold = (int)threshold; + x.last_threshold = (int)threshold; + int64_t numerator = threshold * M - N; + int64_t core = numerator <= 0 ? 0 + : (numerator + threshold - 2) / (threshold - 1); + int64_t inside = core > C ? core - C : 0; + int64_t sync = e - inside + K; + x.inside_sum += inside; + ++x.lines; + x.packing = x.inside_sum + - (int64_t)x.lines * (x.lines - 1) * C / 2; + if (x.packing > e) { + x.finish = 2; + x.upper = upper; + return x; + } + if (sync - 1 < upper) upper = (int)sync - 1; + } + x.failure = 8; + return x; +} + +static void print_row(const char *name, const struct row *x) { + printf("%s e=%d failure=%d finish=%d cutoff=%d lines=%d upper=%d " + "piece=%lld bound=%lld slack=%lld inside=%lld packing=%lld " + "first_threshold=%d last_threshold=%d groups=%lld base=%lld " + "target=%lld\n", + name, x->e, x->failure, x->finish, x->cutoff, x->lines, + x->upper, (long long)x->piece, (long long)x->bound, + (long long)x->slack, (long long)x->inside_sum, + (long long)x->packing, x->first_threshold, x->last_threshold, + (long long)x->groups, (long long)x->base, + (long long)x->target); +} + +int main(void) { + struct row first = {0}, first_packing = {0}, last = {0}, wall = {0}; + int paid = 0, profile = 0, packing = 0, max_lines = 0; + int line_count[MAX_LINES + 1] = {0}; + for (int e = 124806; e <= 130199; ++e) { + struct row x = record(e); + if (x.failure) { wall = x; break; } + if (!paid) first = x; + if (x.finish == 2 && !first_packing.e) first_packing = x; + last = x; + ++paid; + ++line_count[x.lines]; + if (x.lines > max_lines) max_lines = x.lines; + if (x.finish == 1) ++profile; + else if (x.finish == 2) ++packing; + else reject("finish", e, x.finish); + } + if (!(paid == 5393 && profile == 3837 && packing == 1556 && + max_lines == 5 && line_count[1] == 3534 && + line_count[2] == 397 && line_count[3] == 1397 && + line_count[4] == 59 && line_count[5] == 6 && + first.e == 124806 && first.bound == 2603990 && + first.first_threshold == 433 && + first_packing.e == 128340 && first_packing.lines == 2 && + first_packing.packing == 129391 && + last.e == 130198 && last.cutoff == 65504 && last.lines == 5 && + last.packing == 133160 && last.first_threshold == 34 && + last.last_threshold == 19 && + wall.e == 130199 && wall.failure == 5 && wall.lines == 9 && + wall.groups == 269019 && wall.base == 8154082 && + wall.target == 7947054 && wall.inside_sum == 126232 && + wall.packing == 126052)) { + reject("census", wall.e, paid); + } + print_row("FIRST", &first); + print_row("FIRST_PACKING", &first_packing); + print_row("LAST_PAID", &last); + print_row("FIRST_WALL", &wall); + printf("MCA_FULL_LIFT_RECURSIVE_LINE_PEELING_CORE_PACKING_V1_SCAN_PASS " + "paid=%d profile=%d packing=%d max_lines=%d " + "line_counts=1:%d,2:%d,3:%d,4:%d,5:%d\n", + paid, profile, packing, max_lines, line_count[1], line_count[2], + line_count[3], line_count[4], line_count[5]); + return 0; +} diff --git a/experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.py b/experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.py new file mode 100644 index 000000000..63f325df9 --- /dev/null +++ b/experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.py @@ -0,0 +1,206 @@ +#!/usr/bin/env python3 +"""Verify selected exact records for recursive M31 line peeling.""" + +from __future__ import annotations + +import copy + + +R, D, K = 1048576, 67448, 6 +N, M, C = R + K, D + K, K - 1 +BUDGET, LINE = 16777215, N - M + 1 + +EXPECTED = { + "first": { + "e": 124806, "cutoff": 65304, "certificate": "profile", + "lines": 1, "upper": 59633, "piece": 1622861, + "bound": 2603990, "slack": 14173225, "inside_sum": 65178, + "packing": 65178, "thresholds": [433], + }, + "first_packing": { + "e": 128340, "cutoff": 65304, "certificate": "packing", + "lines": 2, "upper": 65523, "piece": 0, "bound": 0, + "slack": 0, "inside_sum": 129396, "packing": 129391, + "thresholds": [213, 1122], + }, + "last": { + "e": 130198, "cutoff": 65504, "certificate": "packing", + "lines": 5, "upper": 67454, "piece": 0, "bound": 0, + "slack": 0, "inside_sum": 133210, "packing": 133160, + "thresholds": [34, 30, 26, 22, 19], + }, + "adjacent": { + "e": 130199, "cutoff": 65504, "certificate": "wall", + "lines": 9, "upper": 67454, "piece": 8154082, + "bound": 7947054, "slack": 0, "inside_sum": 126232, + "packing": 126052, "thresholds": [33, 29, 25, 22, 18, 14, 11, 7, 3], + }, + "paid_count": 5393, +} + + +class Reject(ValueError): + pass + + +def raw_cap(e: int, h: int) -> int: + shortened, agreement = N - e, M - h + if agreement <= C: + raise Reject("prefix agreement") + johnson = agreement * agreement - shortened * C + if johnson > 0: + return shortened * (agreement - C) // johnson + gap = -johnson + balance = 2 * agreement * agreement - shortened * C + tangent = (shortened - agreement) ** 2 - (shortened - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("undefined prefix") + return ((shortened - 1) * shortened * shortened * (agreement - C) + // (agreement * tangent)) + + +def prefix(e: int, cutoff: int) -> int: + if not 1 <= cutoff <= M: + raise Reject("prefix cutoff") + values = [0] + [raw_cap(e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def choose_cutoff(e: int) -> int: + initial = 65304 + cutoff = initial + while cutoff < M: + h = cutoff + 1 + agreement = 2 * h - e + if (2 * h > e and agreement > C and + agreement * agreement > e * C): + break + cutoff += 1 + if cutoff > initial and cutoff < M: + cutoff += 2 + if cutoff >= M: + raise Reject("no bank") + return cutoff + + +def record(e: int) -> dict[str, object]: + H = e - (e - K) // 3 - 1 + if H < M: + raise Reject("scope") + cutoff = choose_cutoff(e) + bank_prefix = prefix(e, cutoff) + upper = M + steps: list[dict[str, int]] = [] + inside_sum = 0 + + while len(steps) < 32: + target = BUDGET - len(steps) * LINE + if target < 0: + raise Reject("budget") + try: + whole_prefix = prefix(e, upper) + except Reject: + whole_prefix = None + if whole_prefix is not None and whole_prefix <= target: + bound = len(steps) * LINE + whole_prefix + return { + "e": e, "cutoff": cutoff, "certificate": "profile", + "lines": len(steps), "upper": upper, "piece": whole_prefix, + "bound": bound, "slack": BUDGET - bound, + "inside_sum": inside_sum, + "packing": inside_sum - len(steps) * (len(steps) - 1) * C // 2, + "thresholds": [step["threshold"] for step in steps], + } + groups = 0 + for h in range(cutoff + 1, upper + 1): + agreement = 2 * h - e + denominator = agreement * agreement - e * C + if 2 * h <= e or agreement <= C or denominator <= 0: + raise Reject("bank guard") + groups += e * (agreement - C) // denominator + base = bank_prefix + upper - cutoff - groups + required = target - base + 1 + if required <= 0: + return { + "e": e, "cutoff": cutoff, "certificate": "wall", + "lines": len(steps), "upper": upper, "piece": base, + "bound": target, "slack": 0, "inside_sum": inside_sum, + "packing": inside_sum - len(steps) * (len(steps) - 1) * C // 2, + "thresholds": [step["threshold"] for step in steps], + } + threshold = (required + groups - 1) // groups + if not 2 <= threshold <= LINE: + raise Reject("threshold") + numerator = threshold * M - N + core = 0 if numerator <= 0 else ( + numerator + threshold - 2) // (threshold - 1) + inside = max(core - C, 0) + sync = e - inside + K + inside_sum += inside + line_number = len(steps) + 1 + packing = inside_sum - line_number * (line_number - 1) * C // 2 + steps.append({"threshold": threshold, "inside": inside, + "sync": sync, "packing": packing}) + if packing > e: + return { + "e": e, "cutoff": cutoff, "certificate": "packing", + "lines": len(steps), "upper": upper, "piece": 0, + "bound": 0, "slack": 0, "inside_sum": inside_sum, + "packing": packing, + "thresholds": [step["threshold"] for step in steps], + } + upper = min(upper, sync - 1) + raise Reject("recursion") + + +def validate(expected: dict[str, object]) -> int: + checks = 43 + for name in ("first", "first_packing", "last", "adjacent"): + got = record(expected[name]["e"]) + if got != expected[name]: + raise Reject(name) + checks += 10 + len(got["thresholds"]) + if (expected["paid_count"] != + expected["last"]["e"] - expected["first"]["e"] + 1 or + expected["adjacent"]["e"] != expected["last"]["e"] + 1): + raise Reject("summary") + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(EXPECTED)) + mutations = [] + for name, key, delta in ( + ("first", "bound", 1), + ("first_packing", "packing", -1), + ("last", "inside_sum", 1), + ("adjacent", "piece", -1)): + changed = copy.deepcopy(EXPECTED) + changed[name][key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + changed = copy.deepcopy(EXPECTED) + changed["paid_count"] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_RECURSIVE_LINE_PEELING_CORE_PACKING_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 8e627a27b64be6a8c8928958bb5e03a1216fc505 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 10:56:50 +0100 Subject: [PATCH 21/40] MCA: jointly charge recursive line cores --- agents.md | 2 +- experimental/agents-log.md | 19 ++ ..._full_lift_joint_core_charge_peeling_v1.py | 64 +++++ experimental/grande_finale.tex | 84 ++++++ ..._full_lift_joint_core_charge_peeling_v1.md | 99 +++++++ ...a_full_lift_joint_core_charge_peeling_v1.c | 222 ++++++++++++++++ ..._full_lift_joint_core_charge_peeling_v1.py | 242 ++++++++++++++++++ 7 files changed, 731 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_joint_core_charge_peeling_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_joint_core_charge_peeling_v1.md create mode 100644 experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c create mode 100644 experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.py diff --git a/agents.md b/agents.md index 170294fe2..936984794 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e130219`. +- **What to do next:** Add structure for the zero-core line population at + the adjacent method wall `e=130220` or bridge to the high-support interval. + ### 2026-08-13 - Mersenne recursive line-peeling core-packing interval - **Agent/model:** Codex. diff --git a/experimental/audit_mca_full_lift_joint_core_charge_peeling_v1.py b/experimental/audit_mca_full_lift_joint_core_charge_peeling_v1.py new file mode 100644 index 000000000..e4156ab18 --- /dev/null +++ b/experimental/audit_mca_full_lift_joint_core_charge_peeling_v1.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python3 +"""Independent rational audit of the M31 joint-core charge endpoints.""" + +from __future__ import annotations + +from fractions import Fraction + + +N, M, C = 1048582, 67454, 5 +BUDGET, Q = 16777215, N - M + 1 + + +def charge(e: int, r: int) -> tuple[int, int, int, int]: + if r == 0: + return 0, 0, 0, 0 + budget = min(r * (M - 1), e + r * (r + 1) * C // 2) + full, remainder = divmod(budget, M - 1) + if full == r: + value = Fraction(r * Q, 1) + else: + left = r - full + value = Fraction(full * Q, 1) + value += Fraction(N - remainder, M - remainder) + value += (left - 1) * Fraction(N, M) + return value.numerator // value.denominator, budget, full, remainder + + +def inside(threshold: int) -> int: + numerator = threshold * M - N + core = 0 if numerator <= 0 else ( + numerator + threshold - 2) // (threshold - 1) + return max(core - C, 0) + + +def main() -> None: + assert charge(130199, 3) == (981355, 130229, 1, 62776) + first_inside = [inside(33)] + [inside(29)] * 3 + first_packing = sum(first_inside) - 6 * C + assert first_inside == [36789, 32409, 32409, 32409] + assert first_packing == 133986 > 130199 + + assert charge(130219, 12) == (981513, 130609, 1, 63156) + last_inside = [inside(21)] + [inside(18)] * 12 + last_packing = sum(last_inside) - 78 * C + assert last_inside == [18393] + [9736] * 12 + assert last_packing == 134835 > 130219 + + assert charge(130220, 42) == (987456, 134735, 1, 67282) + assert charge(130220, 43) == (1962895, 134950, 2, 44) + wall_inside = [inside(20)] + [inside(16)] * 42 + wall_packing = sum(wall_inside) - 903 * C + assert wall_inside == [15811] + [2041] * 42 + assert wall_packing == 97018 <= 130220 + assert inside(13) == 0 + assert BUDGET - charge(130220, 43)[0] == 14814320 + + print( + "MCA_FULL_LIFT_JOINT_CORE_CHARGE_PEELING_V1_AUDIT_PASS " + "first=130199 last=130219 wall=130220" + ) + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 5ad50de79..60f06c805 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -3012,6 +3012,90 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_full_lift_recursive_line_peeling_core_packing_v1.c}. \end{remark} +\begin{corollary}[Joint-core charge for peeled explanation lines] +\label{cor:full-lift-joint-core-charge} +In the setting of +Corollary~\ref{cor:full-lift-recursive-line-peeling}, suppose \(r\) +distinct parameterized affine explanation lines have been removed, and let +\(g_i\) be their actual total common-core sizes. Put +\[ + S_r=\min\left\{r(m-1),e+\binom{r+1}{2}c\right\}, + \quad q_r=\left\lfloor\frac{S_r}{m-1}\right\rfloor, + \quad z_r=S_r-q_r(m-1), +\] +and \(f(g)=(N-g)/(m-g)\), \(Q=N-m+1\). Their total number of assigned +slopes is at most +\[ + L_r= + \begin{cases} + rQ,&q_r=r,\\ + \left\lfloor + q_rQ+f(z_r)+(r-q_r-1)f(0) + \right\rfloor,&q_r130219. +\] +At adjacent \(e=130220\), the first threshold is \(20\) and the next 42 are +\(16\). Their packing lower bound is only +\[ + 15811+42\cdot2041-\binom{43}{2}\cdot5 + =97018. +\] +For \(r=43\), the joint core allowance is +\(134950=2(m-1)+44\), the exact joint charge is \(1962895\), and the next +threshold drops to \(13\), whose forced-core lower bound is zero. Later +thresholds cannot increase. This is a method wall, not an unsafe +certificate; the Mersenne residual is now +\(130220\le e\le1044241\). + +The endpoint replay, independent rational audit, and full interval replay +are +\path{experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.py}, +\path{experimental/audit_mca_full_lift_joint_core_charge_peeling_v1.py}, +and +\path{experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_joint_core_charge_peeling_v1.md b/experimental/notes/thresholds/mca_full_lift_joint_core_charge_peeling_v1.md new file mode 100644 index 000000000..13f5664d9 --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_joint_core_charge_peeling_v1.md @@ -0,0 +1,99 @@ +# Full-lift joint-core charge peeling + +## Scope + +This packet strengthens the recursive Mersenne-31 affine-line peeling +route. It pays `130199<=e<=130219`; it does not certify the adjacent row +unsafe or close the full Mersenne row. + +## Joint core budget + +For `r` distinct peeled parameterized explanation lines, let `g_i` be +their actual total common-core sizes and `I_i` their parts inside the fixed +`e`-coordinate gauged direction support. Distinct codeword pairs give + +```text +|I_i intersect I_j| <= c=K-1. +``` + +Each line direction is a nonzero degree-` 130219. +``` + +At adjacent `e=130220`, thresholds `20,16,...,16` for the first 43 lines +give only + +```text +15811+42*2041-C(43,2)*5 = 97018. +``` + +The joint allowance at `r=43` is `134950=2(m-1)+44`, so the joint charge +jumps to `1962895` and the next threshold falls to `13`, with zero forced +core. Later thresholds cannot increase. This is a method wall, not an +unsafe certificate. The residual is + +```text +130220<=e<=1044241. +``` + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.py +python3 experimental/audit_mca_full_lift_joint_core_charge_peeling_v1.py +cc -O2 -std=c11 -Wall -Wextra -Werror \ + experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c \ + -o /tmp/verify_m31_joint_charge +/tmp/verify_m31_joint_charge +``` diff --git a/experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c b/experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c new file mode 100644 index 000000000..d9a6e896e --- /dev/null +++ b/experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c @@ -0,0 +1,222 @@ +#include +#include +#include + +/* Full constant-memory replay of the M31 joint-core charge interval. */ + +typedef __int128 i128; + +enum { + R = 1048576, D = 67448, K = 6, N = R + K, M = D + K, + C = K - 1, BUDGET = 16777215, Q = N - M + 1, + INITIAL_CUTOFF = 65304, MAX_LINES = 256, +}; + +static int64_t caps[M + 1]; + +static void reject(const char *what, int e, int value) { + fprintf(stderr, "REJECT %s e=%d value=%d\n", what, e, value); + exit(1); +} + +static int raw_cap(int e, int h, int64_t *out) { + int64_t shortened = N - e, agreement = M - h; + if (agreement <= C) return 0; + int64_t johnson = agreement * agreement - shortened * C; + if (johnson > 0) { + *out = shortened * (agreement - C) / johnson; + return 1; + } + int64_t gap = -johnson; + int64_t balance = 2 * agreement * agreement - shortened * C; + int64_t tangent = (shortened - agreement) * (shortened - agreement) + - (shortened - 1) * gap; + if (balance < 0 || tangent <= 0) return 0; + *out = (int64_t)(((i128)(shortened - 1) * shortened * shortened + * (agreement - C)) + / ((i128)agreement * tangent)); + return 1; +} + +static int64_t prefix(int e, int cutoff) { + caps[0] = 0; + for (int h = 1; h <= cutoff; ++h) { + if (!raw_cap(e, h, &caps[h])) reject("prefix", e, h); + } + for (int h = cutoff - 1; h >= 1; --h) { + if (caps[h + 1] < caps[h]) caps[h] = caps[h + 1]; + } + int64_t total = 0; + for (int h = 1; h <= cutoff; ++h) { + total += (caps[h] - caps[h - 1]) * (e / h); + } + return total; +} + +static int choose_cutoff(int e) { + int cutoff = INITIAL_CUTOFF; + while (cutoff < M) { + int h = cutoff + 1; + int64_t overlap = 2LL * h - e; + if (2 * h > e && overlap > C && + overlap * overlap > (int64_t)e * C) break; + ++cutoff; + } + if (cutoff > INITIAL_CUTOFF && cutoff < M) cutoff += 2; + if (cutoff >= M) reject("cutoff", e, cutoff); + return cutoff; +} + +struct charge { + int64_t value, core_budget, full, remainder; +}; + +static struct charge joint_charge(int e, int lines) { + struct charge x = {0}; + if (!lines) return x; + int64_t pair_budget = e + (int64_t)lines * (lines + 1) * C / 2; + int64_t trivial = (int64_t)lines * (M - 1); + x.core_budget = pair_budget < trivial ? pair_budget : trivial; + x.full = x.core_budget / (M - 1); + if (x.full > lines) x.full = lines; + x.remainder = x.core_budget - x.full * (M - 1); + if (x.full == lines) { + x.value = (int64_t)lines * Q; + return x; + } + int64_t left = lines - x.full; + i128 denominator = (i128)(M - x.remainder) * M; + i128 numerator = (i128)(N - x.remainder) * M + + (i128)(left - 1) * N * (M - x.remainder); + x.value = x.full * Q + (int64_t)(numerator / denominator); + return x; +} + +struct row { + int e, failure, cutoff, lines, first_threshold, middle_threshold; + int last_threshold, positive; + int64_t prefix, groups, base, target, charge, core_budget; + int64_t full, remainder, packing; +}; + +static struct row record(int e) { + struct row x = {0}; + x.e = e; + x.cutoff = choose_cutoff(e); + x.prefix = prefix(e, x.cutoff); + for (int h = x.cutoff + 1; h <= M; ++h) { + int64_t overlap = 2LL * h - e; + int64_t denominator = overlap * overlap - (int64_t)e * C; + if (2 * h <= e || overlap <= C || denominator <= 0) { + x.failure = 1; + return x; + } + x.groups += (int64_t)(((i128)e * (overlap - C)) / denominator); + } + x.base = x.prefix + M - x.cutoff - x.groups; + int64_t inside_sum = 0; + for (int removed = 0; removed < MAX_LINES; ++removed) { + struct charge q = joint_charge(e, removed); + x.target = BUDGET - q.value; + int64_t required = x.target - x.base + 1; + if (required <= 0) { + x.failure = 2; + return x; + } + int64_t threshold = (required + x.groups - 1) / x.groups; + if (threshold < 2) { + x.failure = 3; + return x; + } + if (!removed) x.first_threshold = (int)threshold; + else if (removed == 1) x.middle_threshold = (int)threshold; + x.last_threshold = (int)threshold; + int64_t numerator = threshold * M - N; + int64_t core = numerator <= 0 ? 0 + : (numerator + threshold - 2) / (threshold - 1); + int64_t inside = core > C ? core - C : 0; + ++x.lines; + if (inside > 0) { + inside_sum += inside; + ++x.positive; + } + x.packing = inside_sum + - (int64_t)x.positive * (x.positive - 1) * C / 2; + x.charge = q.value; + x.core_budget = q.core_budget; + x.full = q.full; + x.remainder = q.remainder; + if (x.packing > e) return x; + if (inside == 0) { + x.failure = 4; + return x; + } + } + x.failure = 5; + return x; +} + +static void print_row(const char *name, const struct row *x) { + printf("%s e=%d failure=%d cutoff=%d prefix=%lld groups=%lld base=%lld " + "lines=%d positive=%d thresholds=%d,%d,%d packing=%lld " + "charge=%lld target=%lld core_budget=%lld full=%lld rem=%lld\n", + name, x->e, x->failure, x->cutoff, (long long)x->prefix, + (long long)x->groups, (long long)x->base, x->lines, x->positive, + x->first_threshold, x->middle_threshold, x->last_threshold, + (long long)x->packing, (long long)x->charge, + (long long)x->target, (long long)x->core_budget, + (long long)x->full, (long long)x->remainder); +} + +int main(void) { + struct row first = {0}, last = {0}, wall = {0}; + int paid = 0, max_lines = 0; + int line_count[MAX_LINES + 1] = {0}; + for (int e = 130199; e <= 130220; ++e) { + struct row x = record(e); + if (x.failure) { + wall = x; + break; + } + if (!paid) first = x; + last = x; + ++paid; + ++line_count[x.lines]; + if (x.lines > max_lines) max_lines = x.lines; + } + if (!(paid == 21 && max_lines == 13 && + line_count[4] == 2 && line_count[5] == 10 && + line_count[6] == 3 && line_count[7] == 2 && + line_count[8] == 1 && line_count[10] == 1 && + line_count[13] == 2 && + first.e == 130199 && first.cutoff == 65504 && + first.prefix == 8421151 && first.groups == 269019 && + first.base == 8154082 && first.lines == 4 && + first.first_threshold == 33 && first.middle_threshold == 29 && + first.last_threshold == 29 && first.packing == 133986 && + first.charge == 981355 && + last.e == 130219 && last.cutoff == 65514 && + last.prefix == 11445963 && last.groups == 269400 && + last.base == 11178503 && last.lines == 13 && + last.first_threshold == 21 && last.middle_threshold == 18 && + last.last_threshold == 18 && last.packing == 134835 && + last.charge == 981513 && + wall.e == 130220 && wall.failure == 4 && wall.lines == 44 && + wall.positive == 43 && wall.cutoff == 65515 && + wall.prefix == 11904256 && wall.groups == 260559 && + wall.base == 11645636 && wall.first_threshold == 20 && + wall.middle_threshold == 16 && wall.last_threshold == 13 && + wall.packing == 97018 && wall.charge == 1962895 && + wall.target == 14814320 && wall.core_budget == 134950 && + wall.full == 2 && wall.remainder == 44)) { + reject("census", wall.e, paid); + } + print_row("FIRST", &first); + print_row("LAST_PAID", &last); + print_row("FIRST_WALL", &wall); + printf("RATE_HALF_MCA_M31_JOINT_CORE_CHARGE_PEELING_SCAN_PASS " + "paid=%d max_lines=%d line_counts=4:%d,5:%d,6:%d,7:%d,8:%d,10:%d,13:%d\n", + paid, max_lines, line_count[4], line_count[5], line_count[6], + line_count[7], line_count[8], line_count[10], line_count[13]); + return 0; +} diff --git a/experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.py b/experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.py new file mode 100644 index 000000000..a5c1676c3 --- /dev/null +++ b/experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.py @@ -0,0 +1,242 @@ +#!/usr/bin/env python3 +"""Verify endpoints of the M31 joint-core charge peeling payment.""" + +from __future__ import annotations + +import copy +from fractions import Fraction + + +R, D, K = 1048576, 67448, 6 +N, M, C = R + K, D + K, K - 1 +BUDGET, LINE = 16777215, N - M + 1 + + +class Reject(ValueError): + pass + + +def raw_cap(e: int, h: int) -> int: + shortened, agreement = N - e, M - h + if agreement <= C: + raise Reject("prefix agreement") + johnson = agreement * agreement - shortened * C + if johnson > 0: + return shortened * (agreement - C) // johnson + gap = -johnson + balance = 2 * agreement * agreement - shortened * C + tangent = (shortened - agreement) ** 2 - (shortened - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("prefix cap") + return ((shortened - 1) * shortened * shortened * (agreement - C) + // (agreement * tangent)) + + +def prefix(e: int, cutoff: int) -> int: + if not 1 <= cutoff <= M: + raise Reject("cutoff") + values = [0] + [raw_cap(e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def choose_cutoff(e: int) -> int: + initial = 65304 + cutoff = initial + while cutoff < M: + h = cutoff + 1 + overlap = 2 * h - e + if (2 * h > e and overlap > C and + overlap * overlap > e * C): + break + cutoff += 1 + if cutoff > initial and cutoff < M: + cutoff += 2 + if cutoff >= M: + raise Reject("no bank") + return cutoff + + +def joint_charge(e: int, lines: int) -> tuple[int, int, int, int]: + if lines == 0: + return 0, 0, 0, 0 + core_budget = min( + lines * (M - 1), e + lines * (lines + 1) * C // 2) + full, remainder = divmod(core_budget, M - 1) + full = min(full, lines) + if full == lines: + charge = lines * LINE + else: + left = lines - full + rational = Fraction(full * LINE, 1) + rational += Fraction(N - remainder, M - remainder) + rational += (left - 1) * Fraction(N, M) + charge = rational.numerator // rational.denominator + return charge, core_budget, full, remainder + + +def runs(values: list[int]) -> list[list[int]]: + answer: list[list[int]] = [] + for value in values: + if answer and answer[-1][0] == value: + answer[-1][1] += 1 + else: + answer.append([value, 1]) + return answer + + +def record(e: int) -> dict[str, object]: + cutoff = choose_cutoff(e) + p = prefix(e, cutoff) + groups = 0 + for h in range(cutoff + 1, M + 1): + overlap = 2 * h - e + denominator = overlap * overlap - e * C + if 2 * h <= e or overlap <= C or denominator <= 0: + raise Reject("bank guard") + groups += e * (overlap - C) // denominator + base = p + M - cutoff - groups + thresholds: list[int] = [] + insides: list[int] = [] + + for removed in range(256): + charge, core_budget, full, remainder = joint_charge(e, removed) + target = BUDGET - charge + required = target - base + 1 + if required <= 0: + return { + "e": e, "cutoff": cutoff, "prefix": p, + "groups": groups, "base": base, + "certificate": "base_wall", "lines": removed, + "target": target, "charge": charge, + "core_budget": core_budget, "full": full, + "remainder": remainder, + "threshold_runs": runs(thresholds), + "inside_runs": runs(insides), + } + threshold = (required + groups - 1) // groups + if threshold < 2: + raise Reject("threshold") + numerator = threshold * M - N + core = 0 if numerator <= 0 else ( + numerator + threshold - 2) // (threshold - 1) + inside = max(core - C, 0) + thresholds.append(threshold) + insides.append(inside) + positive = [value for value in insides if value > 0] + packing = (sum(positive) + - len(positive) * (len(positive) - 1) * C // 2) + if packing > e: + return { + "e": e, "cutoff": cutoff, "prefix": p, + "groups": groups, "base": base, + "certificate": "core_packing", "lines": removed + 1, + "target": target, "charge": charge, + "core_budget": core_budget, "full": full, + "remainder": remainder, "packing": packing, + "threshold_runs": runs(thresholds), + "inside_runs": runs(insides), + } + if inside == 0: + return { + "e": e, "cutoff": cutoff, "prefix": p, + "groups": groups, "base": base, + "certificate": "zero_core_wall", "lines": removed + 1, + "target": target, "charge": charge, + "core_budget": core_budget, "full": full, + "remainder": remainder, "packing": packing, + "threshold_runs": runs(thresholds), + "inside_runs": runs(insides), + } + raise Reject("recursion") + + + +EXPECTED = { + "first": { + "e": 130199, "cutoff": 65504, "prefix": 8421151, + "groups": 269019, "base": 8154082, "certificate": "core_packing", + "lines": 4, "target": 15795860, "charge": 981355, + "core_budget": 130229, "full": 1, "remainder": 62776, + "packing": 133986, "threshold_runs": [[33, 1], [29, 3]], + "inside_runs": [[36789, 1], [32409, 3]], + }, + "last": { + "e": 130219, "cutoff": 65514, "prefix": 11445963, + "groups": 269400, "base": 11178503, "certificate": "core_packing", + "lines": 13, "target": 15795702, "charge": 981513, + "core_budget": 130609, "full": 1, "remainder": 63156, + "packing": 134835, "threshold_runs": [[21, 1], [18, 12]], + "inside_runs": [[18393, 1], [9736, 12]], + }, + "adjacent": { + "e": 130220, "cutoff": 65515, "prefix": 11904256, + "groups": 260559, "base": 11645636, + "certificate": "zero_core_wall", "lines": 44, + "target": 14814320, "charge": 1962895, + "core_budget": 134950, "full": 2, "remainder": 44, + "packing": 97018, + "threshold_runs": [[20, 1], [16, 42], [13, 1]], + "inside_runs": [[15811, 1], [2041, 42], [0, 1]], + }, + "line_counts": {4: 2, 5: 10, 6: 3, 7: 2, 8: 1, 10: 1, 13: 2}, + "paid": 21, +} + + +def validate(expected): + checks = 31 + for name in ("first", "last", "adjacent"): + if record(expected[name]["e"]) != expected[name]: + raise Reject(name) + checks += 14 + line_counts = {} + paid = 0 + for e in range(expected["first"]["e"], expected["adjacent"]["e"]): + got = record(e) + if got["certificate"] != "core_packing": + raise Reject("interval") + line_counts[got["lines"]] = line_counts.get(got["lines"], 0) + 1 + paid += 1 + checks += 1 + if (paid != expected["paid"] or line_counts != expected["line_counts"] or + expected["last"]["e"] - expected["first"]["e"] + 1 != paid or + expected["adjacent"]["e"] != expected["last"]["e"] + 1): + raise Reject("census") + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(EXPECTED)) + mutations = [] + for name, key, delta in ( + ("first", "packing", -1), ("last", "charge", 1), + ("adjacent", "packing", 1), ("adjacent", "target", -1)): + changed = copy.deepcopy(EXPECTED) + changed[name][key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + changed = copy.deepcopy(EXPECTED) + changed["paid"] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_FULL_LIFT_JOINT_CORE_CHARGE_PEELING_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 4f9ed736ecb5421f60f2c06297f0b998e55103c3 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 11:22:06 +0100 Subject: [PATCH 22/40] MCA: retain forced line-core lower bounds --- agents.md | 2 +- experimental/agents-log.md | 18 ++ ...l_lift_lower_aware_joint_core_charge_v1.py | 91 +++++++ experimental/grande_finale.tex | 68 +++++ ...l_lift_lower_aware_joint_core_charge_v1.md | 77 ++++++ ...l_lift_lower_aware_joint_core_charge_v1.py | 232 ++++++++++++++++++ 6 files changed, 487 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_lower_aware_joint_core_charge_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_lower_aware_joint_core_charge_v1.md create mode 100644 experimental/verify_mca_full_lift_lower_aware_joint_core_charge_v1.py diff --git a/agents.md b/agents.md index 936984794..e73e5e65a 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e tuple[int, list[int], int]: + count = len(lower) + budget = min(count * (M - 1), e + count * (count + 1) * C // 2) + values = sorted(lower, reverse=True) + excess = budget - sum(values) + assert excess >= 0 + for index, value in enumerate(values): + addition = min(excess, M - 1 - value) + values[index] += addition + excess -= addition + if excess == 0: + break + assert excess == 0 + total = sum((Fraction(N - value, M - value) for value in values), + Fraction()) + return total.numerator // total.denominator, values, budget + + +def core(threshold: int) -> tuple[int, int]: + numerator = threshold * M - N + total = (0 if numerator <= 0 else + (numerator + threshold - 2) // (threshold - 1)) + return total, max(total - C, 0) + + +def audit_paid(e: int, prefix: int, groups: int, allocation_head: int) -> int: + base = prefix + M - 65515 - groups + thresholds = [20] + [16] * 33 + [20] * 4 + cores: list[int] = [] + insides: list[int] = [] + for threshold in thresholds: + old_charge = 0 if not cores else charge(e, cores)[0] + target = BUDGET - old_charge + forced = (target - base + 1 + groups - 1) // groups + assert forced == threshold + total, inside = core(threshold) + cores.append(total) + insides.append(inside) + packing = sum(insides) - len(insides) * (len(insides) - 1) * C // 2 + assert packing == 142893 > e + final_charge, allocation, budget = charge(e, cores[:-1]) + assert final_charge == 609 + assert allocation == [allocation_head] + [15816] * 3 + [2046] * 33 + assert budget == e + 37 * 38 * C // 2 + return len(thresholds) * 4 + 8 + + +def main() -> None: + checks = audit_paid(130220, 11904256, 260559, 18769) + checks += audit_paid(130221, 11903751, 269440, 18770) + + wall_thresholds = [18] + [14] * 42 + [11] * 127 + [7] * 66 + [3] * 52 + wall_cores = [core(value)[0] for value in wall_thresholds] + wall_insides = [core(value)[1] for value in wall_thresholds] + assert wall_cores == [9741] + [0] * 287 + assert wall_insides == [9736] + [0] * 287 + wall_charge, allocation, budget = charge(130222, wall_cores) + assert allocation == [67453] * 5 + [1037] + [0] * 282 + assert budget == 338302 and wall_charge == 4910044 + assert BUDGET - wall_charge == 11867171 < 12148280 + + first_q = sum((Fraction(N - x, M - x) + for x in [18769] + [15816] * 3 + [2046] * 33), + Fraction()) + last_q = sum((Fraction(N - x, M - x) + for x in [18770] + [15816] * 3 + [2046] * 33), + Fraction()) + wall_q = sum((Fraction(N - x, M - x) for x in allocation), Fraction()) + assert (first_q.numerator, first_q.denominator) == ( + 894348212835561, 1468173681520) + assert (last_q.numerator, last_q.denominator) == ( + 1565078288323625, 2569251168624) + assert (wall_q.numerator, wall_q.denominator) == ( + 379266425096056, 77242971) + print("MCA_FULL_LIFT_LOWER_AWARE_JOINT_CORE_CHARGE_V1_AUDIT_PASS " + f"checks={checks + 13} exact_rational_replay=1") + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 60f06c805..c0b9cb2d4 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -3096,6 +3096,74 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_full_lift_joint_core_charge_peeling_v1.c}. \end{remark} +\begin{corollary}[Lower-aware joint-core charge] +\label{cor:full-lift-lower-aware-joint-core-charge} +In the setting of +Corollary~\ref{cor:full-lift-joint-core-charge}, suppose the actual total +cores also have certified lower bounds +\(0\le\ell_i\le g_i\). Sort the \(\ell_i\) decreasingly, start with +\(x_i=\ell_i\), and distribute +\[ + S_r-\sum_i\ell_i +\] +units by filling \(x_1\) to \(m-1\), then \(x_2\), and so on. Whenever +the lower bounds are feasible, the removed lines contain at most +\[ + L_r^{\rm low}=\left\lfloor\sum_{i=1}^r + \frac{N-x_i}{m-x_i}\right\rfloor \tag{LA1} +\] +assigned slopes. Consequently the residual peeling target may be +strengthened to \(B-L_r^{\rm low}\). +\end{corollary} + +\begin{proof} +Since line labels do not occur in the objective, sort both the lower bounds +and a feasible core vector decreasingly. This preserves the constraints: +if \(\ell_i\ge\ell_j\) but \(g_ie. +\] +Thus both supports are safe. + +At adjacent \(e=130222\), the compiler reaches 288 removed lines. The +maximizing vector is \(67453\cdot5,1037,0\cdot282\), its charge is +\(4910044\), and the residual target \(11867171\) lies below the certified +base \(12148280\). This is a method wall, not an unsafe certificate; the +Mersenne residual is now \(130222\le e\le1044241\). + +The endpoint replay and independent rational audit are +\path{experimental/verify_mca_full_lift_lower_aware_joint_core_charge_v1.py} +and +\path{experimental/audit_mca_full_lift_lower_aware_joint_core_charge_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_lower_aware_joint_core_charge_v1.md b/experimental/notes/thresholds/mca_full_lift_lower_aware_joint_core_charge_v1.md new file mode 100644 index 000000000..497265abf --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_lower_aware_joint_core_charge_v1.md @@ -0,0 +1,77 @@ +# Full-lift lower-aware joint-core charge + +## Scope + +This packet strengthens the recursive Mersenne-31 affine-line peeling +charge by retaining every forced total-core lower bound. It pays +`e=130220,130221`; it does not certify the adjacent row unsafe or close the +full Mersenne row. + +## Majorization envelope + +For `r` previously peeled lines, let `g_i` be the actual total-core sizes +and `ell_i<=g_i` the lower bounds forced when those lines were selected. +The preceding joint-core theorem gives + +```text +0<=ell_i<=g_i<=m-1, +sum_i g_i<=S_r:=min(r(m-1),e+C(r+1,2)c). +``` + +Sort the lower bounds decreasingly. Starting from that vector, spend all +available excess `S_r-sum ell_i` by filling the first coordinate to `m-1`, +then the second, and so on; call the result `x`. Since + +```text +f(g)=(N-g)/(m-g)=1+(N-m)/(m-g) +``` + +is increasing and convex, an exchange from a smaller nonterminal coordinate +to a larger one cannot reduce the sum. Therefore + +```text +sum_i f(g_i) <= sum_i f(x_i). +``` + +The floor of the right side is a valid charge for the removed lines. Unlike +the preceding endpoint envelope, it cannot erase already forced lower bounds +by moving their mass to unrelated coordinates. + +## Exact payment + +At both new supports, 37 removed lines have lower-bound runs + +```text +15816*4, 2046*33. +``` + +The maximizing allocations and charges are + +```text +e=130220: 18769,15816*3,2046*33; S_37=133735; charge=609, +e=130221: 18770,15816*3,2046*33; S_37=133736; charge=609. +``` + +One final threshold `20` is forced. The 38 inside-core lower bounds then +have runs `15811*5,2041*33`, and + +```text +5*15811+33*2041-C(38,2)*5 = 142893 > e. +``` + +At adjacent `e=130222`, the compiler reaches 288 removed lines. Its +maximizing allocation is `67453*5,1037,0*282`, with charge `4910044`. +The target `11867171` is below the certified base `12148280`, so the line +bank no longer forces another line. This is a method wall, not an unsafe +certificate. The residual is + +```text +130222<=e<=1044241. +``` + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_lower_aware_joint_core_charge_v1.py +python3 experimental/audit_mca_full_lift_lower_aware_joint_core_charge_v1.py +``` diff --git a/experimental/verify_mca_full_lift_lower_aware_joint_core_charge_v1.py b/experimental/verify_mca_full_lift_lower_aware_joint_core_charge_v1.py new file mode 100644 index 000000000..fd58bb7e5 --- /dev/null +++ b/experimental/verify_mca_full_lift_lower_aware_joint_core_charge_v1.py @@ -0,0 +1,232 @@ +#!/usr/bin/env python3 +"""Verify the M31 lower-aware joint-core charge payment.""" + +from __future__ import annotations + +import copy +from fractions import Fraction + + +R, D, K = 1048576, 67448, 6 +N, M, C = R + K, D + K, K - 1 +BUDGET = 16777215 + + +class Reject(ValueError): + pass + + +def raw_cap(e: int, h: int) -> int: + shortened, agreement = N - e, M - h + if agreement <= C: + raise Reject("prefix agreement") + johnson = agreement * agreement - shortened * C + if johnson > 0: + return shortened * (agreement - C) // johnson + gap = -johnson + balance = 2 * agreement * agreement - shortened * C + tangent = (shortened - agreement) ** 2 - (shortened - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("prefix cap") + return ((shortened - 1) * shortened * shortened * (agreement - C) + // (agreement * tangent)) + + +def prefix(e: int, cutoff: int) -> int: + values = [0] + [raw_cap(e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def choose_cutoff(e: int) -> int: + initial = 65304 + cutoff = initial + while cutoff < M: + h = cutoff + 1 + overlap = 2 * h - e + if (2 * h > e and overlap > C and + overlap * overlap > e * C): + break + cutoff += 1 + if cutoff > initial and cutoff < M: + cutoff += 2 + if cutoff >= M: + raise Reject("no bank") + return cutoff + + +def runs(values: list[int]) -> list[list[int]]: + answer: list[list[int]] = [] + for value in values: + if answer and answer[-1][0] == value: + answer[-1][1] += 1 + else: + answer.append([value, 1]) + return answer + + +def lower_aware_charge(e: int, lower: list[int]) -> tuple[int, int, int, list[list[int]]]: + count = len(lower) + if count == 0: + return 0, 0, 0, [] + core_budget = min(count * (M - 1), + e + count * (count + 1) * C // 2) + lower_sum = sum(lower) + if lower_sum > core_budget: + raise Reject("infeasible lower bounds") + allocation = sorted(lower, reverse=True) + excess = core_budget - lower_sum + for index in range(count): + addition = min(excess, M - 1 - allocation[index]) + allocation[index] += addition + excess -= addition + if excess == 0: + break + if excess: + raise Reject("allocation") + rational = sum((Fraction(N - value, M - value) + for value in allocation), Fraction()) + return (rational.numerator // rational.denominator, core_budget, + lower_sum, runs(allocation)) + + +def record(e: int) -> dict[str, object]: + cutoff = choose_cutoff(e) + weighted_prefix = prefix(e, cutoff) + groups = 0 + for h in range(cutoff + 1, M + 1): + overlap = 2 * h - e + denominator = overlap * overlap - e * C + if 2 * h <= e or overlap <= C or denominator <= 0: + raise Reject("bank guard") + groups += e * (overlap - C) // denominator + base = weighted_prefix + M - cutoff - groups + thresholds: list[int] = [] + cores: list[int] = [] + insides: list[int] = [] + + for removed in range(512): + charge, core_budget, lower_sum, allocation_runs = ( + lower_aware_charge(e, cores)) + target = BUDGET - charge + required = target - base + 1 + positive = [value for value in insides if value > 0] + packing = (sum(positive) + - len(positive) * (len(positive) - 1) * C // 2) + common = { + "e": e, "cutoff": cutoff, "prefix": weighted_prefix, + "groups": groups, "base": base, "lines": removed, + "target": target, "charge": charge, + "core_budget": core_budget, "lower_sum": lower_sum, + "allocation_runs": allocation_runs, + "threshold_runs": runs(thresholds), + "core_runs": runs(cores), "inside_runs": runs(insides), + "packing": packing, + } + if required <= 0: + common["certificate"] = "base_wall" + return common + threshold = (required + groups - 1) // groups + if threshold < 2: + raise Reject("threshold wall") + numerator = threshold * M - N + core = (0 if numerator <= 0 else + (numerator + threshold - 2) // (threshold - 1)) + inside = max(core - C, 0) + thresholds.append(threshold) + cores.append(core) + insides.append(inside) + positive = [value for value in insides if value > 0] + packing = (sum(positive) + - len(positive) * (len(positive) - 1) * C // 2) + if packing > e: + common.update({ + "certificate": "core_packing", "lines": removed + 1, + "threshold_runs": runs(thresholds), + "core_runs": runs(cores), "inside_runs": runs(insides), + "packing": packing, + }) + return common + raise Reject("recursion limit") + + +EXPECTED = { + "first": { + "e": 130220, "cutoff": 65515, "prefix": 11904256, + "groups": 260559, "base": 11645636, + "certificate": "core_packing", "lines": 38, + "target": 16776606, "charge": 609, + "core_budget": 133735, "lower_sum": 130782, + "allocation_runs": [[18769, 1], [15816, 3], [2046, 33]], + "threshold_runs": [[20, 1], [16, 33], [20, 4]], + "core_runs": [[15816, 1], [2046, 33], [15816, 4]], + "inside_runs": [[15811, 1], [2041, 33], [15811, 4]], + "packing": 142893, + }, + "last": { + "e": 130221, "cutoff": 65515, "prefix": 11903751, + "groups": 269440, "base": 11636250, + "certificate": "core_packing", "lines": 38, + "target": 16776606, "charge": 609, + "core_budget": 133736, "lower_sum": 130782, + "allocation_runs": [[18770, 1], [15816, 3], [2046, 33]], + "threshold_runs": [[20, 1], [16, 33], [20, 4]], + "core_runs": [[15816, 1], [2046, 33], [15816, 4]], + "inside_runs": [[15811, 1], [2041, 33], [15811, 4]], + "packing": 142893, + }, + "adjacent": { + "e": 130222, "cutoff": 65516, "prefix": 12406922, + "groups": 260580, "base": 12148280, + "certificate": "base_wall", "lines": 288, + "target": 11867171, "charge": 4910044, + "core_budget": 338302, "lower_sum": 9741, + "allocation_runs": [[67453, 5], [1037, 1], [0, 282]], + "threshold_runs": [[18, 1], [14, 42], [11, 127], [7, 66], [3, 52]], + "core_runs": [[9741, 1], [0, 287]], + "inside_runs": [[9736, 1], [0, 287]], + "packing": 9736, + }, +} + + +def validate(expected: dict[str, object]) -> int: + checks = 23 + for name in ("first", "last", "adjacent"): + if record(expected[name]["e"]) != expected[name]: + raise Reject(name) + checks += 17 + if (expected["first"]["e"] != 130220 or + expected["last"]["e"] != 130221 or + expected["adjacent"]["e"] != 130222): + raise Reject("interval") + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(EXPECTED)) + mutations = [] + for name, key, delta in ( + ("first", "packing", -1), + ("last", "charge", 1), + ("adjacent", "target", -1), + ("adjacent", "lower_sum", 1)): + changed = copy.deepcopy(EXPECTED) + changed[name][key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise Reject("mutation controls") + print("MCA_FULL_LIFT_LOWER_AWARE_JOINT_CORE_CHARGE_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)} " + "paid=130220..130221 wall=130222") + + +if __name__ == "__main__": + main() From 42ff3c97401d52211c4cecc194b2892d6c081a26 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 11:37:10 +0100 Subject: [PATCH 23/40] MCA: split high and capped line cores --- agents.md | 2 +- experimental/agents-log.md | 17 ++ ...ll_lift_core_dichotomy_capped_charge_v1.py | 80 +++++++ experimental/grande_finale.tex | 67 ++++++ ...ll_lift_core_dichotomy_capped_charge_v1.md | 81 +++++++ ...ll_lift_core_dichotomy_capped_charge_v1.py | 226 ++++++++++++++++++ 6 files changed, 472 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_core_dichotomy_capped_charge_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_core_dichotomy_capped_charge_v1.md create mode 100644 experimental/verify_mca_full_lift_core_dichotomy_capped_charge_v1.py diff --git a/agents.md b/agents.md index e73e5e65a..43a0b3c3e 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e int: + count = len(lower) + if count == 0: + return 0 + budget = min(count * cap, e + count * (count + 1) * C // 2) + values = sorted(lower, reverse=True) + excess = budget - sum(values) + for index, value in enumerate(values): + addition = min(excess, cap - value) + values[index] += addition + excess -= addition + if excess == 0: + break + assert excess == 0 + value = sum((Fraction(N - x, M - x) for x in values), Fraction()) + return value.numerator // value.denominator + + +def core(threshold: int) -> tuple[int, int]: + numerator = threshold * M - N + total = (0 if numerator <= 0 else + (numerator + threshold - 2) // (threshold - 1)) + return total, max(total - C, 0) + + +def paid(e: int, cap: int, base: int, groups: int, + threshold: int, lines: int, expected_packing: int, + absorption_prefix: int) -> int: + assert absorption_prefix + LINE < BUDGET + cores: list[int] = [] + insides: list[int] = [] + for _ in range(lines): + target = BUDGET - charge(e, cores, cap) + assert (target - base + 1 + groups - 1) // groups == threshold + total, inside = core(threshold) + cores.append(total) + insides.append(inside) + packing = sum(insides) - len(insides) * (len(insides) - 1) * C // 2 + assert packing == expected_packing > e + return 7 + 4 * lines + + +def main() -> None: + checks = paid(130222, 64781, 12148280, 260580, 18, 14, 135849, 4180178) + checks += paid(130223, 64782, 12138824, 269480, 18, 14, 135849, 4180156) + checks += paid(130224, 64783, 12702685, 260602, 16, 70, 130795, 4180145) + checks += paid(130225, 64784, 12693152, 269520, 16, 70, 130795, 4180124) + + e, cap, lines = 130226, 64785, 14763 + budget = min(lines * cap, e + lines * (lines + 1) * C // 2) + full, remainder = divmod(budget, cap) + value = full * Fraction(N - cap, M - cap) + value += Fraction(N - remainder, M - remainder) + value += (lines - full - 1) * Fraction(N, M) + wall_charge = value.numerator // value.denominator + assert (budget, full, remainder) == (545032556, 8412, 61136) + assert wall_charge == 3199542 + target = BUDGET - wall_charge + assert target == 13577673 + assert (target - 13317279 + 1 + 260627 - 1) // 260627 == 1 + assert core(14) == (0, 0) + assert 4180114 + LINE == 5161243 < BUDGET + print("MCA_FULL_LIFT_CORE_DICHOTOMY_CAPPED_CHARGE_V1_AUDIT_PASS " + f"checks={checks + 15} exact_rational_replay=1") + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index c0b9cb2d4..7b4aea03d 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -3164,6 +3164,73 @@ \section{Higher-codimension support-flag recursion} \path{experimental/audit_mca_full_lift_lower_aware_joint_core_charge_v1.py}. \end{remark} +\begin{corollary}[Core-dichotomy capped charge] +\label{cor:full-lift-core-dichotomy-capped-charge} +Fix an integer absorption cutoff $b$. At every line selected by the +recursive bank, split on its actual total core $g$. + +If $g\ge e+10-b$, every explanation of deficit at least $b+1$ lies on +that line, and the original family has size at most +\[ + P_e(b)+(N-m+1), \tag{CD1} +\] +where $P_e(b)$ is the exact weighted prefix through $b$. + +In the complementary branch, every peeled core has the individual cap +\[ + g_i\le G_e:=e+9-b. \tag{CD2} +\] +The lower-aware convex envelope remains valid after replacing its fill +ceiling by $G_e$ and its common budget by +\[ + S_r=\min\left\{rG_e,e+\binom{r+1}{2}c\right\}. \tag{CD3} +\] +\end{corollary} + +\begin{proof} +A selected line has at least $g-c$ core coordinates inside the gauged +direction support. An explanation with inside agreement size $h$ and +two line anchors shares at least $(g-c)+h-e$ of those coordinates. +Restriction injectivity therefore puts it on the line whenever +\[ + h\ge e-g+c+K=e-g+11. +\] +The high-core threshold in the statement makes this at most $b+1$, +proving (CD1). + +The integer complement of the high-core branch is (CD2). Under that cap, +sort the forced core lower bounds and greedily spend (CD3), filling each +coordinate only to $G_e$. The convex exchange in +Corollary~\ref{cor:full-lift-lower-aware-joint-core-charge} applies +unchanged and proves the capped joint charge. +\end{proof} + +\begin{remark}[Exact Mersenne core-dichotomy payment] +Take $b=65450$. On $130222\le e\le130226$, the high-core bound (CD1) +is at most $5161307<16777215$. In the complementary branch, +$e=130222,130223$ force fourteen threshold-18 lines and +\[ + 14\cdot9736-\binom{14}{2}\cdot5=135849>e. +\] +The next two supports force seventy threshold-16 lines and +\[ + 70\cdot2041-\binom{70}{2}\cdot5=130795>e. +\] +Thus every support $130222\le e\le130225$ is safe. + +At adjacent $e=130226$, the first threshold is 14 and has zero forced +core. After 14763 zero-lower-bound peels, the capped budget is +$545032556=8412\cdot64785+61136$, the charge is $3199542$, and the +next pigeonhole threshold is one. This is a method wall, not an unsafe +certificate; the Mersenne residual is now +$130226\le e\le1044241$. + +The endpoint replay and independent rational audit are +\path{experimental/verify_mca_full_lift_core_dichotomy_capped_charge_v1.py} +and +\path{experimental/audit_mca_full_lift_core_dichotomy_capped_charge_v1.py}. +\end{remark} + \begin{theorem}[Full-explanation lifted-rank gauge dichotomy] \label{thm:full-explanation-lifted-rank-dichotomy} Let \(C\leq\F^D\) have dimension \(K\), let diff --git a/experimental/notes/thresholds/mca_full_lift_core_dichotomy_capped_charge_v1.md b/experimental/notes/thresholds/mca_full_lift_core_dichotomy_capped_charge_v1.md new file mode 100644 index 000000000..b3b5c65dd --- /dev/null +++ b/experimental/notes/thresholds/mca_full_lift_core_dichotomy_capped_charge_v1.md @@ -0,0 +1,81 @@ +# Full-lift core-dichotomy capped charge + +## Scope + +This packet adds an exhaustive actual-core split to the Mersenne recursive +line bank. It pays `130222<=e<=130225`; it does not certify the adjacent row +unsafe or close the full Mersenne row. + +## High-core branch + +Fix `b_abs=65450`. A selected affine explanation line with actual total +core `g` has at least `g-c` core coordinates inside the gauged direction +support. Two line anchors synchronize every explanation of inside deficit + +```text +h>=e-g+c+K=e-g+11 +``` + +onto the same line. Thus, when `g>=e+10-b_abs`, the exact weighted prefix +through `b_abs` pays the low explanations and one absolute line cap pays the +rest. Across the five endpoint rows needed for payment and wall audit, + +```text +P_e(65450)+(N-m+1) <= 5161307 < 16777215. +``` + +## Capped complementary branch + +If no selected line enters the high branch, every actual core obeys + +```text +g_i<=G_e:=e+9-b_abs. +``` + +Retain the forced lower bounds and greedily spend the common core budget + +```text +S_r=min(rG_e,e+C(r+1,2)c) +``` + +up to the individual ceiling `G_e`. The same convex exchange as in the +lower-aware theorem proves that this maximizes the total removed-line charge. + +At `e=130222,130223`, fourteen threshold-18 lines give + +```text +14*9736-C(14,2)*5 = 135849 > e. +``` + +At `e=130224,130225`, seventy threshold-16 lines give + +```text +70*2041-C(70,2)*5 = 130795 > e. +``` + +## Adjacent wall + +At `e=130226`, the first threshold is 14 and its forced total core is zero. +All later thresholds are no larger. After 14,763 zero-lower-bound peels, + +```text +S_r=545032556=8412*64785+61136, +charge=3199542, +target=13577673, +base=13317279, +next threshold=1. +``` + +The bank no longer forces an actual line. This is a method wall, not an +unsafe certificate. The residual is + +```text +130226<=e<=1044241. +``` + +## Replay + +```bash +python3 experimental/verify_mca_full_lift_core_dichotomy_capped_charge_v1.py +python3 experimental/audit_mca_full_lift_core_dichotomy_capped_charge_v1.py +``` diff --git a/experimental/verify_mca_full_lift_core_dichotomy_capped_charge_v1.py b/experimental/verify_mca_full_lift_core_dichotomy_capped_charge_v1.py new file mode 100644 index 000000000..7763d38f0 --- /dev/null +++ b/experimental/verify_mca_full_lift_core_dichotomy_capped_charge_v1.py @@ -0,0 +1,226 @@ +#!/usr/bin/env python3 +"""Verify the M31 high-core/low-core capped-charge dichotomy.""" + +from __future__ import annotations + +import copy +from fractions import Fraction + + +R, D, K = 1048576, 67448, 6 +N, M, C = R + K, D + K, K - 1 +BUDGET, LINE, ABS_CUTOFF = 16777215, N - M + 1, 65450 + + +class Reject(ValueError): + pass + + +def raw_cap(e: int, h: int) -> int: + shortened, agreement = N - e, M - h + if agreement <= C: + raise Reject("prefix agreement") + johnson = agreement * agreement - shortened * C + if johnson > 0: + return shortened * (agreement - C) // johnson + gap = -johnson + balance = 2 * agreement * agreement - shortened * C + tangent = (shortened - agreement) ** 2 - (shortened - 1) * gap + if balance < 0 or tangent <= 0: + raise Reject("prefix cap") + return ((shortened - 1) * shortened * shortened * (agreement - C) + // (agreement * tangent)) + + +def prefix(e: int, cutoff: int) -> int: + values = [0] + [raw_cap(e, h) for h in range(1, cutoff + 1)] + for h in range(cutoff - 1, 0, -1): + values[h] = min(values[h], values[h + 1]) + return sum((values[h] - values[h - 1]) * (e // h) + for h in range(1, cutoff + 1)) + + +def choose_cutoff(e: int) -> int: + initial = 65304 + cutoff = initial + while cutoff < M: + h = cutoff + 1 + overlap = 2 * h - e + if (2 * h > e and overlap > C and + overlap * overlap > e * C): + break + cutoff += 1 + if cutoff > initial and cutoff < M: + cutoff += 2 + if cutoff >= M: + raise Reject("no bank") + return cutoff + + +def bank(e: int) -> tuple[int, int, int, int]: + cutoff = choose_cutoff(e) + weighted_prefix = prefix(e, cutoff) + groups = 0 + for h in range(cutoff + 1, M + 1): + overlap = 2 * h - e + denominator = overlap * overlap - e * C + if 2 * h <= e or overlap <= C or denominator <= 0: + raise Reject("bank guard") + groups += e * (overlap - C) // denominator + return cutoff, weighted_prefix, groups, weighted_prefix + M - cutoff - groups + + +def runs(values: list[int]) -> list[list[int]]: + answer: list[list[int]] = [] + for value in values: + if answer and answer[-1][0] == value: + answer[-1][1] += 1 + else: + answer.append([value, 1]) + return answer + + +def capped_charge(e: int, lower: list[int], cap: int) -> tuple[int, int, list[list[int]]]: + count = len(lower) + if count == 0: + return 0, 0, [] + budget = min(count * cap, e + count * (count + 1) * C // 2) + allocation = sorted(lower, reverse=True) + excess = budget - sum(allocation) + if excess < 0: + raise Reject("lower bounds") + for index, value in enumerate(allocation): + addition = min(excess, cap - value) + allocation[index] += addition + excess -= addition + if excess == 0: + break + if excess: + raise Reject("allocation") + rational = sum((Fraction(N - value, M - value) + for value in allocation), Fraction()) + return rational.numerator // rational.denominator, budget, runs(allocation) + + +def zero_charge(e: int, lines: int, cap: int) -> tuple[int, int, int, int]: + if lines == 0: + return 0, 0, 0, 0 + budget = min(lines * cap, e + lines * (lines + 1) * C // 2) + full, remainder = divmod(budget, cap) + if full == lines: + rational = lines * Fraction(N - cap, M - cap) + else: + rational = full * Fraction(N - cap, M - cap) + rational += Fraction(N - remainder, M - remainder) + rational += (lines - full - 1) * Fraction(N, M) + return rational.numerator // rational.denominator, budget, full, remainder + + +def paid(e: int) -> dict[str, object]: + cap = e + 9 - ABS_CUTOFF + absorb = prefix(e, ABS_CUTOFF) + cutoff, weighted, groups, base = bank(e) + thresholds: list[int] = [] + cores: list[int] = [] + insides: list[int] = [] + for removed in range(256): + charge, budget, allocation = capped_charge(e, cores, cap) + target = BUDGET - charge + threshold = (target - base + 1 + groups - 1) // groups + if threshold < 2: + raise Reject("wall") + numerator = threshold * M - N + core = (0 if numerator <= 0 else + (numerator + threshold - 2) // (threshold - 1)) + inside = max(core - C, 0) + thresholds.append(threshold) + cores.append(core) + insides.append(inside) + packing = sum(insides) - len(insides) * (len(insides) - 1) * C // 2 + if packing > e: + return { + "e": e, "core_cap": cap, "absorption_prefix": absorb, + "absorption_bound": absorb + LINE, "cutoff": cutoff, + "prefix": weighted, "groups": groups, "base": base, + "certificate": "core_packing", "lines": removed + 1, + "target": target, "charge": charge, "core_budget": budget, + "allocation_runs": allocation, + "threshold_runs": runs(thresholds), + "core_runs": runs(cores), "inside_runs": runs(insides), + "packing": packing, + } + raise Reject("limit") + + +def wall(e: int) -> dict[str, object]: + cap = e + 9 - ABS_CUTOFF + absorb = prefix(e, ABS_CUTOFF) + cutoff, weighted, groups, base = bank(e) + first = (BUDGET - base + 1 + groups - 1) // groups + if first * M - N > 0: + raise Reject("positive first core") + for lines in range(20000): + charge, budget, full, remainder = zero_charge(e, lines, cap) + target = BUDGET - charge + threshold = (target - base + 1 + groups - 1) // groups + if threshold < 2: + return { + "e": e, "core_cap": cap, "absorption_prefix": absorb, + "absorption_bound": absorb + LINE, "cutoff": cutoff, + "prefix": weighted, "groups": groups, "base": base, + "certificate": "threshold_one_wall", "lines": lines, + "target": target, "charge": charge, "core_budget": budget, + "full_caps": full, "remainder": remainder, + "first_threshold": first, "next_threshold": threshold, + "packing": 0, + } + if threshold > first: + raise Reject("monotonicity") + raise Reject("limit") + + +EXPECTED = { + "e130222": {"e": 130222, "core_cap": 64781, "absorption_prefix": 4180178, "absorption_bound": 5161307, "cutoff": 65516, "prefix": 12406922, "groups": 260580, "base": 12148280, "certificate": "core_packing", "lines": 14, "target": 16776980, "charge": 235, "core_budget": 130677, "allocation_runs": [[13785, 1], [9741, 12]], "threshold_runs": [[18, 14]], "core_runs": [[9741, 14]], "inside_runs": [[9736, 14]], "packing": 135849}, + "e130223": {"e": 130223, "core_cap": 64782, "absorption_prefix": 4180156, "absorption_bound": 5161285, "cutoff": 65516, "prefix": 12406366, "groups": 269480, "base": 12138824, "certificate": "core_packing", "lines": 14, "target": 16776980, "charge": 235, "core_budget": 130678, "allocation_runs": [[13786, 1], [9741, 12]], "threshold_runs": [[18, 14]], "core_runs": [[9741, 14]], "inside_runs": [[9736, 14]], "packing": 135849}, + "e130224": {"e": 130224, "core_cap": 64783, "absorption_prefix": 4180145, "absorption_bound": 5161274, "cutoff": 65517, "prefix": 12961350, "groups": 260602, "base": 12702685, "certificate": "core_packing", "lines": 70, "target": 16776111, "charge": 1104, "core_budget": 142299, "allocation_runs": [[3171, 1], [2046, 68]], "threshold_runs": [[16, 70]], "core_runs": [[2046, 70]], "inside_runs": [[2041, 70]], "packing": 130795}, + "e130225": {"e": 130225, "core_cap": 64784, "absorption_prefix": 4180124, "absorption_bound": 5161253, "cutoff": 65517, "prefix": 12960735, "groups": 269520, "base": 12693152, "certificate": "core_packing", "lines": 70, "target": 16776111, "charge": 1104, "core_budget": 142300, "allocation_runs": [[3172, 1], [2046, 68]], "threshold_runs": [[16, 70]], "core_runs": [[2046, 70]], "inside_runs": [[2041, 70]], "packing": 130795}, + "adjacent": {"e": 130226, "core_cap": 64785, "absorption_prefix": 4180114, "absorption_bound": 5161243, "cutoff": 65518, "prefix": 13575970, "groups": 260627, "base": 13317279, "certificate": "threshold_one_wall", "lines": 14763, "target": 13577673, "charge": 3199542, "core_budget": 545032556, "full_caps": 8412, "remainder": 61136, "first_threshold": 14, "next_threshold": 1, "packing": 0}, +} + + +def validate(expected: dict[str, object]) -> int: + checks = 29 + for e in range(130222, 130226): + if paid(e) != expected[f"e{e}"]: + raise Reject(str(e)) + checks += 21 + if wall(130226) != expected["adjacent"]: + raise Reject("adjacent") + return checks + 21 + + +def main() -> None: + checks = validate(copy.deepcopy(EXPECTED)) + mutations = [] + for name, key, delta in ( + ("e130222", "packing", -1), + ("e130225", "charge", 1), + ("adjacent", "target", -1), + ("adjacent", "full_caps", 1)): + changed = copy.deepcopy(EXPECTED) + changed[name][key] += delta + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise Reject("mutations") + print("MCA_FULL_LIFT_CORE_DICHOTOMY_CAPPED_CHARGE_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)} " + "paid=130222..130225 wall=130226") + + +if __name__ == "__main__": + main() From 7934f552d155a1e72c7b66a3032fc792000dde31 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 11:54:45 +0100 Subject: [PATCH 24/40] MCA: retain exact-layer line incidence --- agents.md | 2 +- experimental/agents-log.md | 17 ++ ...l_lift_exact_layer_slot_core_packing_v1.py | 86 +++++++++ experimental/grande_finale.pdf | Bin 1066103 -> 1099033 bytes experimental/grande_finale.tex | 80 ++++++++ ...l_lift_exact_layer_slot_core_packing_v1.md | 107 +++++++++++ ...l_lift_exact_layer_slot_core_packing_v1.py | 175 ++++++++++++++++++ 7 files changed, 466 insertions(+), 1 deletion(-) create mode 100644 experimental/audit_mca_full_lift_exact_layer_slot_core_packing_v1.py create mode 100644 experimental/notes/thresholds/mca_full_lift_exact_layer_slot_core_packing_v1.md create mode 100644 experimental/verify_mca_full_lift_exact_layer_slot_core_packing_v1.py diff --git a/agents.md b/agents.md index 43a0b3c3e..8d5758c5c 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson and Gram rungs lead to the exact-layer affine-line theorem, which pays every sparse-direction support `e=ceil((lambda*h-e)/(lambda-1))`; with the capped-core branch, three lines pay every support `e=130226..130236`. At adjacent `e=130237`, the bank forces only size-two shift-pair slots with inside core `807`; first-order packing reaches only `65529 int: + if not lower: + return 0 + count = len(lower) + budget = min(count * cap, e + count * (count + 1) * C // 2) + values = sorted(lower, reverse=True) + excess = budget - sum(values) + for index, value in enumerate(values): + addition = min(excess, cap - value) + values[index] += addition + excess -= addition + if excess == 0: + break + assert excess == 0 + value = sum((Fraction(N - core, M - core) for core in values), + Fraction()) + return value.numerator // value.denominator + + +def main() -> None: + checks = 0 + for (e, cutoff, groups, base, cap, expected_threshold, + expected_inside, expected_packing) in ROWS: + lower: list[int] = [] + for _ in range(3): + target = BUDGET - capped_charge(e, lower, cap) + threshold = (target - base + groups) // groups + assert threshold == expected_threshold + inside = (threshold * (cutoff + 1) - e + threshold - 2) // (threshold - 1) + assert inside == expected_inside + lower.append(inside) + checks += 4 + packing = sum(lower) - len(lower) * (len(lower) - 1) * C // 2 + assert packing == expected_packing > e + checks += 4 + + e, cutoff, groups, base, cap, lower, lines = ( + 130237, 65521, 1933560, 13961576, 64796, 807, 7583) + assert 2 * (cutoff + 1) - e == lower + maximum = max(s * lower - s * (s - 1) * C // 2 + for s in range(1, 1000)) + assert maximum == 65529 < e + budget = min(lines * cap, e + lines * (lines + 1) * C // 2) + lower_sum = lines * lower + full, remainder = divmod(budget - lower_sum, cap - lower) + value = full * Fraction(N - cap, M - cap) + value += Fraction(N - lower - remainder, M - lower - remainder) + value += (lines - full - 1) * Fraction(N - lower, M - lower) + charge = value.numerator // value.denominator + assert (budget, lower_sum, full, remainder) == ( + 143903917, 6119481, 2153, 16119) + assert charge == 882245 + target = BUDGET - charge + assert target == 15894970 + assert (target - base + groups) // groups == 1 + print("MCA_FULL_LIFT_EXACT_LAYER_SLOT_CORE_PACKING_V1_AUDIT_PASS " + f"checks={checks + 17} exact_rational_replay=1") + + +if __name__ == "__main__": + main() diff --git a/experimental/grande_finale.pdf b/experimental/grande_finale.pdf index a1ffc745c59c57aa5da8b005b7e3833268611d6f..d70e08e9587119db73ab80527c6543db5ec7376d 100644 GIT binary patch delta 495999 zcmV(`K-0hXh(Vc(L$Im|m+?#iDYMfF_zVNX+QO60794*o92|ln8+k68jy>Yh;G! z&`lmhX}(8FCfB^*F{Fz;_-GQ??^JlCPtk^|RTFgK!uE@iX!&^uY}FCHq9c!nG)G}B zNxd(-+uwinSxjGOaKptOhEuc26~S}NBD$qv>ImoRM;Y7PDa1`G>=i>8|ER*?8NOzS z!>TBYSQtvvKcB@W`1(pH?*XYEucbMFwP)4TzTKF1?Cuc!!+~&?1Tx7MOI&MRnr~*h zx@(LJ=J6CC%sr!mg}6vs*h^tkFd&1P^p6Pzov42{GYBWbBUivM`V5Gs+v_4~vN!eo z&B7eFr1aGi^EkVIWp%zSnuh&R8UJ!OvhK?Vx&@+x_h*5-%Athx@cW{r&Y~d7rXds) z?VirhGb28>v_1Av&wdxu+r4<^tD`(!r9qyCCOIa;xPwdJj9(6&@&B#~&prxeZe(+G za%GbVIt!C?7#0CEm+?#iDSxe5TXUnv5`N!bp+{7;D9i;I&YR=1bKP2J<4USdavsJo 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zI6JF#ZKb-2BB!`&Vr%J46O`uu9Dt@Cb^QuCP2LKYWplo(e!1fh4X;S&9!F6-ka z=GmBgB@OamT~&g?`&PbB{19TEXspdsoGX^~n~wVF$!7riw|NlUI>vo~0M6W~3*zn? zK?%lhH*Os$hbRQgC>Sg*7!cFC-tDCkJDR~S)`ar8AYUFuOh9s8Kx{=LfMjS5@-|(3 z@kjTM;k}_Rn|C)LUKu@?M3L9Pz5DsME;1%G=r8_R6Oei8u6K`Vd7zyTLQp;)Gl&#~ z6#$}Y=2f_S3!um_C-wkF-2~Hkn&quTwL=a48&vu@1lSbr*%a2<5_<+2|BQU!8WCJZ zbC(dbiWZ`N%-k{$O@tTz{||`%hml6{lw639+6QFUZf4B?dj{0q!_>{g$Ia3T UfsK`on}dxVfr?5>MH=D%0%5wYga7~l diff --git a/experimental/grande_finale.tex b/experimental/grande_finale.tex index 42ce965d0..1dcfb94fc 100644 --- a/experimental/grande_finale.tex +++ b/experimental/grande_finale.tex @@ -1968,6 +1968,117 @@ \section{Higher-codimension support-flag recursion} \path{experimental/verify_mca_sparse_direction_terminal_deficit_line_payment_v1.py}. \end{remark} +\begin{theorem}[Top-third exact-layer affine-line cap] +\label{thm:top-third-affine-line} +Retain the sparse-direction notation of +\cref{thm:sparse-direction-punctured-johnson-mca}. Assume \(e\ge K\), put +\[ + s=\left\lfloor\frac{e-K}{3}\right\rfloor, + \qquad H=e-s-1, +\] +and assume \(N-m>s\). For \(0\le r\le s\), let \(L_r\) be the number of +selected explanations with exact outside deficit \(h=e-r\). Then +\begin{equation} + L_r\le + \left\lfloor\frac{N-e-(K-1)}{m-e+r-(K-1)}\right\rfloor. + \label{eq:top-third-affine-line} +\end{equation} +Every explanation counted by \(L_r\) owns at most one selected slope. +\end{theorem} + +\begin{proof} +For an explanation \(a_i\) in this layer, let \(S_i\subseteq E\) be its +inside agreement set with its assigned slope. Then +\[ + |E\setminus S_i|\le r. +\] +For explanations at distinct slopes \(\gamma_i,\gamma_j\), the normalized +difference +\[ + p_{ij}=\frac{a_i-a_j}{\gamma_i-\gamma_j} +\] +is a degree-\(s\) gives \(N-e>A_r\), so the ratio increases with +\(g\); substituting \(g\le K-1\) proves +\eqref{eq:top-third-affine-line}. Finally \(r=K`, any three +explanations share at least `K` exceptional agreements. Restriction +injectivity synchronizes all normalized pair differences, so the entire +exact layer lies on one affine codeword line. + +With `c=K-1` and `n=N-e`, outside-core packing gives + +```text +L_r <= floor((n-c)/(m-e+r-c)). +``` + +Take all layers `0<=r<=floor((e-K)/3)` this way. The remaining lower +deficits retain the positive punctured-Johnson cumulative profile. A +uniform two-threshold estimate gives + +```text +prefix <= (e-1)J_floor(e/2)+J_H, +H=e-floor((e-K)/3)-1. +``` + +On both official rows, conservative endpoint substitution proves +`J_floor(e/2)<=31` and `J_H<=47` for every `e=d` or prove either deployed row. + +## Replay + +```bash +python3 experimental/verify_mca_sparse_direction_top_third_affine_line_payment_v1.py +``` + +The verifier checks the endpoint indices, raw Johnson fractions, grouped +floor sums, exact row comparisons, a sharp triple-overlap model, and hostile +mutations. diff --git a/experimental/verify_mca_sparse_direction_top_third_affine_line_payment_v1.py b/experimental/verify_mca_sparse_direction_top_third_affine_line_payment_v1.py new file mode 100644 index 000000000..5330e687d --- /dev/null +++ b/experimental/verify_mca_sparse_direction_top_third_affine_line_payment_v1.py @@ -0,0 +1,119 @@ +#!/usr/bin/env python3 +"""Audit the sparse-direction top-third affine-line branch payment.""" + +from __future__ import annotations + +import copy +from fractions import Fraction + + +ROWS = { + "KoalaBear": { + "R": 1048576, "d": 67472, "K": 14, + "budget": 274980728111395087, + "e": 67471, "s": 22485, "H": 44985, "u": 33735, + "A_u": 33751, "num_u": 35377329420, "den_u": 1125498331, + "A_H": 22501, "num_H": 23580691920, "den_H": 492663331, + "n": 981119, "A": 15, "line": 9405342, + "total": 11496959, "floor": 67472, "ceiling": 1044238, + }, + "Mersenne-31": { + "R": 1048576, "d": 67448, "K": 6, + "budget": 16777215, + "e": 67447, "s": 22480, "H": 44966, "u": 33723, + "A_u": 33731, "num_u": 35364476532, "den_u": 1132537451, + "A_H": 22488, "num_H": 23575269106, "den_H": 500467234, + "n": 981135, "A": 7, "line": 9405365, + "total": 11496238, "floor": 67448, "ceiling": 1044241, + }, +} + + +class Reject(ValueError): + pass + + +def grouped_floor_sum(numerator: int, first: int, last: int) -> int: + total = 0 + x = first + while x <= last: + quotient = numerator // x + end = min(last, numerator // quotient) + total += quotient * (end - x + 1) + x = end + 1 + return total + + +def finite_control() -> int: + E = set(range(8)) + sets = [E - missed for missed in ({0, 1}, {2, 3}, {4, 5})] + if len(set.intersection(*sets)) != 2: + raise Reject("triple overlap") + blocks = [{0, 1, 2}, {0, 3, 4}, {0, 5, 6}] + core = set.intersection(*blocks) + stripped = [block - core for block in blocks] + if any(left & right for i, left in enumerate(stripped) for right in stripped[i + 1:]): + raise Reject("outside packing") + return len(sets) * len(blocks) + + +def validate(rows: dict[str, dict[str, int]]) -> int: + checks = 0 + for name, row in rows.items(): + R, d, K = row["R"], row["d"], row["K"] + N, m, c = R + K, d + K, K - 1 + e = d - 1 + s = (e - K) // 3 + H = e - s - 1 + u = e // 2 + if (row["e"], row["s"], row["H"], row["u"]) != (e, s, H, u): + raise Reject(f"{name}: indices") + if N - m <= s: + raise Reject(f"{name}: outside slack") + for h, cap, suffix in ((u, 31, "u"), (H, 47, "H")): + A = m - h + numerator = N * (A - c) + denominator = A * A - N * c + value = Fraction(numerator, denominator) + if row[f"A_{suffix}"] != A: + raise Reject(f"{name}: A_{suffix}") + if row[f"num_{suffix}"] != numerator or row[f"den_{suffix}"] != denominator: + raise Reject(f"{name}: Johnson fraction") + if value.numerator // value.denominator > cap: + raise Reject(f"{name}: Johnson cap") + checks += 4 + n, A = N - e, m - e + line = grouped_floor_sum(n - c, A - c, A + s - c) + if (row["n"], row["A"], row["line"]) != (n, A, line): + raise Reject(f"{name}: line sum") + total = (d - 2) * 31 + 47 + line + if row["total"] != total or total > row["budget"]: + raise Reject(f"{name}: total") + if row["floor"] != d: + raise Reject(f"{name}: residual floor") + checks += 5 + return checks + + +def main() -> None: + checks = validate(copy.deepcopy(ROWS)) + finite_control() + mutations = [] + for name, key in (("KoalaBear", "line"), ("Mersenne-31", "total")): + changed = copy.deepcopy(ROWS) + changed[name][key] += 1 + try: + validate(changed) + except Reject: + mutations.append(True) + else: + mutations.append(False) + if not all(mutations): + raise AssertionError("mutation controls") + print( + "MCA_SPARSE_DIRECTION_TOP_THIRD_AFFINE_LINE_PAYMENT_V1_PASS " + f"checks={checks} mutations={sum(mutations)}/{len(mutations)}" + ) + + +if __name__ == "__main__": + main() From 1913917f0a8fa3dd36668230f5f51279d7daffd4 Mon Sep 17 00:00:00 2001 From: AllenGrahamHart Date: Thu, 13 Aug 2026 06:17:19 +0100 Subject: [PATCH 10/40] MCA: extend full-lift wall by common-core lines --- agents.md | 2 +- experimental/agents-log.md | 16 ++ experimental/grande_finale.pdf | Bin 1027758 -> 1035591 bytes experimental/grande_finale.tex | 80 +++++++++ ...l_lift_top_third_common_core_payment_v1.md | 52 ++++++ ...l_lift_top_third_common_core_payment_v1.py | 162 ++++++++++++++++++ 6 files changed, 311 insertions(+), 1 deletion(-) create mode 100644 experimental/notes/thresholds/mca_full_lift_top_third_common_core_payment_v1.md create mode 100644 experimental/verify_mca_full_lift_top_third_common_core_payment_v1.py diff --git a/agents.md b/agents.md index 1efe65c48..818ac4b71 100644 --- a/agents.md +++ b/agents.md @@ -80,7 +80,7 @@ At this snapshot: - The cumulative M31 rank-seven compiler now reaches `Q=147594` locally. Its adjacent `Q=147595`, `k=4981` survivor is reduced to a varying proper-`G`, zero-excess cross-cofactor incidence terminal requiring an aggregate cap `2157928`; no v4 atom, higher-rank theorem, or row closure follows. - The KoalaBear equality-wall PR stack proves local normalization, a `Q=6,u=2` geometry reduction, and exclusion of 60 labelled `P3+C3` cases, but retains 405 labelled conic cases and lacks global ownership/add-back. The separate column-far transverse-secant certificate is per fixed union only. No large `U_paid` claimed by an abstract candidate-record compiler is banked; the live active value remains `null`. - The exact adjacent fixed-`G` Hahn relaxation has optimum `20737821.0968...`, above the list target, and its complementarity identity shows the proposed selection-gap hypothesis is target-equivalent. This cuts the ordinary pairwise-distribution route without proving an unsafe list or moving the M31 row. -- The affine-span transverse MCA compiler `thm:affine-span-mca` is refuted even under its printed direction-separation hypothesis. An exact `GF(1009)` rank-one family has 31 pair-noncontained bad slopes against the claimed bound 23, with direction agreement `20=992852` for KoalaBear and `e>=1037876` for Mersenne. In the full-lift branch, `W=C+` has `d_1(W)=e` and every higher generalized weight MDS-sharp. The punctured Johnson, centered Gram, mean-centered, and terminal-line rungs first move the low-support walls to `e=64048` and `e=65455`. The top-third affine-line theorem then pays every remaining sparse-direction support `e` has `d_1(W)=e` and every higher generalized weight MDS-sharp. 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