[MCA] Route rank eleven through anchored rich flats

[scottdhughes]

Summary

This is a one-commit successor to PR #1172, stacked on exact head
193b7bf99a5cc7ccea042f25677e698d9f988eee.

It attacks the rank-eleven rank-two/common-factor terminal by anchoring one
actual low-margin record and partitioning every represented minimizing pair by
the rank-one or rank-two row space of its coefficient-matrix difference from
the anchor.

For each represented row space, the anchor supplies at least c=2A-n labeled
evaluation columns in its annihilator. If no proper annihilator flat is heavy,
a greedy ordered-basis count bounds the number of row spaces. Rank-one groups
are paid by #1171's common-core-aware ray cap; rank-two groups are paid by the
exact dimension-two interleaved pair-list cap.

At the optimized cell

tau = 1547
h   = 42452

the exact accounting is

rank-one groups        60,010,642,445,729,852
rank-two groups       146,093,034,425,737,644
anchor pair                         982,651
low total             206,103,676,872,450,147
high tail              68,875,044,016,173,272
near add-back                     134,944
------------------------------------------------
total                  274,978,720,888,758,363
budget                 274,980,728,111,395,087
slack                    2,007,222,636,724

The adjacent threshold h=42453 is over budget by
17,108,854,816,460. An exhaustive exact scan of every legal cutoff finds
that 42452 is the global maximum payable threshold, attained at cutoffs
1547, 1548, and 1549.

Exact successor terminal

Every over-budget line emits an actual represented row space U of rank one
or two and a strictly larger direction subspace W such that

U < W <= C'
dim(W) >= dim(U)+1

and every polynomial in W vanishes on at least 42,453 common actual
coordinates contained in the anchor's good size-m support.

Consequently:

• rank-one case: dim W >= 2 and common-factor degree at least 42,453;
• rank-two case: dim W >= 3 and common-factor degree at least 42,453.

In the rank-two case the original plane additionally retains the stronger
anchor-overlap floor 131,850. Division by the emitted squarefree locator
places the residual direction space inside degree <1,006,123 polynomials.

Scope

• pays the complete anchored 42,452-transverse branch;
• active-v4 ledger movement: 0;
• rank-eleven payment: no;
• KoalaBear closure: no;
• result: direct branch payment and structural route cut.

The packet does not assert that all remaining pair types lie in the emitted
larger subspace, that different emitted locators synchronize, or that the new
terminal already has a chronology owner.

Verification

• canonical payload:
  f08059cdb1593c8b22355e8f512da326d882106e2c1cd7c75a02263bf59351cc;
• primary Python normal and optimized modes: PASS;
• independent Python normal and optimized modes: PASS;
• hostile mutations: 6/6 rejected;
• primary finite multiset/matroid controls: 2,077;
• independently implemented finite controls: 1,286;
• exact selected-cell Wolfram replay: PASS;
• exhaustive all-cutoff scan reproduced independently;
• manifest and all shipped file hashes: PASS;
• adversarial mathematics review: GREEN for the stated result;
• certificate/custody review: GREEN.

No floating-point quantity, external binary, mutable web source, or external
literature theorem is load-bearing. Guruswami--Kopparty subspace designs were
reviewed as motivation for the next Wronskian attack, but their construction-
specific hypotheses are not imported here.

Dependency and next theorem

This PR must integrate after #1172.

The next exact target is factor-flag synchronization: either route many emitted
spaces through one common locator to a chronology-safe shortened correction
owner, or prove that varying locators force a Wronskian/subspace-design
multiplicity collision. A per-edge cancellation or sum of local factor charges
is not authorized.

[AllenGrahamHart]

I compared this terminal against the common-core/K' machinery and audited the most direct proposed continuation: serially repeat the ordered-basis rich-flat census on the emitted factor flag.

Two scope points are load-bearing:

• the 42,453-point zero set is common to the anchor and one represented U-group, not to the whole residual family, so it cannot be inserted directly as a line-global shortening core;
• a nontransverse U-group can be promoted to a larger affine W-container, but independently charging those containers consumes a fresh ordered-basis gap.

At tau=1547 the complete low-record budget after high, anchor, and near charges is

L_low = 206105684094104220.

For dimension q and gap Delta, I used exactly the PR's census

floor(m_fall_(10-q) / Delta^(10-q))

times R_1=8147918, or for q>=2

R_q=(n-A) floor(C(n-K+q,q)/C(A-K+q,q)),

with the existing sub-square common-support interleaving collapse.

The exact shared first-rung scan independently reproduces the PR boundary:

Delta=89398, h=42452,
charge=206103676871467496,
slack=2007222636724;
h=42453 is over by 17108854816460.

But exhaustive scans of every positive two-rung gap allocation (the two gaps sum to at most c0+1=131851) give:

q=1 -> 2:
  minimum 2539543014780268202 at (64305,67546)
  excess over L_low 2333437330686163982

q=2 -> 3:
  minimum 3232479920013973566 at (66671,65180)
  excess over L_low 3026374235919869346

Each scan grants that one survivor branch the entire low budget and ignores its sibling, so self-similar independent iteration fails decisively. This does not rule out the PR's live successor ideas: common-factor synchronization, a Wronskian/subspace-design collision, or chronology-safe cross-bucket ownership. It says one of those genuinely new couplings is necessary.

Proof packet and two independent exact replays:
AllenGrahamHart/rs-mca-prize-dag@9add2d131

Modal exact scan: ap-QtAAm9XKW8KPLLjrG6ShS5 (one 256 MiB worker).

[AllenGrahamHart]

A quantitative strengthening drops out of the same envelope and may be the more useful successor interface.

An unsafe line has at least B_*+1 records, while the complete paid #1173 envelope is 274978720888758363. Therefore the union of non-42452-transverse groups has record mass at least

E_rich = 274980728111395088 - 274978720888758363
       = 2007222636725.

Two consequences are exact.

1. Every original rank-one/rank-two row-space group has at most R_2=247628052 records (rank one is smaller), so there are at least

ceil(E_rich/R_2)=8106

distinct represented nontransverse row spaces.

2. For each rich U, extend the selected proper flat to a hyperplane of U^perp. Its orthogonal is a dimension-(rank(U)+1) space W that still vanishes on the same >=42453 actual coordinates. Assign the complete U-group to W and merge identical W before charging. Dimension-two and dimension-three merged buckets have caps

R_2=247628052,  R_3=3953204973,

so their distinct counts satisfy

R_2 B_2 + R_3 B_3 >= 2007222636725,
B_2+B_3 >= ceil(E_rich/R_3)=508.

All pair types in one merged bucket agree with the anchor/received pair on the union of its assigned rich zero sets. Thus every bucket is an exact group-local input to the common-core shortening adapter; no varying-core summation is used.

This turns the successor problem from “one rich flat exists” into a typed family of at least 508 distinct dimension-two/three rich containers (and 8106 original U-spaces). A synchronization/Wronskian theorem can target that finite population directly.

Proof packet and independent replays:
AllenGrahamHart/rs-mca-prize-dag@66e4c65f7

Modal replay: ap-vICnkUlgtQNsXJtC9zXZQH (one 256 MiB worker).

[AllenGrahamHart]

Correction after dependency audit: the first version of this comment used the code dimension K=1,048,576 as the incidence universe. The zero sets are only proved to lie in the anchor-good set G_0, with |G_0|<=m=1,116,048. The qualitative result survives, but the correct weaker constants are:

• From any 508 promoted dimension-2/3 containers, choosing 42,453 actual zero coordinates in each gives total incidence 21,566,124 = 19m + 361,212. Hence one coordinate lies in at least 20 containers.
• The exact balanced-degree lower bounds are 197,707,236 pair incidences and 1,143,217,764 triple incidences. Averaging forces a pair with at least 1,536 common actual zeros and span dimension at most 6, and a triple with at least 53 common actual zeros and span dimension at most 9.
• Pigeonholing a same-dimension subfamily of 254 containers gives typed thresholds: one coordinate in 10 containers, a pair with at least 1,458 common zeros and span dimension at most 2r, and a triple with at least 45 common zeros and span dimension at most 3r.

Important nonclaim: these are pair/triple intersections of labelled actual zero sets. They do not synchronize all 508 locators and do not supply a global chronology/owner.

Corrected proof node and two exact-integer verifiers: AllenGrahamHart/rs-mca-prize-dag@279e0a74c

The superseded constants 21/1640/61 and 11/1562/52 should not be used.

[AllenGrahamHart]

A stronger collective payment follows by charging the complete nontransverse union once inside its total correction span.

For cutoff tau=1549, threshold h=42451, every unsafe line already has collective nontransverse span at least seven. The exact ledger is

transverse envelope                 274947501264373505
dimension-six union cap              15909196289385
total                               274963410460662890
budget                              274980728111395087
slack                                   17317650732197

and h=42452 is over by 1804196591101. This also strengthens the survivor population to at least 134181 represented row spaces and 8406 promoted rich containers.

Reoptimizing at dimension nine gives the sharper terminal. At (tau,h)=(1679,38384),

transverse envelope                 209812758437679617
M_9                                    66298487937
R_9=(n-A)M_9                         65157026870188671
total                               274969785307868288
slack                                   10942803526799

The field guard M_9^2 < 2130706433^6 holds, so ordinary affine-span plus the common-support interleaving collapse applies to the complete nontransverse pair family whenever its total direction span has dimension at most nine. Therefore every unsafe survivor satisfies

dim(V_nt)=dim(C')=10.

Its nontransverse mass is at least 65167969673715471, forcing at least 262093370 represented row spaces and 16384884 distinct promoted dimension-two/three containers, each with at least 38385 common actual zeros. The adjacent h=38385 ledger is over by 2062328934603.

This exhausts this payment family: the globally best dimension-ten one-container formula is tau=872,h=0, still over by 773076621594690156. Thus the next theorem can assume full ten-dimensional span and must use factor-flag synchronization, Wronskian/subspace-design collision, or chronology; further bounded-span tuning cannot close the branch.

One exact multiplicative subclass is already excluded: if every promoted container lies in g_i B for g_i in one two-dimensional factor space and fixed dim B<=3, their total span is at most six, contradicting the survivor theorem. This is not identified with the general base-field-normalized split-pencil census.

Proof nodes, independent exact scans, and hostile mutations:

• AllenGrahamHart/rs-mca-prize-dag@0badb9680
• AllenGrahamHart/rs-mca-prize-dag@31f1ba744

Nonclaims: rank eleven and KoalaBear remain unpaid; no global locator, first-match owner, or chronology is inferred.

[AllenGrahamHart]

A scoped factor-flag continuation is now available on top of the full-span terminal.

Assume the exact model

Cprime=span(PB),  dim P=2,  dim B=5,
W_i <= g_i B

for every promoted dimension-two/three rich container. In the primitive base-free branch, splitting at factor-root cutoff T=408 and counting residual subspaces by ordered labelled bases gives

factor-heavy classes       2763267104042675
residual dim-two classes  11330947785633956
residual dim-three classes 51071925374444624
union                     65166140264121255
transverse envelope      209812758437679617
------------------------------------------------
total                    274978898701800872
slack                         1829409594215

Thus every unsafe exact 2 x 5 factor flag emits either:

1. a common base coordinate of P or B, which is exactly the existing nonempty line-global-core route (C) because all of Cprime vanishes there; or
2. a deeper residual subspace vanishing on at least 18166 actual anchor-good coordinates.

The threshold is sharp for this charge family. A genuine second residual band aimed at 18167 or above contributes at least

187184 R_4 + 3381 R_6 = 66303977459889028,

already over the entire residual allowance by 1136007786173558 before factor and transverse charges. If that band is empty, the original all-cutoff optimizer gives the same 18166 maximum. Serial ordered-basis iteration is therefore exhausted; the remaining useful theorem is genuinely the factor-presentation/synchronization, Wronskian/subspace-design collision, or chronology coupling named in this PR.

Proof nodes, two independent exact replays, and hostile mutations:

• AllenGrahamHart/rs-mca-prize-dag@dc29ef241
• AllenGrahamHart/rs-mca-prize-dag@fe73b5407
• AllenGrahamHart/rs-mca-prize-dag@4fb9f49ab

Nonclaims: no existence theorem for the 2 x 5 presentation, no payment of (C), no general split-pencil census, and no rank-eleven closure.