MCA: route rank-eleven components and close rank-nine prefix through K'=87 (post-#1169)

[AllenGrahamHart]

Stack and review boundary

Stacked on exact PR #1169 head b4bad860750f91955dbaead8f2b5a0fdef1f1343.

b4bad860750f91955dbaead8f2b5a0fdef1f1343..d296510e806d2bafbb08cb0c2fc25be6a3b03a50

This is one cumulative rank-eleven dense-locator / base-field-normalized split-pencil packet. It refines the finite component ledger and introduces no new first-match atom. Concurrent PRs #1171, #1172, and #1173 address the complementary minimizing-pair/rich-flat router; this packet neither depends on nor supersedes them.

Latest extension: raw-clipped coupling and K'=87

Commit d296510e imports two proved prize-DAG nodes from public source commit b869fb9dd2e740e323bee74e7ee72475905901f5: a reusable raw-clipped fixed-union adjacent-support theorem and the exact K'=87 payment applying it. The closed rank-nine prefix moves from K'=10..86 to K'=10..87; K'=88 is the first open rank-nine row.

Reusable structural theorem

Retain the proved base-field-normalized fixed-union census. If independent arguments bound selected eleven-set incidences from support-d and support-d+1 circuits by U_d,U_(d+1), divide by the exact extension factors and floor:

X = floor(U_d / C(m'-d,11-d))
Y = floor(U_(d+1) / C(m'-d-1,10-d)).

Impose these circuit caps simultaneously with every fixed-union stratum inequality:

b_i x_i + a_i y_i <= A_i,    x_i <= L_i,
sum x_i + x_0 <= X,          sum y_i + y_0 <= Y.

The selected-incidence contribution is bounded by the floor of the resulting rational weighted maximum. The proof gives two descriptions of the same piecewise-linear envelope: fix the support-d total and fill in increasing b_i/a_i, or fix the support-d+1 total and fill in increasing a_i/b_i. Only support-disjoint adjacent-pair replacements may be added; overlapping pairs are not composed.

The compact verifiers exercise both orientations on a K'=87, (u,g,d)=(34,6,5) specialization and independently recover

42623778408284696740784253522991572305773969600 / 3.

Exact finite census

The support-2/3 partition is one ordinary lane plus offsets 1..76.

ordinary source units                         542,840
ordinary raw rows                           3,799,880
ordinary geometry rows per implementation  2,940,875

offset source units per implementation     17,801,784
offset raw rows per implementation        124,612,488
raw-safe units                              17,290,107
raw-unsafe units                               511,677
unsafe offsets                                    1..43
fully safe offsets                               44..76

Six contiguous paired raw-clipped shards exhaust all 43 unsafe offsets. The two implementations agree on 14,388,660 source units, all 511,677 unsafe units, and 77,179,660 carrier profiles each. Every raw-clipped value is at most the global raw-safe leader

s2=55/s3=46/s4=37/s5=30/offset9/c6F/c7F/c8F/c9F

with premium 41460899125475443837881046685022762331499044695.

Exact K'=87 payment

Quantity	Exact value
Completion premium	41460899125475443837881046685022762331499044695
Safe premium ceiling	41460914669043067085305042221812436226076443389
Strict premium margin	15543567623247423995536789673894577398694
Rank-nine marks	39251822161232243278275316442674108186111685491076612666319890850
Kernel capacity	101428358503090898644576011950562401247281266078921010827210
Full-rank capacity	920959461652781618758411787796949507196105181499688874690276771
Total capacity	921060890011284709657056363808900069597352462765767795701103981
Required incidence	921060967723676391242250303252610946492991464556444164407248882
Strict component gap	77712391681585193939443710876895639001790676368706144901

Cumulative finite payments

10; 11; 12; 13; 14..21; 22..40; 41; 42; 43..44; 45;
46..53; 54..59; 60..70; 71; 72..73; 74..78; 79..82; 83; 84; 85; 86; 87.

The current route boundary is

10..21:            no rank-eleven component target
22..87:            rank eight only
88..15528:         rank nine or rank eight
15529..22525:      rank eight only, with chart-local payment fenced
22526..37995:      dense-owner chronology terminal only
37996..1048576:    no rank-eleven component target

No adjacent-row extrapolation is claimed. The next finite action is to test the raw-clipped envelope at K'=88 while seeking a symbolic continuation in K'.

Verification

• K'=87 compact primary replay: PASS.
• K'=87 hostile-mutation suite: PASS, 10/10 rejected.
• K'=87 independent replay: PASS; reverse theorem orientation, coverage, ceiling remainder, margin, gap, prefix, and first-open row reconstructed.
• Parent K'=86 primary, hostile-mutation, and independent replays: PASS unchanged.
• Strict JSON parse, public source pin, node trees, source contract, stale-boundary scan, and git diff --check: PASS.
• Grande Finale: three 1 GB RAM-guarded pdflatex passes, 181 pages.
• PDF SHA-256: 0686647b70fb2155b28591ea381d6a9683947591dda1fb4a5a9a56859797b207.
• Manifest SHA-256: ec81aa702181ba7a683e9f2a1afeb942a968c7d0232d6e3e7f3e44669d3f0f8a.
• Primary verifier SHA-256: da8e97ab5a3b8778acd4415b598e5086f8bd5942c275ad4644c45843cfbc0b40.
• Independent verifier SHA-256: 4f0c45d41bd6282fb167fe577e2428ecc9df11aab03ec81f37d1156bff9ad9be.

The final PDF retains pre-existing duplicate equation-destination and underfull-box warnings. This extension adds no unresolved-reference or rerun warning after the final pass.

Latest provenance

Object	Pin
Prize-DAG source commit	b869fb9dd2e740e323bee74e7ee72475905901f5
Raw-clipped theorem-node tree	f1e826876806f3e30287f4e64d86f655714af081
K'=87 row-node tree	1c00acd2b2293b97623bbbec695d98aa8689d218
Source-contract SHA-256	fc0699db0b33a5ff6a6fe04b918e5cbd5eefe3e9b29b9acd7816880b31cf7c88
Immediate K'=86 packet SHA-256	d0869e5755252d08a59bfe763fd33c5032796ebc67ca410c7432d98d05762072
Merged raw-clipped capture SHA-256	6f8064320850e0009c18c967e2b61ec5b4d77c51e1c2afb4bee6fc41921e5cd8

Scope and nonclaims

This is a proved reusable theorem plus a proved one-row extension with zero deployed-v4 ledger movement. It does not pay rank nine on 88..15528, rank eight on 22..22525, construct chronology, pay aggregate error rank eleven, close KoalaBear, close LIST, close MCA, or close either prize problem.

[AllenGrahamHart]

Quotient-pair split-pencil extension pushed in 24f23478 (research source AllenGrahamHart/rs-mca-prize-dag@f557775d5).

This is deliberately a conditional route cut: it consumes the fully printed 520-type coprime-direction interface and does not claim that the present rank-eleven compiler already reaches that interface.

The exact conclusions are:

affine scalar-plane occupancy <= 218
dimension-three common received-pair core >= 407831
minimum shortened incidence slack = 178

at plane occupancy 218:
  2044 <= K_prime <= 5025
  represented projective directions >= 210
  aggregate unused direction degree <= 41736
  aggregate root saturation > 0.9618
  dual non-rich line-pair budget <= 1603

The equality branch is not excluded over the official field. Its next theorem is a quotient-periodic rational-fiber classification/payment or a finite-characteristic arrangement theorem valid for this represented polynomial pencil. No rank-eleven payment, active-v4 movement, or row closure is claimed.

Replay:

python3 experimental/scripts/verify_kb_mca_rank11_quotient_pair_plane218_v1.py
python3 -O experimental/scripts/verify_kb_mca_rank11_quotient_pair_plane218_v1.py
python3 experimental/scripts/verify_kb_mca_rank11_quotient_pair_plane218_v1.py --tamper-selftest
python3 experimental/scripts/verify_kb_mca_rank11_quotient_pair_plane218_v1_independent.py

Both implementations scan all 2,982 endpoint states; normal/optimized runs agree, 12/12 hostile mutations are rejected, and the optional source replay verifies the frozen source-node trees and file hashes.

[AllenGrahamHart]

Follow-up e66f6da3 extends the conditional plane-218 packet with an exact pure-power quotient-periodic router. Under the additional hypothesis that the residual direction pencil is projectively equivalent to (X^e,1), endpoint capacity leaves only e=2048,4096: degree 2048 forces the singleton shortened row K-prime=2049, all 218 directions and lines, and at most 72 missing root slots; degree 4096 is confined to 4097..4237, has at least 211 directions, and at most 28744 missing slots. The contract keeps both actual-pure-power and survivor-payment claims false. Normal/optimized primary and independent replays pass, source replay checks the separately pinned extension node, and 16/16 hostile mutations are rejected. No source-interface proof, active-v4 movement, rank-eleven payment, or prize closure is claimed.

[AllenGrahamHart]

Follow-up extension: dimension-three rich-plane recurrence sharpening

Packet commit: 23b60807
Source prize-DAG commit: d79701f94add274e6c7fbf2f4744980d77817f4b
Source node tree: 40b4d860ac0cd50dbe2c43e883ede75b0b85ac84

Under the PR's existing conditional 520-type coprime-direction split-pencil interface, three affine owner planes containing at least 189 types would cover at least 3*189-3*15=522>520 types. Hence at most two such planes exist. Each spans two independent residual directions, so its coordinate recurrence is at most K'-2. The exact ledger

520(67470+K') <= 188(1048576+K')+60(K'-2)

forces K'<=595763 and raises the dimension-three common-core floor from 407831 to 452813. Capacity slack is 232 at K'=595763; the adjacent row has deficit 40.

Replay: normal/optimized primary PASS; normal/optimized independent PASS; pinned source replay PASS; 20/20 hostile mutations rejected.

Status remains PROVED CONDITIONAL ROUTE CUT / ZERO DEPLOYED LEDGER MOVEMENT. This does not derive the 520-type interface, strengthen the dimension-four 407831 branch, pay the shortened family, exclude the 218-plane endpoint, pay rank eleven, or close KoalaBear.

[AllenGrahamHart]

Extended the packet at 6b1037e1 with the independent dimension-three pair-overlap moment floor.

New conditional output under the packet source interface:

• balanced sum_x binom(d_x,2) excludes every residual dimension 3 <= k' <= 4835;
• exact adjacency is gap(4835)=-2110, gap(4836)=115260;
• together with the rich-plane ceiling, 4836 <= k' <= 595763, equivalently 452813 <= |J| <= 1043740.

The machine-readable contract explicitly sets shared-pair-core payment transport to false. The 87 rows 4836..4922 only overlap that separate threshold numerically: its premise quantifies over every low-margin minimizing pair in the full post-near family, while this packet controls the retained synchronized quotient-type interface.

Replay: normal/optimized primary PASS, normal/optimized independent PASS, source custody PASS at prize-DAG 473f41afc6b76d747e534cb8e509a0353dcde3aa, and 26/26 hostile mutations rejected. No deployed ledger movement.

[AllenGrahamHart]

Extended the same certificate at 6186c7b1 with the complete type-population consequence (conditional on the packet family being the complete retained quotient-type family carrying M=255011043).

The plane-incidence and pair-overlap inequalities have doubled compatibility cross-product
-109 q(q-218)(619q-1962831).
It changes sign between q=3170 and q=3171, proving:

• 520 <= q <= 3170;
• one retained pair type owns at least ceil(255011043/3170)=80446 records;
• at q=3170, 4960 <= k' <= 4982 and at least 985,788 residual coordinates have owner multiplicity 218;
• q=3171 has exact doubled cross-product deficit 18,372,095,406.

This remains a dense-owner/saturated-plane router, not a payment. Normal/optimized primary and independent replay pass, source custody passes at prize-DAG 1d52ff3013b6ab4e94f39cf9d6627f7562d65cf8, and 32/32 hostile mutations are rejected.

[AllenGrahamHart]

Pushed dab75a23, extending the finite (Q) / base-field-normalized split-pencil packet through the exact q=3170 population endpoint.

New exact reduction:

• 4960 <= k' <= 4982 and the full-coordinate floor force 339..358 distinct 218-point owner planes.
• Balanced plane incidence forces at least 22,752 plane pairs meeting in a saturated 15-point line, hence at least 217 distinct saturated lines; every represented line recurs on at least k'-2609 >= 2351 residual coordinates.
• Using the internal direction bank of each full plane gives 41,746 <= R <= 47,836 represented saturated projective directions.
• Every represented direction has at least 2351 roots, aggregate unused direction degree is at most 30,203,244, and aggregate root saturation is at least 5,750,430 / 6,589,409 > 0.8726.

This is deliberately still a router, not a payment claim: the packet does not yet classify the near-full bank as quotient-periodic, force a common polynomial factor, or close rank eleven.

Verification rerun after final manifest hashes: primary under python3 and python3 -O; independent recomputation under both modes; 40/40 hostile mutations rejected; source replay passed against prize-DAG commits f0a13cc6e33399aa8192bf4879b9a9e7941371e3 and 1db90bbbef8c8e31b881de04dc9cedb387728c0f with frozen node-tree hashes.

[AllenGrahamHart]

Pushed e3039cf8, adding an exact projective-image router after the saturated q=3170 direction bank.

The normalization point is explicit: the shortened common received core need not be the full polynomial gcd. If G is the residual scalar-space gcd, every remaining official root of G has owner multiplicity zero. The exact 218-fold occupancy deficit therefore gives at most 310 such roots. After dividing by G, every represented primitive direction retains at least 2,041 official roots.

For the resulting basepoint-free map phi:P^1 -> P^2, if d is polynomial degree, c image-curve degree, and e map degree, pullback degree gives d=ec and scalar dimension three gives c>=2.

• c=2: after a base-field projective change, W_hom=span(A^2,AB,B^2) for one rational map A/B of degree 1021..2490; all 41,746 or more represented directions become binary quadratics in that same map.
• c>=3: one normal has at most floor((K'-1)/3) full preimages, forcing 597..633 distinct full evaluation normals on the 23 endpoint rows.

Neither branch is excluded, quotient-classified, or paid. This narrows the next theorem to a one-function split-fiber classification in the conic branch and a distinct-normal curve-incidence theorem in the higher-image branch.

Verification after final manifest hashes: primary and independent normal/optimized replays pass; 47/47 hostile mutations rejected; source replay passed against prize-DAG commit 121e75fa14d2b58968ca398f352437e1357b16fb, node tree 0537b44d0ff8240f47d08942467febeb6ca57cd6.