[MCA] Route affine error rank by support-local transversality

[scottdhughes]

Summary

This is a ready successor to draft PR #1165, restacked on exact #1165 head
b6d30ef4f5ff966665b7672e1780a637509873a4.

It replaces the false final occupancy factor in the old affine-span MCA
compiler by the exact support-local transversality margin, proves a reversible
arbitrary-rank error-to-explanation gauge, and conditionally pays every
intrinsic post-near KoalaBear family through affine error rank 9. At ranks
10, 11, and 12 it emits actual identical-support direction-exception
terminals of size at most 12, 387, and 12049.

The packet also preserves the smooth GF(257) regression that refutes the old
formula. It makes zero active-v4 ledger movement and does not claim
KoalaBear closure.

Latest-parent reconciliation

The restack preserves all later #1165 full-lift/common-factor work. It also
repairs the inherited status summary to match #1165's own current results:
Koala full lift is paid through e=96150; Mersenne-31 through e=130236,
with e=130237 routed but not paid.

Verification

• canonical payload 31ab611e5623641fcda5d786b3fd1a14b7e4400ed69a8bcf4e8716c95e68696b;
• normal and optimized verifier PASS;
• 32/32 hostile mutations rejected;
• Sage and FLINT independent replays PASS;
• 13 source pins and 18 packet hashes match;
• all 7 pinned predecessor verifiers, 36 added parent Python checks, and 4 C replays PASS;
• campaign audit PASS;
• full TeX compile PASS (118 pages);
• separate mathematics and custody reviews GREEN.

The unrelated Vercel authorization check may remain red.

[AllenGrahamHart]

Stack note: dependency PR #1165 has advanced from the pinned d4d653723 to 9c708e2f7. The later commits are compatible support-profile continuations: terminal and top-third affine-line payments, per-layer total-core repair, and now a cross-layer global-line theorem paying Mersenne-31 full-lift supports through e=97908. They do not use the invalid direction-separated factor identified here. No need to widen this PR immediately, but a future rebase should retain those commits rather than stopping at the mean-centered profile.

[AllenGrahamHart]

Dependency update: #1165 is now at c09423b8b. The new post-9c708e2f7 commit composes the already-pinned mean-centered Gram theorem with the cross-layer global-line cap, with an explicit full-lift A_H>K-1 guard. It advances the compatible support walls to KoalaBear e<=96150 and Mersenne e<=98229; it does not use the retracted direction-separated factor.

[AllenGrahamHart]

Dependency update from #1165 head b04c8f1d: the proved mean-centered/global-line route now absorbs the first Mersenne boundary layer by a two-anchor case split. With q=e-K-3 floor((e-K)/3)>=1, it proves |Z| <= max(P_H+1, P_(H-1)+(N-m+1)); at e=98230 this is 16487313, leaving slack 289902. The residual starts at e=98231, where the same legal bound is over budget. This is compatible with support-local transversality and does not use the retracted direction-separated factor.

[AllenGrahamHart]

I proved a successor payment for the rank-10 terminal in this PR. It closes error rank 10 without first classifying the full direction-exception-12 forest.

After the reversible gauge, let the explanation flat be c_0+C' with s=9, and define the margin separately for each slope:

theta_gamma=min(w+1,min_(b in C') |{x in S_gamma:r_1(x)!=b(x)}|).

Split the post-near family at an arbitrary integer threshold T.

• High part theta_gamma>=T: apply this PR's support-local transversality theorem to the affine span of that subfamily. Taking the maximum over its possible ranks 0<=r<=s gives max_r ST_r(T); rank zero has the direct cap floor(n/T).
• Low part theta_gamma<=T-1: choose b_gamma in C' witnessing the margin and put a_gamma=h_gamma-gamma b_gamma. Then (a_gamma,b_gamma) in (c_0+C') x C' agrees with (r_0,r_1) on at least A=m-T+1 common coordinates. The ordinary affine-span theorem gives

  M_s(T)=floor(C(n-K+s,s)/C(w-T+1+s,s)).
  

  When M_s(T)^2<|F|, the proved sub-square-root interleaving theorem bounds the pair list by M_s(T). For a fixed pair, same-support noncontainment supplies an exception coordinate x, and

  gamma=-(r_0(x)-a(x))/(r_1(x)-b(x))
  

  injects its owning slopes into the at most n-A coordinates outside the pair common core.

Hence

|Z| <= 2w + max_(0<=r<=s) ST_r(T) + (n-m+T-1) M_s(T).

For the official KoalaBear row, T=667 gives:

A                    = 1115382
M_9                  = 57781140652, with M_9^2 < 2130706433^6
max_r ST_r           = 5143522968716559
(n-A)M_9             = 56727790457914040
2w + high + low      = 61871313426765543
budget               = 274980728111395087
slack                = 213109414684629544

An exact scan shows T=16 is already the first paying threshold and T=667 is the unique minimizer of this formula. The analogous explanation-rank-10 minimum is 1040506078215897711 at T=876, so this mechanism honestly stops before error rank 11.

Proof packet and two independent exact verifiers are in campaign commit c2c37ceb8, under background/nodes/rate_half_mca_rank10_margin_interleaving_split_payment/. No v4 ledger atom or Prize closure is claimed.

[AllenGrahamHart]

Dependency update: #1165 is now at bc6f8286. Its new recursive full-lift line-peeling theorem repeatedly removes a forced parameterized affine explanation line; the residual either enters the exact weighted prefix or violates the pairwise K-1 inside-core intersection packing bound for distinct peeled lines. Exact replay pays Mersenne supports through e=130198 (5,393 new rows, at most five lines), leaving e=130199 as a method wall. This remains compatible with the support-local transversality repair in this PR and does not use the retracted direction-separated occupancy factor.

[scottdhughes]

Successor payment is now packaged and independently GREEN in #1167.

It is a one-commit successor to exact current #1166 head b67078c7c. The
margin/interleaving split pays the KoalaBear post-near affine-error-rank-10
branch at T=667 with total 61,871,313,426,765,543 and slack
213,109,414,684,629,544. The exact same-formula rank-11 minimum remains
1,040,506,078,215,897,711 at T=876, and the packet includes an actual
GF(11) fixed-pair multiplicity sharpness control. It makes zero active-v4
ledger movement and claims no KoalaBear closure.

PR: #1167
Canonical payload:
642641809784eba3e4323f331bc28cc3d09192a287bd2708752a179080896f53.