[MCA] Repair support-wise near-rational reduction by 2w

[scottdhughes]

Stack and review boundary

This is a ready scoped MCA source-repair packet stacked on #1159 at exact head e603e0cedc5220ec2f29bd53836e732e3ec14934.

Review only:

e603e0cedc5220ec2f29bd53836e732e3ec14934..c5f4ea7a0c78828c901ae5f3428894a8b2e2806b

Upstream main is still 93fba1be3f3299b0ba4708d88715377bbb656e45; #1157--#1159 remain open and ready. A refreshed open-PR audit found no duplicate two-anchor/common-support repair.

Exact result

The active support-wise near-rational reduction formerly used a displayed +1 charge after deleting a common-support branch. This packet gives an actual deployed KoalaBear falsifier to that inference.

Let w=m-K=67,472. The construction has:

• a common support of size 2,029,680 > m;
• 67,472 distinct finite affine slopes;
• for each slope, its own identical size-m=1,116,048 noncontained witness support;
• word distance at most w from the RS code;
• balanced-slope count zero.

Thus the former one-slope allowance is false even though a different common support exists.

The replacement theorem is uniform: for a finite-field Reed--Solomon evaluation domain, if

w = m-K >= 1
3w <= n-K,

then the number of distinct finite affine slopes that are both support-wise MCA-bad and within distance w of the code is at most 2w.

The proof uses two anchors, minimum-distance collapse for every third near-codeword, and an injective same-witness coordinate-ratio map. It gives exact local charges:

KoalaBear: 2w = 134,944
Mersenne-31: 2w = 134,896

Source repairs

The packet repairs the foundation and Grande Finale source so that:

• exact-support shrinking preserves the identical bad witness;
• a common support translates to an exact sparse pair with the same bad-slope set, rather than deleting those slopes;
• rank-one injection, global code-line counting, owner localization, triple collapse, and correction-ray prose use the same noncontained witness support;
• the partial U_paid common-support zero is explicitly only the supplied pair-contained witness, never a slope-global deletion;
• identically split kernels receive the tangent payment only under the existing guard n-a'<=t; arbitrary deficiency remains unpaid in the rank-one/spread residual;
• the old four-row v3 compiler is marked historical because its additive MCA/LIST interface is not the active v4 S/A/E chronology.

The first mathematics audit caught the missing tangent-threshold qualification. It was fixed, resealed, and independently re-reviewed GREEN.

Ledger effect and scope

This is a proved local distinct-slope charge, not a regenerated v4 first-match ledger.

global ledger movement = 0
U_Q    = null
U_BC   = null
U_K3   = null
U_new  = null

It does not prove SEM-QBC, the actual-record K3 compiler, KoalaBear closure, any LIST-row closure, or the universal smooth-domain result. The active maximum-type S/A/E reserve and add-back chronology remain open.

Verification

Canonical payload:

1eb7b0aff6a89affe99ff0b723952d26eeb1c8712ff859ab11891d9dd20f7446

Passed:

• primary Python verifier in normal and optimized modes;
• 34/34 hostile mutations rejected in both modes;
• exhaustive GF(7) census: 117,649 syndrome pairs, exact maximum 2=2w;
• Sage exact finite-field replay;
• FLINT exact polynomial/root and matrix replay;
• exact original/repaired/public-DAG blob and SHA bindings;
• structural schema plus canonical JSON/payload and all seven packet hashes;
• partial U_paid generator and independent checker, payload c7f6b33a007ce43de949284431c24f3ea06fd21d37e6cc0620a2ca72a5a23eaa;
• both modified TeX sources compile;
• git diff --check.

Wolfram cloud independently confirmed the deployed integer arithmetic earlier in the run. A final wrapper repeat returned Insufficient credits; no proof or acceptance gate depends on it.

Independent current-byte reviews are GREEN for source/custody, mathematics, and certificate integrity. A second cross-model mathematics audit is also GREEN for this scoped repair.

Maximal next attack

Regenerate the active v4 S/A/E first-match chronology with the proved 2w local charge, route the common-factor branch separately before applying thm:partial-relative, and then attack the largest surviving S/A/E atom. Do not promote this local theorem to Q/BC/K3 ownership without an actual source-bound adapter.