[aliromman]

This enabling proof now gives a complete adverse repair test for the frozen PR4948 caller: at least 104 additional paid field returns are required at every width f>=72, even with feedback among new calls. The cheapest such repair fails the complete optimistic invoice for every positive concave power tariff r^p, 0<p<=1. A reusable error-versus-return theorem also rules out small-error repairs within the budget. The complete kappa calculation is unchanged.

Repair proof and exact contract, repair receipt, complete optimistic invoice, pointwise tensor proof, source pins. Contribution head: 45f7de73b168721cdf7ec8c1769b1b594493716a. Canonical base: 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7.

The underlying theorem applies to the full n-axis projective Clifford group. At every width f>=4n+4 its normalized tensor Gram matrix lies between I/9 and 17I/9. Coefficients, finite word choices and program length may depend on f. For one fixed Gaussian-dyadic scalar register program, four widths 4n+4 through 4n+7 decide equality at every positive width. The proof retains central phases raised to f. A separate stabilizer-map theorem includes rectangular maps and projections. These are sufficient bounds; the prior sharp one-axis window 5..8 remains stronger than the uniform window 8..11.

The paid-repair theorem combines this Gram estimate with the existing relative-history dimension method. Ordinary operations are scalar row combinations and the fixed columnwise monomial Clifford algebra. The entire retained history is granted at the start. Each new paid return is one field at the same 17-axis capacity and may be an arbitrary linear map using earlier new returns. Every original paid input expression, the original targets and the source-declared invoice stay fixed. All storage, COPY and owned CLEAR costs are granted zero. A primitive with several field returns must count its actual outputs; free outputs from a new supplier are not automatically charged as queries.

For the actual 104 independent source columns, the target cosets have grade 9 and are absent from the retained grades 0, 1 and 7. The quotient needs 104 full ordinary modules; one new return adds at most one. With q returns, the selected block error satisfies:

2^(-17f) ||E||_HS^2 >= lambda_f (104-q)_+
||E||_op^2 >= lambda_f (1-q/104)_+
lambda_f = 1/9 for f>=72; lambda_f = 125/126 for f>=84.

The source invoice has only 96 units of rank-mass surplus. Adding the necessary 104 rank-one returns makes all nine exact concave-order hinges positive, including an eight-unit deficit at rank mass. At the reference tariff p=1-605457647/200000000000, exact interval arithmetic puts the original margin strictly between 36 and 37. Positive margin permits at most 36 additional returns; from width 84 every such approximate repair has squared operator norm error at least 2125/3276. This reference calculation uses the pinned selected value as a tariff, not as a selected-caller replay. Histograms remain the frozen source declarations; no new physical word was recounted.

Changed original paid inputs, added free observations, native internal prefixes, larger capacity, noncolumnwise maps, nonlinear processing or changed targets need a new analysis. The result does not close active-feed constructions or the separate P104 repair work. It supplies no new caller, selected-frontier update or global lower bound.

Python 3.12 and the standard library suffice:

python3.12 -B scripts/research_package.py --package research/clifford-pointwise-faithfulness
python3.12 -I -W error -B research/clifford-pointwise-faithfulness/audit_mutations.py

All six registered normal/optimized checks pass. The repair verifier replays the unchanged author and independent source audits, including all 5,408 late words, and extracts the actual diagonal target and invoice through the retained loader. Further exact checks cover 576 group products, 18,432 phase covariance identities, 48 history prefixes and 12 physical least-squares cases. Controls include cached targets, wrong input versions, complementary singular targets that share one return, and small-width dependence. All twelve deliberate-defect checks fail at their intended diagnostic. The original complete stabilizer/Clifford censuses and native/projection residual checks also pass. See the registered commands, mutation receipts and research record.

The committed-byte check and source-pinned package export pass. All six checks also pass in a fresh read-only 86-file export with Git absent from PATH; its bytes and inventory remain unchanged. The separate full-history preflight refuses the shared checkout's current shallow state; no full-history pass is claimed for this revision. The declared source base is the verified two-parent ancestor and all consumed bytes are in the published tree. The written universal proofs have author review and finite corroboration, not independent human acceptance or proof-kernel certification. No selected cold replay, physical emission, constructive full price, precision application or outer admission was run. Hosted status for this revision is reported separately when available.

Credit Dugongue for PR #4948, source head 86e474038e05f99a76d41f71d81946ad2327e4a4, and Chris R / Clear1nsight for PR #4978, its retained passive audit, uniform-alphabet addendum and owned-reset clarification. Chris R's PR #5425 and PR #5446 supply the paid-history method; Ali Romman's PR #5449 supplies the relative-history quotient formulation. Those principles are not claimed as new.

Ali Romman's PR #5376 supplies the consumed native frontend and exact quadratic kernel, with its retained lineage notices. The invoice comparator credits classical concave order, mightyOil and predecessors. The consumed interval engine is zxv's higher-moment-strictification arithmetic at 20bb93cd990d2ff683e4d0fc56b87a42cd5d3296. Stabilizer normal forms, symplectic residual forms, character orthogonality and rank approximation are classical methods; their sources and the new contribution boundary are in the proofs. Predecessor files and notices are unchanged.

The canonical investigation records the prospective scope and final handoff. The final overlap refresh covered 398 open PRs and 4,826 Discussion records; a separate main-tree note scan supplements the sparse local search. PR #5531 applies the original theorem to a different negative E endpoint and does not overlap this repair contract. Unpublished work and arbitrary historical diffs remain outside coverage. Open next questions are changed-input constructions, complete retained-history extraction for other callers, and sharper spectral thresholds.

Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. One research agent; no independent human or proof-assistant review is claimed.