Expose an A5 block for conditional κ = 1.1447067e-5 > 2^-17

[rohanarun]

The A5 projector corner in icekylinx's #18 basis contains a contiguous block of 21 pivots that the previous certificate charged as separate singleton swaps. Using that block, with Zhihao Chen's #21 translated frames and #23 semantic/bulk assembly, gives

$$T(n)=O(n(\log n)^{1-\kappa}),\qquad \kappa=\frac{11447067}{10^{12}}=1.1447067\times10^{-5}>2^{-17}.$$

This is about 4.159% above #23's 1.099e-5 witness in asymptotic exponent saving. It is not a practical runtime claim.

New geometric block

The profile changes from 47 singleton swaps + 481 to 26 singleton swaps + 21 + 481. After the first scalar pivot in the prescribed 47-square null corner, the rank-one term cancels exactly on rows 1–21 and columns 25–45. Entries to the right of column i+24 vanish; each pivot at (i,i+24) is the generically nonzero product p_i ξ_(i+2). These consecutive, equally ordered pivots form one contiguous recursive interchange under the retained triangular conjugation. The old full-rank tree proof supplies the remaining pivots, and the middle block of width 481 is unchanged.

There are 2N occurrences: subtract 42N singleton children and add 2N children of width 21. Rank mass stays fixed, and the recursive child count falls by 40N. This general identity uses the existing simultaneous basis; finite samples are controls, not its proof.

Compatible assembly

The exact bit saving is 1144733/10^11 = 11.44733e-6, with moment gap greater than 3e-14. The physical graph, role count, dimension, weighted rank sum and maximum bit child are unchanged. Therefore #23's bit halving depth 494, wire exponent 43, complex scalar charge, semantic guard C1=1 and product reservoir p^89000 remain valid.

The complex saving stays 18e-6 and beta=1/4, giving leaf saving 13.5e-6 above the new bit saving. The #23 assembly recipe is recomputed at the new bit exponent, including every strict constraint and final margin.

Review and validation

• Four new regression tests passed, including four exact rational corner cases, both projector signs, every stated Schur entry, and two negative controls per case.
• Both moments, all 47 strict constraints and seven margins pass. The final absorption gap exceeds 5e-13. Negative controls reject omitting the new block, the next certificate grid point, and reusing the old guard or separate exposure bounds.
• An independent parallel mathematical audit checked the general pivot proof and semantic/bulk compatibility; it separately reproduced the exact assembly.
• The four-page manuscript compiled with resolved references and no overfull boxes.
• The focused patch forward-applied to pinned Semantic precision and bulk resampling for a conditional 1099/10^8 > 2^-17 bound #23 and reconstructed all 12 source files exactly. Inherited theorem and source hashes remain unchanged.
• Detailed evidence is in research/a5-semantic/validation.json.

Full repository rerun: passed. make verify exited 0 at 61f81dc: all 180 tests, 18 retained upstream patch checks, and the focused 12-file PR23 patch check passed. All three retained bit producers (h25, h23, h57) and the complex histogram were freshly rebuilt and reproduced exactly. The completion time and log digest are recorded in research/a5-semantic/validation.json. This PR is ready for review.

This branch starts at #23 commit 661ebabc1076c9f525d7ce4967c876b203fa9827. The main-branch diff includes its pending predecessors; the final commit isolates this geometric addition. Pinned inherited sources are preserved. The new proof explicitly incorporates the retained finite, analytic, routing, bulk, precision and tape dependencies. This is conditional mathematical work with exact executable checks, not formal verification or completed external expert review. No global-optimality claim is made.

Attribution

The additional A5 contiguous-block observation, proof, controls and compatible witness are by Rohan Arun with substantial OpenAI Codex assistance, including parallel reviews.

Credit icekylinx for #18's prescribed basis, partial swaps and producers, and #10's batching/tape interfaces; Zhihao Chen (jacklightChen) for #21's translated frames and phase proof and #23's graph-specific semantic/bulk composition; RaD/hipotures for the semantic precision, routing, phase-cell inverse and bulk arguments exposed in #20; eumemic for #13 source frames and preceding work; and Dominik Scholz for #22's parameter tightening. Earlier #7/#16 work, Douglas Colkitt, OpenAI and all prior contributors remain credited. We do not claim their constructions as new.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: A5 Schur blocks preserve rank mass with a different child distribution.

Review and limits: Fresh semantic tests; finite-interface and analytic assumptions remain explicit.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: 003f366cfa99e19fbeacd9300a219f2d44c8cff0. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.