Exact data-corner certificates for conditional κ = 1.5878574e-5 > 2^-16

[rohanarun]

PR #29 charges all 107 data-corner pivots separately. Exact rank-partition identities expose two compatible, contiguous blocks of widths 51 and 45 in that same prescribed basis. Keeping its physical producers, topology, frames and paid endpoint correction gives

$$T(n)=O(n(\log n)^{1-\kappa}),\qquad \kappa=\frac{7939287}{500000000000}=1.5878574\times10^{-5}>2^{-16}.$$

This conditional exponent saving is 2.20503% above PR #29's 1.5536e-5 witness. This compares asymptotic exponent savings, not practical runtimes.

Geometric proof

At the retained dimensions (55,53), the 107-square null corner is diagonally equivalent to

$$M=F,\operatorname{diag}(x_0,\ldots,x_{54},y_0,\ldots,y_{52},-1),G^T,$$

where F and G are the actual prescribed incidence matrices augmented by a constant column. The last-corner residue offset is 52, and is explicitly retained.

For an augmented k-square minor, partition its coefficient columns into X and Y. The bound

$$\operatorname{rank}(U D V^T)\leq\operatorname{rank}(U_X)+\operatorname{rank}(V_Y)<k$$

proves an exact zero identity for every diagonal weight. The package supplies 2,265 such certificates, verifies all 4,530 ranks with integer elimination, and gives a normalized rational witness with all 107 ordered pivots nonzero. These identities establish the required Schur zeros for all weights; the witness establishes nonvanishing on the actual normalized projector family. The written finite-product argument places the new conditions in PR29's same free GL55 × GL53 basis family.

An independent worker separately reconstructed every required augmented minor, checked every partition rank without the search oracle, reproduced the rational pivot sequence, and checked the physical order. The two runs occupy rows1..51/columns2863..2913 and rows58..102/columns2812..2856; the width2701 middle remains coordinates107..2807. No separated coordinates are gathered.

Exact cost and assembly

The data profile changes from 107 singletons + 2701 to 11 singletons + 51 + 45 + 2701, preserving rank2808. Across 2N data macros, the histogram removes 192N singleton calls and adds 2N calls of each new width. The paid N endpoint-copy corrections remain.

The original m=2915, W=31753659079 and rank sum=92561585391935 are unchanged. The exact bit saving becomes 15879079/10^12. PR21's complex saving 18e-6, PR23's semantic C1=1 and PR29's beta=1/10 are retained. All 47 strict constraints and seven final margins pass. The product row degree is recomputed and remains 47000; no producer, routing, phase or bulk mechanism is newly claimed.

Validation and scope

The written proof, exact coefficient certificates, rational witness, full histogram and final assembly are included with source provenance and negative controls. The inherited PR29 exact verifier was replayed against pinned sources. Six new focused tests and the combined ten-test parent/new suite passed, including import isolation and negative controls. The four-page manuscript compiled with both the native editor and Tectonic; the exported PDF has no reported warnings or overfull boxes. The focused source patch forward-applied against pinned PR29 and reconstructed all 18 source files. Results are recorded in research/two-stage-corners/validation.json.

Full repository rerun: PASS. make verify completed successfully on commit 02f68f95369c39955c5ffcfddbd75cd83e26b305: 186 tests (63.162 seconds for the suite), four additional parent-focused tests, 18 historical upstream patch checks, the focused 18-source patch reconstruction, fresh h53/h55 producer and label audits, and inherited h25/h23/h57 and complex h28 reconstruction. Full log SHA256: a1dd068151d4ebff1e7b1ee31a26f22e5e4204cf82f3d7b4f7be8bc711601625. The validation receipt records completion time and the harmless last-decimal change in an inherited floating estimate; all exact rational values matched and the pinned parent file is retained unchanged. Ready for external review under the stated mathematical assumptions.

The branch is based on PR29 commit 9d963275075fa98f1da821e27757b238dafd6b3c. Its pending dependencies appear in the main-branch diff; commit 02f68f9 isolates this corner refinement, followed by a validation-receipt-only commit. The focused patch targets that pinned PR29 source, not the original OpenAI manuscript.

This remains conditional on the inherited basis/climb, physical, analytic, routing, bulk, fixed-tape and recovery arguments. Exact finite identities and independent agent audits are not formal verification or completed external expert review. No global optimality claim is made.

Attribution

The coefficient rank certificates, ordered-corner refinement and assembled witness are by Rohan Arun with substantial OpenAI Codex assistance, including parallel proof and implementation reviews.

Credit Zhihao Chen (jacklightChen) for PR29's compatible unequal two-stage composition and PR21/23 predecessor interfaces; Aurel Prosz (Paureel) for the two-stage topology and paid correction; Swapnil Jain for the linked two-stage batching development; icekylinx for the retained-total producers and inner basis; RaD/hipotures for semantic/bulk mechanisms; eumemic for source frames and earlier work; and all earlier contributors. Their sources, licenses and assistance disclosures remain intact. The rank-partition lemma is standard linear algebra; the contribution here is its exact application to these prescribed corners.

[Mister-Meeseeks]

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

Contribution: Ordered 51/45 corner blocks and their rank-cut argument.

Review and limits: Focused corner tests; preserves the actual order rather than treating dimensions as interchangeable.

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: 122e6453e80330633af701eabd1a6a72dfefd286. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.