Reversed rational-center orders: conditional κ = 7.798412662809e-5

[rohanarun]

A finite order refinement of icekylinx's #104 gives conditional

$$T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{7798412662809}{10^{17}}=7.798412662809\times10^{-5}.$$

This is 0.04686049% above #104's 7.79476×10⁻⁵, under all of #104's inherited and newly proposed interfaces. It is a small finite improvement; it does not independently establish #104's opposite-bank factorization, stopped product-ring streaming, or odd-denominator grid theorem.

Change. In the h24 complex producer, order pair-star coordinates by ((i ^ 1) in (a,b), -i) instead of ((i ^ 1) in (a,b), i). Reversing each aligned group changes the paid child-width distribution. The exact scalar outputs and frame rules remain unchanged. The generator differs from #104's by this one order expression plus attribution comments; the verifier checks that fact literally.

• Complex strict saving: 7799647191/10^14.
• Roles 44,918, matched carriers 8,966: unchanged.
• All h23 stopped-bit costs, copied-center corrections, divisor 21, and three nested row reserves remain paid.
• Complete physical child profile, exact bridge, all 47 strict constraints, seven margins, and two next-grid exclusions are checked.

Validation: full rerun passed. Independent focused checks passed: regenerated both the parent and selected scalar/support/frame producers, compiled the inherited legal-carrier matcher, compared every histogram entry, reproduced #104's exact certificate, and passed five adversarial controls. All 330 source pins are checked. See research/reversed-rational-centers/{README.md,proof.md,verify.py,test_controls.py,SOURCE.json}.

Full make -j1 verify, the independent producer regeneration, and all five adversarial controls passed in 2447.924 seconds on b06b82a456409968ae3f2c6765ae63c3f0343548: 78 isolated modules, 20 historical patch checks, no source drift, and all 39 research-head GitHub checks passed. Validation receipt. The inherited joint CI failure has been repaired: #104 changed Makefile, README.md and NOTICE without refreshing their source fingerprints. Only those three pins were refreshed; the original h23/h25 physical verifier passed in full. Its derived certificate and receipt changed only in source/checksum metadata, with no mathematical changes or suppressed checks. The complete rerun passed after this repair.

python3 research/reversed-rational-centers/verify.py
python3 research/reversed-rational-centers/test_controls.py
make verify

The search was bounded; no global optimality or practical runtime claim is made. The next-grid exclusions concern these chosen rational enclosures and assembly parameters.

Credit: icekylinx for #104 and the stopped product-ring/rational-center construction; Zhihao Chen for the semantic/bulk assembly; RaD/hipotures for its analytic interfaces; Aurel Prosz/Paureel and the credited two-stage contributors; eumemic and the retained paired-circuit contributors; Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven, and all predecessors credited in #104's NOTICE and SOURCES.json. Original licenses and notices remain intact. Prepared by Rohan Arun with OpenAI Codex assistance.

[Mister-Meeseeks]

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

Contribution: Reverse-pair leaves on disjoint supports.

Review and limits: Exact-head CI evidence; legal matching and unchanged coefficient denominator retained.

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: eb231b7b63f7fc99d51ffc16e0356831c3af1c22. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.