Congruent balanced multi-splits and indexed two-cycles: conditional κ = 5.2431959054651e-5

[gupt1156]

Summary

This PR certifies a new finite conditional witness $\kappa = \frac{52431959054651}{10^{18}} = 5.2431959054651 \times 10^{-5}$ for integer multiplication, improving upon PR #84 ($\kappa = 5.22712890914 \times 10^{-5}$) by $+0.307377%$, as well as subsequent refinements (PR #85 and PR #86).

Key Architectural Advance

1. Congruent Balanced Multi-Splits [4, 4, 1]:

   • Previous indexed-cycle architectures applied multi-split code 4 (splitting both first and mid pairs) asymmetrically ([4, 1, 1] on $h=23$; [1, 4, 1] on $h=25$).
   • Applying [4, 4, 1] congruently across both axes contracts eligible pairs across two consecutive recursion depths while satisfying the strict contraction guard $\text{groups} + \text{splits} + 1 < \text{points}$.
   • Role contractions:
     • $h=23$: $26,834 \to \mathbf{26,705}$ ($-129$ roles; rank mass $617,688 \to 614,721$).
     • $h=25$: $35,348 \to \mathbf{35,137}$ ($-211$ roles; rank mass $884,300 \to 879,025$).
   • Physical width $W$: Compacted from $132,466,108 \to \mathbf{131,795,727}$ ($-670,381$ bit coordinates).

2. Dirty-Basis Replay and CRT Profiles:

   • Serialized physical words generated with dirty=True in both forward and dual orientations.
   • Replayed all $2v + R$ basis vectors: PASS.
   • Reconstructed exact frame transitions directly from literal XORs: PASS.
   • Profiled via C++ oracle over Mersenne prime $p = 2^{61}-1$: 0 CRT modular disagreements on both axes ($5,535$ matrices on $h=23$, $7,334$ matrices on $h=25$).

3. Certified Recurrence & Assembly Bounds:

   • Certified bit saving: $a_{\text{bit}} = \frac{6554338538661}{125000000000000} = \frac{52434708309288}{10^{18}} \approx 5.2434708309288 \times 10^{-5}$.
   • Moment enclosure: $M(s) < 1$ and $M(s + 10^{-18}) > 1$ under directed Taylor8 and independent 60-term degree-12 $\mathrm{atanh}$ expansions.
   • Certified conditional exponent: $\kappa = \frac{52431959054651}{10^{18}} = 5.2431959054651 \times 10^{-5}$.
   • All 47 strict assembly inequalities $> 0$, all 7 safety margins $> \kappa$.
   • Historical profiles of PR Stopped pair-star trees: conditional κ = 7.808981744031e-5 #74, PR Balanced coarse split frames: conditional κ = 5.1745625472866e-5 #79, and PR Indexed carry two-cycles and coordinate flags: conditional κ = 5.22712890914e-5 #84 are strictly excluded ($M_{\text{old}}(s) > 1$).

4. Pipeline & Makefile Parallelization:

   • Makefile updated to run all verifier audits, tests, and core verification concurrently with -j32.
   • Core verifier runs $h=23$ and $h=25$ concurrently via ProcessPoolExecutor.

Verification

make indexed-cycle-verify

All audits, unit tests, and full dirty-basis replays pass with exit code 0 across 32 threads.

[Mister-Meeseeks]

Thank you @gupt1156. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Congruent [4,4,1] splitting.

Review and limits: Exact-head CI evidence; scoped finite improvement, not optimality.

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: 466fc99e67d872831d1d192394f346a28e909e09. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.