Alternating pair-block networks for conditional κ = 4.034168124 × 10⁻⁵ > 2⁻¹⁵

[hipotures]

Result

We obtain the following conditional improvement to the integer multiplication bound:

T(n) = O(n (log n)^(1−κ))

κ = 1008542031 / (25 × 10¹²) = 4.034168124 × 10⁻⁵ > 2⁻¹⁵

This improves the conditional exponent saving reported in #37 by approximately 4.76%.

Construction

The improvement comes from changing the computation DAG topology through alternating complete-pair addition trees, together with optimized retained-controller allocations.

The resulting networks at dimensions (23, 25) require:

• 36,685 physical roles at dimension 23.
• 48,479 physical roles at dimension 25.

The construction preserves the inherited copied-center framework, reversed data geometry and required endpoint corrections.

Verification

Independent finite checks confirm scalar identities, rational-frame compatibility and physical role transitions.

Exact rational arithmetic verifies the recursive moments, precision bounds, product row reservations, 47 strict assembly constraints and seven final margins.

Reproducible source code, input records and certificates are included in this pull request.

The result remains conditional on the inherited analytic and fixed-tape multiplication framework. Formal verification and external mathematical peer review have not been completed.

Attribution

This work builds on the original OpenAI manuscript and Douglas Colkitt's framework, with contributions from:

• icekylinx — copied-center networks and physical compiler (Copied retained centers and two-stage assembly for conditional kappa = 3.84569e-5 #36).
• James Chang — reversed tensor geometry (Reversed two-stage basis and balanced assembly for conditional κ = 1.639226629e-5 #34).
• Rohan Arun — reversed data-corner construction (Copied centers and reversed corners for conditional κ = 3.850771033e-5 > 2^-15 #37).
• Zhihao Chen and earlier contributors — semantic precision, routing and bulk-resampling methods.

The new contribution is the alternating computation-DAG construction, optimized controller allocation and resulting improved conditional bound.

Research and verification were conducted with substantial OpenAI Codex assistance.

[Mister-Meeseeks]

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

Contribution: RaD association and alternating producer arithmetic with exact composition.

Review and limits: Both own arithmetic drivers replayed. Arithmetic does not by itself certify a changed producer; later #42 replay provides separate integration evidence for the alternating construction. Preserve this distinction.

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: 03bde70c0c2a453f1ceae20c619b602841405467. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.