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Replace the complex motif's per-pair side wires with a shared-sum circuit (3,693,800 -> 108,195 side roles per invocation at h=25). Binary coordinate and pair-star labels give every frame residual an orthonormal basis; injected disjoint pieces must leave an outside point uncovered, since otherwise the residual is an alternating hyperbolic plane. The certified complex saving rises from 418/10^12 to 14/10^9, so the paired bit network becomes the only binding motif. With the compact-control recurrence this gives the conditional witness kappa = 59/10^11 > 2^-31, 590/83 (about 7.11) times the preceding 83/10^12. - scripts/complex_circuit.py: circuit, F2 label algebra, role compiler, both-direction frame checks, exact dirty-scratch simulator - scripts/complex_network.py: full h=25 checks, counts, guard, gate bound, parameter witness -> certificates/complex-network.json - scripts/make_complex_circuit_patch.py -> patches/complex-circuit-31.patch - notes/complex-circuit-*.tex and artifacts/complex-circuit-note.pdf - tests/test_complex_network.py; README, status and reproducibility docs Contributed by eumemic. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
The upstream workflow limits each job to ten minutes. Cache residual and label verdicts by canonical basis (same checks, entry/exit checks now also require full label validity), and let the patch generator recompute only the counts and witness instead of the full finite checks, which scripts/complex_network.py and the tests already run. The certificate drops its diagnostic count of distinct cached checks; the patch is unchanged. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
- Evaluate the Gaussian maps S' and E as chirped block correlations with the established multiplier: O(p^(1+delta)) per output - Bound powers of the resampling correction by a potential argument, so the Neumann series needs O(p/alpha^2 + 1/theta) terms - alpha = floor(sqrt(b/(8d))): Gaussian cost O(d^2 p^delta), epsilon may approach 1/2; beta = 19/25 fits the compressed complex saving - Conditional witness kappa = 1479/10^12 > 2^-30 (1479/590 above 59/10^11) Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
- Feed each center wire from its group total inside the side circuit: it rises only to the star span H_i, so it loses h-1 instead of h dimensions per invocation in both stage directions - Align the paired blocks across common points and share each top-level pair-star chain between its two groups: 494196 roles vs 509244 - Bit saving 296/10^11 -> 325/10^11; witness 1479 -> 1624/10^12 Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
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Thank you @rohanarun. This contribution is now recorded on main in the scoped reconciliation review and contributor credits. Contribution: Compatible source boundaries improve the nested stream. Review and limits: Fresh nested-stream tests; entrance work is not counted twice. 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: |
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Conditional improvement by composing the recent geometric refinements
This combines #15's smaller h30 physical producer with the data-corner profiles of #14/#16 and #16's nested auxiliary-exit basis. The resulting exact conditional witness is
That is 15.94% above #15 and 27.39% above #16 in exponent saving, not practical runtime. This is a composition of credited contributions, not a claim of priority for the nested-basis or data-corner lemmas, or a claim of global optimality.
The key compatibility point is that #15's internal wire reuse preserves every auxiliary D0/D1 boundary, data endpoint, and interstage partner permutation. The dimension-parametric nested-basis lemma therefore applies at h30 to the same four physical edge classes. Its h-wide auxiliary-exit block and the two data blocks replace their own singleton pivots; the source-relocated zero entrance is not counted a second time.
The bit saving is 2496689/10^12 with a strictly positive rational moment gap greater than 19/10^15. The h28 complex network uses the conservative saving 4/10^6, all proper residuals, and C1=3749/2500. All 29 constraints and seven final margins are recomputed, with a strict absorption gap above the claimed κ.
Included
research/nested-stream/.Validation
make verifyrerun: 308 unit tests, 21 inherited patch-application checks, and all preceding certificate/circuit rebuild commands. The regenerated inherited certificates are unchanged as JSON values. Seeresearch/nested-stream/validation.jsonfor the validation record.The finite controls do not prove the general rational finite-family basis lemma. The compiler, source-frame, mixed-width tape, precision, and inherited multiplication arguments remain dependencies requiring independent mathematical review. This is not formal verification or an OpenAI-endorsed result.
Attribution and review base
This branch starts at #15 (
a17cab396ce1c79c288bde9d06002cefdc0af8e6), so the main-branch diff also includes that pending dependency. The focused addition is the final composition commit. It imports #16 ata80f5e676c84b59def9791495df0655efe89a04fwithout changing those reference files.Credit to eumemic for #13/#15 source frames, producer and all-complex batching; Zhihao Chen (jacklightChen) for #16's nested basis and prior five-subset work; icekylinx for #10's batching/tape/precision interfaces; Rohan Arun for #14's data corners and earlier dimension/matching work; and Douglas Colkitt, OpenAI and earlier contributors for the retained constructions. This composition was prepared by Rohan Arun with substantial OpenAI Codex assistance.