Conditional κ = 9.54395132518163e-4: coupled complex retiming after #375 (+0.01588%)

[rohanarun]

Combining #365's four coupled complex retimings with #375's seven-stage E8 construction raises the conditional saving to κ = 954395132518163 / 1000000000000000000 = 0.000954395132518163, approximately 0.01588% above #375 at 79874623cf86d92325c26ccafc31413d12e9d248.

The complex supplier rises from 955155017966616/10^18 to 955306874505079/10^18. Its seven-stage recursive child count falls from 29,498 to 29,491, with rank mass, deficit, scalar-operation counts and full rational scalar map preserved. The bit word, banks, geometry and finite bill are unchanged; the complex side still binds. This contribution is the checked composition of the two existing results, not a new E8 word or new retiming discovery.

Validation

• PASS locally: the retimed complex circuit through κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's unsplit B3 label, scalar, splice and precision checkers, including halving degree two and the 13,967-row coefficient.
• PASS locally: both rational moment engines and adjacent-grid rejection; the unchanged bit saving; all 47 strict assembly constraints with the new complex supplier.
• PASS: the complete Reduce E8 recursive cost with four coupled frame changes #365 reconstruction and full 1,023-square rational scalar-map comparison, including all initial helper dependencies.
• PASS locally: the full standalone packaged κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375 replay and composition at 28f3c5404edf36f4c679de2825f70a362ed0f1ca, including every stage listed below.
• PASS on GitHub: Python 3.11 and 3.14 full replay at 28f3c5404edf36f4c679de2825f70a362ed0f1ca.
• PASS: all 58 GitHub checks at 28f3c5404edf36f4c679de2825f70a362ed0f1ca.

Reproduce:

python -m pip install -r research/e8-seven-stage-coupled-retiming/requirements.txt
python -B research/e8-seven-stage-coupled-retiming/verify.py /path/to/fresh/output

Both upstream packages are vendored unchanged and source-bound. The verifier runs the complete #375 replay, the complete #365 replay, and then their composition with frozen pins. This includes producer reconstruction, all 1,585 seven-stage formal columns, negative controls, earlier-stage regressions, E8 geometry, banks and finite bills. A small Bash transport adapter handles spaces in paths and the selected Python interpreter; every upstream scientific check remains enabled. Full provenance, exact values and the concavity argument are in the package README.

Scope and attribution

As in #375, the complex idle climbs remain unsplit, with halving degree two, maximum child 38 at width 63, row coefficient 13,967 < 20,161 and induction gap 49B. The inherited precision/recovery interface at that depth remains an open obligation; the original bridge text stated degree one.

This remains conditional on the interfaces of #375, #365 and their ancestors. There is no new Lean build, unconditional multiplication theorem or global-optimality claim. In particular, the B3 bridge, nine retained bit totals at dimension eight, E8 geometry, frame transfer, recursion, tape compiler, assembly and seven-stage finite-bill proof interfaces remain inherited. Original Lean-data regeneration applies only to the baseline circuit.

Credit: Jacob Sussman for the complex E8 circuit and framework; Dylan Crow / utcorvusvolat-dotcom with Claude assistance for the bit word; DaysSky with Claude assistance for B_K and #375; aliromman for #365 (prepared with OpenAI Codex assistance); and the pricing/assembly authors credited in the preserved notices. This composition and verifier were prepared with substantial OpenAI Codex assistance. Apache-2.0.

An isolated prerequisite commit reuses #318's unchanged three-file joint-dual source-pin repair, as in #366/#368, so the inherited repository check binds the actual base files. It makes no mathematical change.

[ahrazzle]

Independent full-source replay of the composition on third hardware — PASS

I ran this PR's own verify.py end-to-end on new hardware, from a fresh checkout, with no saved PASS inputs:

• Host: Apple M1 Max, 10 cores, 64 GB, macOS 27.0.1; Python 3.14.6, numpy 2.5.4, sympy 1.14.0, mpmath 1.3.0; GCC 16.2.0 (homebrew) via --boost-include /opt/homebrew/include (the package's darwin support/ carries bits/stdc++.h; Boost comes from the include flag).
• Result: PASS_FULL_SOURCE_BOUND_SEVEN_STAGE_COMPOSITION, κ = 954395132518163/10¹⁸ = 9.54395132518163e-4, baseline (this PR's #375) κ = 95424356568133/10¹⁸, relative gain +0.0158835% — matches the PR description exactly.
• What it replayed: the complete pinned PR #375 package (word regeneration from producer parameters, all-column checker with all 6 negative controls rejected, legality, five-stage columns, primes, banks, complex gx/gxcore, precision guard, finite bill, and the generalized-vs-original certify_bit regression reproducing PR #325's p10b κ byte-for-byte), then the complete PR #365 reconstruction with the full 1,023-row rational scalar map (all initial helper columns), then the composition under frozen B3 pins: both rational moment engines with adjacent-tick rejection, 47 strict assembly constraints, unchanged seven-stage bit histogram and finite bill.
• Composition receipts: complex b = 955306874505079/10¹⁸ (from 955155017966616/10¹⁸), seven-stage recursive child count 29,498 → 29,491, rank mass/deficit/scalar-operation counts and full rational scalar map preserved, halving degree 2, row coefficient 13,967 < 20,161, induction gap 49B.

Frontier observation for the record: after this composition the two sides have converged — complex b = 9.5531e-4 now binds against bit c = 9.6820e-4, i.e. the remaining headroom to the non-binding side is 1.35% (down from 9.3% at #352). The E8 bit word (#370) closed the gap that #352's bit side had opened; further κ gains now need a stronger complex supplier (Sussman's next unit above 9.5531e-4) or a bit word above 9.6820e-4 — both within ~1.4% of each other.

No changes to this PR's code — independent replay only. This confirms the draft's local claims on independent hardware, so the only remaining gate is the repository CI matrix.

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Coupled complex retiming in the seven-stage composition.

Review and limits: Full pinned PR375 replay plus changed component checks. No claim that the original local Lean theorem certifies the altered data.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 4316086532cb7424c25777ccfcc272b16c675608. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.