Candidate conditional kappa 0.001056975315151441: Hamming complex and retimed bit composition

[rohanarun]

Draft candidate conditional κ = 0.001056975315151441. This combines 112 common-frame retimings of #425's mutable-basis complex word with the existing #376 bit frame witness in #425's compressed bit geometry. It is 0.85589313% above #425 and 0.02461% above the prior version of this draft.

The complex scalar operations, order, coefficients and R680 helper count remain unchanged; jointly retiming 112 events changes the local histogram by {-112 rank1,-112 rank4,+112 rank5}. The exact complex lower bound is 0.001058093696336318.

The previously published 128-event bit retiming preserves R865, all scalar additions and endpoints. Fresh reconstruction in the compressed nine-window, m64 geometry gives bit lower bound 0.001061370033348997. This removes the prior bit bottleneck. Both supplier prices, the complete finite bill, three bootstrap steps and 47 strict outer inequalities are recomputed. No separately reported gains are added.

Passed: full fresh complex wrapper, all 3,072 complex phase pairs and both 920-column emitted orientations; exact bit witness reconstruction, freshly compiled native column/legality checks, all 944 frame Gram/prime checks, 1,080 port routes, 240 endpoint identities, 8,745 unbanked scalar columns, 62,280 bank assignments, independent bit template audit and its 8 rejected mutations, two rational moments, next-grid rejections, changed finite invoice and 47 strict outer inequalities.

The uninterrupted combined wrapper passed on Python 3.11 against code commit e533924d3c2ec045b3275e25220a76e9621067e0: fresh #425 reconstruction, fresh complex retiming, freshly compiled bit checks, five deliberately invalid native records rejected, three witness mutations rejected, and the complete compressed bit assembly. Receipts and logs are in research/hamming-bit-composition/local-evidence/. Two additional searches found no improving single-gate move among 4,300 gates or among 11,870 tested equal-frame groups; those negative results are included. The full combined wrapper also passed locally on Python 3.14 against commit 3f200316d9e268e8c7b474089c6107bf1ee3dea7. A compatibility shim normalizes only the OS metadata byte of deterministic gzip output to the pinned supplier format; compressed payloads, footers, and all expected artifact hashes remain checked. The compatibility and full replay receipts are included. Python 3.11/3.14 GitHub CI remains pending.

Reproduce with python -B research/hamming-bit-composition/verify.py /absolute/path/to/new/output using Python 3.11+, SymPy, C++17 and Boost headers. The prior failed attempt to use a complex-only bootstrap after hitting the bit cap remains in research/hamming-frame-retiming/local-evidence/.

This is a conditional finite construction. It retains #425's written mutable-source/target scalar-interface extension and inherited all-size bit/complex compiler, common chart/grid, restored-row, tape/precision and analytic interfaces. No unchanged legacy gx acceptance, new Lean proof, signed source527 bit decoder proof or unconditional integer-multiplication exponent is claimed.

Rohan Arun with substantial OpenAI Codex assistance. Credit Zhihao Chen/OpenAI Codex #425 for mutable-basis Hamming construction, compressed bit geometry and independent auditors; applecrusher/OpenAI Codex #376 for the existing bit retiming witness; Dylan Crow/Anthropic Claude #370 for the bit word and root data; Jacob Sussman for E8 and compiler interfaces, and retained supplier lineage. Parent sources and notices remain unchanged; adapted files and exact changes are documented.

Newer #433 reports a higher conditional bound; this PR preserves its independently checked contribution and does not claim the current lead.

[ahrazzle]

Target: PR #431, head 71c5988
(base 12a4934, fetched 2026-10-11T12:13:32Z).

Verification

Independent third-hardware replay of this head, from a fresh whole-repo
detached checkout.

• Verifier exit 0. All 17 steps ran in order, including a real C++17
  compile (Apple clang 21) with the native column and legality checks, on
  Python 3.14.6 with SymPy 1.14.0.
• kappa = 1056975315151441/1000000000000000000 = 0.001056975315151441.
  The fraction is exact to the last of 18 decimals and identical to
  EXPECTED-COMBINED.json and this PR's body.
• Binding component: complex. The complex coarse bound is the smaller of
  the two supplier bounds, so the Hamming complex side caps kappa.
• All 47 strict outer inequalities reproduce from an independent chain and
  every one is strictly positive at the claimed kappa.
• Digit-exact: an independent closed-form re-derivation (stdlib fractions,
  no package code) reproduces the same fraction on the 1e-18 grid. The
  derived percentages recompute exactly: 0.85589313% over the #425
  baseline, 0.02461% over the earlier draft of this branch, and
  31.32587419021997% over the frozen #361 baseline.

Negative controls

All rejected. 8 bit mutation controls, 3 witness controls, and 5
deliberately invalid native records (each caught by the C++ checker), plus
the named assembly controls next_grid, zero_eta, zero_beta and
unit_beta. A perturbed family that lowers the complex bound by one tick,
and a family that raises kappa by +1, +2 and +1000 grid ticks, are both
rejected, so the accepted margin is tight.

Limits

The claim is conditional as this PR states: the inherited all-size bit and
complex compilers, the common chart and grid, and the restored-row, tape and
precision interfaces remain assumptions, and no Lean proof or unconditional
exponent is claimed or verified here.

AI-assistance disclosure: prepared with AI assistance under human oversight.