Conditional kappa 0.001001098432210898: signed-prefix donor gauges

[rohanarun]

This draft adds nine exterior donor gauges to the exact 25-pair signed-prefix word of #395. The emitted construction and exact assembly give conditional κ = 500549216105449/500000000000000000 = 0.001001098432210898, above #395's reported 0.001001090245731051 by 0.000000008186479847.

The complex component is b = 1002101634586199/10^18. It uses 782 banks per stage, 14,718 live and 15,511 physical families, and the independently checked original-word #386 bit component (c = 1057833103534317/10^18). It does not compose #390's early-restoration/prewhitening bracket or #394's padding fusion.

The new local normalization retains distinct I and Cq^-1 entrance operators. Exact donor response cancellation permits nine rank-one entrance gauges, and the actual helper residual F*Cq^-1 is converted to eight coordinate factors by an explicitly charged phase adapter. Whole-bank stock and all padding are included. All owner right factors use their actual inverses; 4,709 of 6,264 routes are nonsymmetric.

Validation:

• Fresh standalone replay of all seven changed-complex/assembly stages, from a 187-file manifest; source reconstruction matches Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395's exact candidate hash.
• Full rational operator equality over 1,023 columns, both dirty scalar orientations and private lifetimes, and 40 donor-omission controls.
• 80 new/reflected exact Clifford phase pairs; unchanged local phase receipts reused only by exact basis binding.
• Complete interleaved schedule, explicit owner inverses, all 6,264 routed Gaussian residual identities, 7,038 stage/bank products, and 144 phase-omission controls.
• Independent emitted rank/profile and finite-invoice recount: row coefficient 15,445 < 20,161, router 7,984 bits, literal 35,284 bits.
• Both exact moment engines with full fallback and adjacent-grid rejection; six finite bootstrap cutoffs, all 47 strict outer constraints, and four outer negative controls.

Run python -B research/e8-signed-donor-gauge/verify.py /fresh/output with Python 3.11+ and SymPy 1.14.0. CI runs Python 3.11 and 3.14. The verifier links immutable inputs and writes generated outputs separately.

Draft scope: the changed complex program and assembly were freshly replayed; the previously checked #386 bit component and unchanged baseline local Clifford proofs are source-bound reuse, not a fresh full baseline replay here. Full inherited supplier replay and CI remain before readiness. This is a finite conditional construction retaining local recursive-child, Clifford/rank-zero compiler, arbitrary-dirt cover/banking, tape, common-grid, precision/recovery and analytic interfaces. It is not a new Lean proof or unconditional multiplication theorem. The six-round grid ceiling is not a global optimality claim.

Credits: Jacob Sussman's gx/E8 construction; #378 far retiming; Dugongue's #390 prefix identity; Rohan Arun's #395 signed extension; #377's compressed complex cover; #370/#386 bit lineage; #315 exact engines; and all notices included with the vendored source. Donor selection, nonzero-entrance normalization, native heterogeneous bank composition, independent audits and this package were prepared by Rohan Arun with OpenAI Codex assistance. Original licenses are retained.

Readiness audit (2026-10-11 08:04 UTC)

The dedicated Python 3.11 and 3.14 jobs passed. All three historical joint jobs stop at AssertionError: Makefile in joint_dual_compose.check_sources: both Makefile and research/joint-dual/SOURCE.json are byte-identical to base 12a4934d15006e8c10f9b2264f13f8f341be802b; their hash mismatch predates this PR. No source check has been waived or repinned.

All 188 published package files match the local source, and the bundled bit receipt matches the checked PR386 receipt chain. At that checkpoint the package lacked the inherited bit replay source closure; the update below now supplies the pinned bit sources. This PR remains draft. The seven changed-complex stages and assembly pass do not constitute a fresh full bit replay. PR398 now reports a higher conditional value; it has not been independently replayed here.

Bit source closure (2026-10-11 09:06 UTC)

The additive research/e8-signed-donor-gauge-bit-closure package supplies all 158 PR370 source files pinned by PR386, the PR386 source, and the complete retained bit record/receipt chain. A fresh isolated run passed compressed-bit geometry, global-word emission, independent template audit and exact finite pricing, then reproduced this PR's unchanged kappa, all 47 strict constraints, six finite cutoffs and four outer controls. Runtime fields and their dependent hashes are preserved; the actual geometry and template objects agree with the retained checked construction.

Run python -B research/e8-signed-donor-gauge-bit-closure/verify_closure.py --out /fresh/output. A dedicated Python 3.11/3.14 workflow now runs this source closure. Local PR370 native checks and unchanged PR377 complex receipts remain explicit source-bound reuse; the full historical pipelines were not rerun. This closes the missing bit-source packaging gap, not every inherited supplier/proof obligation. The unrelated historical joint pin mismatch is unchanged, and the PR remains draft. PR401 now reports a higher conditional value; this update claims reproducibility, not a new frontier.

The new bit-closure workflow passed on both Python 3.11 and 3.14 (run 38127217723). Other checks remain in progress; the observed joint failure on this revision repeats the unchanged Makefile source-pin assertion. Draft status is retained.