Conditional κ = 7.76518649004426·10⁻⁴: the full transcript stage stack on #325's p10b word, priced against #327's p = 10 complex supplier (+0.888% over #325)

[chafreaky]

Conditional κ = 7.76518649004426·10⁻⁴: the full transcript stage stack on #325's p10b word, priced against #327's p = 10 complex supplier (+0.888% over #325)

One self-contained package, research/five-stage-p10b-transcript-stages/ (#325's package with the transcript stages re-derived on its word and #327's p = 10 complex program as supplier), with its own workflow. No existing file is changed.

κ = 388259324502213/(5·10¹⁷) = 7.76518649004426·10⁻⁴: +0.888% over #325 (7.69684039383334·10⁻⁴), +0.546% over #328 (7.72285110892838·10⁻⁴), +0.905% over our #320. Bit root 7.77122·10⁻⁴; complex cap 7.79598·10⁻⁴ (#327's centre-sharing p = 10 program, re-certified here with both moment engines); the bit side binds, 0.32% below the cap. With #315/#325's p = 11 supplier (7.72714·10⁻⁴) the same word would be complex-bound at 7.72118·10⁻⁴ — the supplier swap is what lets the stages count.

Stages (φ ledger reproduces #325's κ exactly; every row verified by a truncated replay)

stage (mechanism credited)	on #325's word	bit root	κ
#325	—	7.70276909563042·10⁻⁴	7.69684039383334·10⁻⁴
descent (#287/#279/#316)	480 gates	7.71206727248877·10⁻⁴	7.70612425424136·10⁻⁴
target squares (#268/#273)	120 groups	7.71873297992310·10⁻⁴	7.71277968783544·10⁻⁴
shared-donor closure kernel (#254/#259/#272; #282/#290/#299/#320; #319's shared donors)	1,339 entries, entrance rank 1,445, φ −2,376.6 (18 seeds on two machines converge to −2,376.0…−2,376.8); the 10 sink helpers kept out	7.75353328869776·10⁻⁴	7.74752621500104·10⁻⁴
early restorations (#280; #283 banking; #295)	240	7.76000852680037·10⁻⁴	7.75399141938020·10⁻⁴
terminal sinks (#283; #295)	10 (R 8,100 → 8,090)	7.76601004205538·10⁻⁴	7.75998362749977·10⁻⁴
second descent after the sinks (#291/#299; = #322's search)	47 gates	7.76674466583451·10⁻⁴	7.76071711153240·10⁻⁴
reorder (#299/#306), two rounds	130 + 2 moves	7.77122099155703·10⁻⁴	7.76518649004426·10⁻⁴

A #323-style late kernel round finds one rank-2 entry (φ −0.19) that the width-100 bank rule trims, so it is not a stage.

Complex supplier swap

#327's gcert1-p10-centre-mw-flow.json.gz (commit 40ed045) is vendored byte for byte with its modules, layer and notice under inputs/complex/centre-mw-p10/; source-pins.json pins it. The five places that pinned the p = 11 program (code/complex_labels.py, code/complex_scalars.py, code/complex_splice.py, portable_complex.py, math_check.complex_supplier()) now read h, v, R, W and the scratch namespaces from the certificate and word_pins.shape(); every former literal is a strict complex_* pin (counts, histogram, scalar bill, local bill, stock bit lengths, W, both roots); every inequality guard is unchanged; the completed-child identity is generalised to (2(m − max child) − 1)·B. Pinned: W = 10,295, R = 6,455, m = 100. The program is re-checked here in Python (gx.check1, gxcore, label/scalar/splice replays, both moment roots); #327's Lean check of the same bytes is #327's claim and is not re-run by this package.

Checks and verify

Every #325 stage runs fresh plus the stages above (17 required stages, each with an omitted-stage control): all-column F₂ replays forward and inverse with omission controls, integer replays, exact source spans, nested chains, restored-helper (σ, E) charts, width-100 bank tiling with the economy fallback (kernel entrance rank and restoration saving multiples of 5), scalar replay of the source program and the emitted word, price and finite invoice. Every word-dependent count is a strict pin recorded by discovery/repin.py.

python -m pip install -r research/five-stage-p10b-transcript-stages/requirements.txt    # sympy 1.14.0
python -B research/five-stage-p10b-transcript-stages/verify.py --output /tmp/p10b-stages-verification   # about 3 minutes
python -B research/five-stage-p10b-transcript-stages/bitword/producer/regenerate.py --work /tmp/p10-regeneration

PASS_IMMUTABLE_P10_TRANSCRIPT_STAGES_FIVE_STAGE_BANKED_CONSTRUCTION; passes on macOS and Linux with the same manifest (c9ccfca5…), and the generator check reproduces the five pinned bit-word files. The workflow runs both on Ubuntu.

Scope

Conditional finite construction under the retained public all-size hypotheses (no Lean certificate for the bit side). #325 and #327 are open; their word, package and complex program are inherited unchanged except the re-pinned literals, the generalised complex-stage pins and the tiling fallback.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The p10b word, its package and the p = 10 centre-sharing complex supplier are DreamingOfClouds' (#325, #327, on #315/#304/#285/#276); the descent and target stages rohanarun's (#287/#279/#273/#316; target-prefix mechanism eumemic's #268); the response-kernel mechanism Dugongue's (#254/#259/#272/#283), shared-donor selection eumemic's idea (#319); early restorations utcorvusvolat-dotcom's (#280); the post-sink retiming search coincides with eumemic's #322. This package contributes the re-derivation of the stack on #325's word (the shared-donor closure kernel, the sink-aware kernel exclusion, the reorder stage), the generalised complex-supplier admission and the checks.

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[Mister-Meeseeks]

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

Contribution: Complete transcript stages on the p10b word.

Review and limits: Every word-dependent selection is re-derived and checked, including the stronger p10 complex supplier and revised tiling.

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: 60ca7b5eeedf45301bcf3326b65f3b6a3ca9bf35. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.