Conditional κ = 7.69553898621543·10⁻⁴: the full transcript stage stack on #315's p = 10 word, with a shared-donor closure kernel (+0.697% over #315)

[chafreaky]

Conditional κ = 7.69553898621543·10⁻⁴: the full transcript stage stack on #315's p = 10 word, with a shared-donor closure kernel (+0.697% over #315)

One self-contained package, research/five-stage-p10-transcript-stages/ (#315's package with the transcript stages of the gen4/gen5/gen5b lineage re-derived on the p = 10 word), with its own workflow. No existing file is changed.

κ = 769553898621543/10¹⁸ = 7.69553898621543·10⁻⁴: +0.697% over #315 (7.64230861320245·10⁻⁴), +0.576% over #316, +0.180% over #317 and +0.169% over #319. (The first version of this PR, with a packed kernel, was 7.69198971896986·10⁻⁴.) Bit root 7.70147·10⁻⁴, complex 7.72714·10⁻⁴ (#315's centre-sharing supplier, unchanged): the bit side binds, 0.33% below the cap.

Stages (each predicted by the φ ledger and verified exactly)

The φ ledger (φ(r) = r·ln(100/r), deficit 2,040 per replica, normalized ×12) reproduces #315's κ exactly; every row below is a full verify.replay of the package cut after that stage, and every prediction matched to all digits.

stage (mechanism credited)	on the p = 10 word	κ
#315	—	7.64230861320245·10⁻⁴
concave-descent retiming (#287/#279; = #316's 480 gates)	480 gates, {1: −960, 2: +480, 16: −480, 17: +960, 18: −480}	7.65146115390441·10⁻⁴
target-prefix squares (eumemic #268; rohanarun #273/#287)	120 groups	7.65802240659923·10⁻⁴
collective response kernels as a shared-donor closure (Dugongue #254/#259/#272; our multi-donor form #282/#290/#299; the shared-donor idea of eumemic's #319)	1,047 entries, entrance rank 1,200: per entrance E a maximum-weight closure where a pivot's donors are its representation in a response basis and a donor pays its split cost once however many entries it serves (1–31 donors per entry, a donor shared by up to 30); 24,203 rank-1 lines + 10,541 exact rank 2–14 frame intersections searched, 16 seeds, the 7 sink helpers kept free; φ −1,601.5 (the earlier restart packing gave −1,356.0, #319's appended rank-1 entries −55.1)	7.68105513378630·10⁻⁴
early helper restorations (utcorvusvolat-dotcom #280; endpoint banking #283; ours #295)	240 helpers	7.68740991856791·10⁻⁴
terminal sinks (Dugongue #283; ours #295)	7 (R 8,230 → 8,223)	7.69153177272007·10⁻⁴
addition reordering (ours, #299/#306)	133 moves, φ −276.09	7.69553898621543·10⁻⁴

A second descent pass and a second reorder round were searched after the sinks and after the reorder and found nothing (0 gates, 0 moves). A second kernel round after the closure finds 4 entries (φ −0.77) and is not a stage. Levers of #309/#312 checked and absent on this word: zero-response singles (none), plateau retiming (covered by the descent2 search, 0), local dirty-response transport (no positive closure).

What changed in #315's package

• Stage transforms and raw-ledger rebinds read h, v, m, centres, scatter reads and deficit from word_pins.shape(); every word-dependent count is a strict word_pins.expect pin recorded by discovery/repin.py (75 pins).
• Downstream checks extended for the stages: lowering completes restored helpers with P_σ + I − P_E on the post-sink register count; geometry checks one restored (σ, E) pair exactly; bank charts restored helpers by (σ, E); the prime check keeps the determinant obligation of frames removed by retiming; the scalar check replays the source program over all 10,150 columns and the emitted word over its 10,143; math/finite price the completion-rank histogram; the finite check re-asserts every stage receipt and exactly one raw-ledger receipt per stage.
• Bank tiling: Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's tile() runs out of rank-4 filler blocks after the restorations; an economy_tile() fallback adds patterns such as (19⁴, 20, 4), (18⁴, 20, 4²), (16⁵, 20), (3²⁰, 20²) — every bank exactly width 100 and full, no padding (85,578 banks per stage, literal stock 658,290); the no-stage word still tiles with the original search. The kernel entrance rank and the restoration saving must be multiples of 5 for the width-100 banks (1,200 and 240 here).
• Search tools in discovery/ (descent and target search, the collective census and restart packing, the restore, sink, reorder and descent2 builders), documented in discovery/README.md; STAGES-PROOF.md; Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's README and PR statement kept as README-PR315.md, PR-STATEMENT-PR315.md.

Verify

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

PASS_IMMUTABLE_P10_TRANSCRIPT_STAGES_FIVE_STAGE_BANKED_CONSTRUCTION, 15 required stages each with an omitted-stage control; passes on macOS and on Linux with the same manifest sha256 (adc6bc29…), and the generator check reproduces the five pinned bit-word files. The workflow (#315's, renamed) runs both on Ubuntu.

Scope

Conditional finite construction under the retained public all-size hypotheses of the source527 lineage (no Lean certificate for the bit side). #315 is open; its word, banks and complex supplier are inherited unchanged except the re-pinned literals and the tiling fallback.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The p = 10 word, its package and the centre-sharing complex supplier are DreamingOfClouds' (#315, on #276/#285/#304); the descent retiming and target stages are rohanarun's (#287/#279/#273/#316; target-prefix mechanism eumemic's #268); the response-kernel mechanism and collective kernels are Dugongue's (#254/#259/#272/#283), shared-donor selection eumemic's idea (#319, read as a description), restart packing rohanarun's (#300); early restorations utcorvusvolat-dotcom's (#280); the reorder stage, the multi-donor kernel stage and the gen5 ports are ours (#290/#295/#299/#306). This package contributes the re-derivation of the whole stack at p = 10, the shape-parametrised stages and pins, the tiling fallback 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: Full p10 transcript stack with shared-donor closure.

Review and limits: Shape-dependent pins, economy tiling, stage omissions and complete finite admission are retained. Bounded failed searches are not global optimality results.

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: e912cbdd649fcbd292293c70511882b06266b276. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.