Conditional κ = 7.54549816771013·10⁻⁴: kernel, target-prefix, plateau and reorder stages on #310's w3 word, priced against #304's complex supplier (+0.138% over #311)

[chafreaky]

Conditional κ = 7.54549816771013·10⁻⁴: kernel, target-prefix, plateau and reorder stages on #310's w3 word, priced against #304's complex supplier (+0.138% over #311)

One self-contained package, research/w3-bitword-stages/ (#310's package with four added transcript stages and a supplier admission), with its own workflow. No existing file is changed.

κ = 377274908385506513145517/(5·10²⁶) = 7.54549816771013·10⁻⁴: +0.2200% over #310 (7.52893499605335·10⁻⁴) and +0.1382% over #311 (7.53508664880763·10⁻⁴). Bit coarse 7.5512·10⁻⁴, complex 7.635·10⁻⁴ (#304's supplier): the bit side binds.

What it adds

#310's w3 word (the PR #249/source527 cohort word with a design-T cube layer, twin condensation and a SAT line-pair module) is admitted by #310 with no transcript stages. This package re-derives the stages of the gen4/gen5 lineage on it and prices against the stronger complex supplier, which becomes necessary: with #233's supplier (7547/10⁷) the complex side would bind at 7.541308566883·10⁻⁴.

step	items	local Σφ	κ after the step
#310	–	1,068,458.57	7.52893499605·10⁻⁴
1. kernel stage	433 response-kernel families (twin pairs, integer and F₂ triples, multi-donor families with shared donors on nested chains; multi-seed packing) + 85 zero-response singles; entrance rank 840; 132 new frames	stock 162,123 → 161,843	7.53675926616·10⁻⁴
2. target-prefix + plateau	45 F₂ target-prefix groups (window dependency R_t = Σ ±R_p, k ≤ 3); 660 identical five-gate components retimed into new dim-15 frames	−1,025.22	–
3. reorder	just-in-time MOVE normalisation + 172 commuting addition relocations (two rounds, fixed point)	−326.55	bit cap 7.54549816771·10⁻⁴
4. complex supplier	#304's program through #309's source-bound adapter, priced at 7635/10⁷	–	7.54549816771·10⁻⁴

The φ ledger (φ(r) = r·ln(120/r), deficit fixed; ≈ −6.45·10⁻¹⁰ κ per φ on #310's ledger) predicted the stage-2+3 gain at +8.72·10⁻⁷ against +8.74·10⁻⁷ verified. Reviewer note: the 85 zero-response singles are helpers that the w3 redesign left without prefix reads; they enter directly at their first frame and are carried as kernel entries without partners (≈ 407 φ). Sinks (582 shape matches), early restorations (440 candidates; they would need final-frame changes that #310's legality checker rejects) and cleanup sandwiches (none on w3) are not composed. #311's 209 squares are bound to record indices on #310's word and were not re-screened on this one; this package's 45 target-prefix groups are its own screen on the kernel word.

Checks

• Kernel: exact rank and 9I − J Gram nondegeneracy of all 132 new frames; the F₂ kernel identity of every family on the original prefix responses; E inside every member's first frame; no source owner or donor key among members, pivots and donors disjoint; GF(2) replays with random dirty inputs (sources unchanged, targets t ⊕ x, helpers restored, every register equal to Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310's); controls omitting the Q and Q⁻¹ gates rejected.
• Target-prefix and plateau: nested chains; forward and inverse F₂ all-column replays; an integer replay mod 2⁶¹ − 1 equal on every register; the exact source-span rule (0 violations).
• Reorder: the screen re-run must reproduce the frozen selection; crossed-interval contract; integer replay; non-vacuous omission control; fixed point.
• A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304 supplier: Conditional κ = 7.51499335697560e-4: local dirty-response transport, commuting schedules and37 additional gauges on weighted gen5b #309's adapter archive pinned by hash and member inventory, extracted fresh, gx.check1 on the actual A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304 program with its label, scalar, splice, cover, router, precision and row guards.
• Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310's native chain on the final word: legality (518 selections); all 23,627 global and 22,172 compact F₂ columns (payload 97 bits < 104); exact charts for 4,265 new frames and 22,118 changed projectors (≤ 342 factors); bank review (made generic: census and width-120 tiling read from stages/bank-tiling.json, an integer-programming tiling; 105,523 banks per stage, stock 809,215 / 161,843, 3,026,400 assignments); price (two moment engines, eight levels, 47 strict constraints, adjacent grid rejected); finite invoice (coefficient 58,364,056,416,171,201 < 2⁸⁰, selector 571,950,117,600 < 2⁴⁰).

Verify

sudo apt-get install -y g++ libboost-dev
python3 -m pip install -r research/w3-bitword-stages/requirements.txt
python3 -B research/w3-bitword-stages/verify.py --output /tmp/w3-stages-proof --cxx g++    # about 20 minutes

Passes on macOS (clang++) and on Linux (g++). The workflow (#310's, renamed, 60-minute limit) runs it on Ubuntu. supplied/base/249-DONOR-OWNERSHIP.json keeps #310's exact bytes through a package .gitattributes (-text), so the manifest matches a fresh checkout.

Scope

Conditional finite construction under #310's inherited all-size interfaces (compiler, common weighted charts, restored rows, selectors, routing, prime supply, complex dirty lifting, precision/recovery, analytic transfer). No Lean proof, unconditional theorem or runtime result is claimed. #310, #309 and #304 are open; their word, admission and supplier are inherited unchanged except the generic bank-tiling input and re-pinned literals.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md (#310's notice kept as NOTICE-PR310.md). The w3 word and its admission are utcorvusvolat-dotcom's (#310, from a supplied archive of two agents' builders); the stronger complex supplier is DreamingOfClouds' (#304) through Dugongue's source-bound adapter (#309); the kernel mechanism is Dugongue's (#254/#259/#272) in our multi-donor form (#266/#282/#290); the target-prefix mechanism is eumemic's (#268) with rohanarun's native stages (#273/#311); the reorder stage is ours (#299/#306). This package contributes the four stages on the w3 word, the generic bank tiling and the composition.

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.

[chafreaky]

CI note: this package's own workflow (W3 with kernel, target-prefix, plateau and reorder stages — full replay on Ubuntu/g++) passes, and every other check passes except verify (3.x, joint). That job fails identically on #311 and #313: scripts/experiments/joint_dual_compose.py pins the sha256 of the top-level Makefile, which main's commit 12a4934 changed. This PR does not touch the Makefile. Locally the package verifies on macOS/clang and on Linux/g++ with the same κ = 377274908385506513145517/(5·10²⁶).

[Mister-Meeseeks]

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

Contribution: Kernel, target, plateau and reorder stages on w3.

Review and limits: The final word and stronger complex supplier are jointly priced. The dedicated full pipeline, not inherited repository jobs, is the execution evidence.

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: 328086061dc09499039a7a33fa6510c37e796522. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.