Conditional κ = 7.44714499357628·10⁻⁴: 450 collective response-kernel entries on #285's gen5 word, after #287's descent and target squares (+0.080% over #287)

[chafreaky]

Conditional κ = 7.44714499357628·10⁻⁴: 450 collective response-kernel entries on #285's gen5 word, after #287's descent and target squares (+0.080% over #287)

One self-contained package, research/five-stage-gen5-collective-kernels/ (#287's package with one added stage), with its own workflow. No existing file is changed.

κ = 186178624839407/(25·10¹⁶) = 7.44714499357628·10⁻⁴, +5.94·10⁻⁷ over #287's 7.44120696397042·10⁻⁴ and +0.259% over #285's gen5 word; the bit side binds (bit coarse 7.45270·10⁻⁴, complex 7.47455·10⁻⁴).

Result

The collective response-kernel stage of #282 (gen4) and #284 (on #283), re-derived on gen5 and inserted after #287's two stages: 450 entries with per-entry cuts — 260 twin pairs (244 on a line, 16 on a rank-3 entrance) and 190 multi-donor families (151 quadruples, 39 triples): the pivot's complete F₂ target response is the XOR of its donors', all members share a nondegenerate common entrance of dimension 1–18 inside their first required frames, 33 donors serve several entries along nested chains. The pivot's compensation reads are deleted, it enters at the common entrance, each donor pays its shear there after the cut and the inverse at the full frame.

one invocation, m = 120	#285	#287	this package
entrances	2,554	2,554	3,004
literal stock, 60 replicas	1,247,070	1,247,070	1,244,515
bit coarse saving	7.43338·10⁻⁴	—	7.45270·10⁻⁴
κ	7.42785663120535·10⁻⁴	7.44120696397042·10⁻⁴	7.44714499357628·10⁻⁴

Census on gen5 (after #287's stages): 13,332 plain dirty helpers, every compensation read before the helper's first other use (gap ≥ 41,240 records), so each entry gets its own cut, bound by content; responses of F₂ rank 1,760 (nullity 11,572); 1,220 twin classes; low-weight response circuits 2,402 triangles and 20,003 quadruples at class level; with a common cut and a nondegenerate common entrance: 292 pairs, 276 triples, 1,103 quadruples. Greedy packing under the deficit-fixed φ ledger (φ(r) = r·ln(120/r)) keeps 450 (entrance rank 1,022); the model reproduces #287's κ exactly and predicted this κ exactly. The members are plain dirty helpers, disjoint from the targets whose reads #287's squares delete and from its retimed source-pair mixes.

Checks

kernel_transform.replay runs all 19,406 formal columns forward and inverse on the emitted word, rejects both omitted-gate controls (2,405 / 757 wrong rows) and checks the prefix relation for every entry; then every stage of #287's verifier runs fresh — virtual, raw, bit (with #287's descent and target stages), scalar, primes (545 charts ≤ 576 factors, normalizer bound 815), banks (pivot residual widths tiled by (r⁴, 4³⁰⁻ʳ) banks on top of gen5's own patterns, every bank width 120 and full; selector calls 932,937,188,400 < 2⁴⁰), complex, math, finite — with ten omitted-stage negative controls; the geometry stage samples an actual entrance basis for every rank present. Every #287 literal the rewrite changes is a recomputed pinned value in expected/kernel-pins.json, asserted by pins.pin; none is dropped (KERNEL-PROOF.md lists them).

Verify

python -m pip install -r research/five-stage-gen5-collective-kernels/requirements.txt    # sympy 1.14.0
python3 -B research/five-stage-gen5-collective-kernels/verify.py --output /tmp/gen5-collective-kernels-verification   # about 4 minutes; not under -I

Prints PASS_IMMUTABLE_GEN5_KERNEL_ENTRIES_FIVE_STAGE_BANKED_CONSTRUCTION with κ. The workflow runs it and #285's generator check on Ubuntu (Python 3.11, 3.13).

Scope and limits

• Conditional finite construction under the retained public all-size hypotheses (no Lean certificate). Conditional κ = 7.42785663120535e-4: gen5 — compile-cost-annealed pair and all-but-one modules on #276's gen4 bit word, five-stage completed banks #285 and Conditional κ = 7.44136146100942e-4: two concave-descent retimings (904 + 111 gates) and 220 target-prefix squares on #285's gen5 five-stage banks #287 are open; their word and stages are inherited unchanged, not re-proved.
• The bit side is now 0.29% below the complex supplier (Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193's word, 7.47455·10⁻⁴ in the five-stage layout). A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word, complex saving 7.0991·10⁻⁴ (+1.28% over #194; no new κ, bit binds) #233's complex word gives 7.547·10⁻⁴ in that layout (Lean-checked through The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256 and Fourth result: the circuit of CrocSwap #233 — Walsh–Hadamard at 1 − 7547361/10¹⁰ and the Fourier transform of every length at 1 − 7547360/10¹⁰, checked in Lean jacobalansussman/wht-power-saving-lean#2); its gcert/1 certificate is a drop-in for this package's complex stage (verified separately, κ unchanged while the bit side binds) and will matter once bit-side gains exceed the margin.
• Not exploited: pivots of residual width 3 or 4; quintuple families (never net-positive under φ); the 928 twin pairs whose first frames do not intersect.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The gen5 word and its package are DreamingOfClouds' (#285, on #276); the descent and target-square stages and the package this extends are rohanarun's (#287); the target-prefix mechanism and the kernel-stage form are eumemic's (#268, on the source527 lineage); the response-kernel mechanism, collective kernels and the tiling rule are Dugongue's (#254/#259/#272/#283); the five-stage layout is hcg890's (#234) with Jacob Sussman's construction. This package contributes the gen5 censuses, the 450-entry selection, the generalised stage on gen5, the tiling, the re-pinned checks and the discovery scripts.

[Mister-Meeseeks]

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

Contribution: 450 collective response-kernel entries on gen5.

Review and limits: Source-bound full Linux pipeline and producer regeneration. Shared donors, nondegenerate entrances and inverses are checked on one emitted transcript.

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: 739c3bb0ad0596f427e0a72a2ebeca23277217a3. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.