Conditional κ = 7.26291692225845·10⁻⁴: 91 collective response-kernel families on helpers untouched by #283, at its own cut (+0.0248% over #283)

[chafreaky]

Conditional κ = 7.26291692225845·10⁻⁴: 91 collective response-kernel families on helpers untouched by #283, at its own cut (+0.0248% over #283)

One self-contained package, research/gen4-collective-families-283/ (PR #283's package with 91 rows appended, one contained block in its bank allocator, regenerated receipts), with its own workflow. No existing file is changed.

κ = 181572923056461298166587/(25·10²⁵) = 7.26291692225845·10⁻⁴ by #283's fixed-prime price, +1.80·10⁻⁷ over #283's 7.26111344123733·10⁻⁴ and +0.473% over #276's gen4 word; the bit side binds.

Result

A census of the 13,944 plain dirty helpers of the gen4 word (12,352 distinct complete F₂ target responses of rank 1,760) enumerates the low-weight F₂ circuits among the responses — 1,616 twin pairs, 5,245 triples, 38,633 quadruples after twin expansion — keeps those with a nondegenerate common entrance inside all members' first required frames (616 / 1,131 / 4,490), prices them under the deficit-fixed ledger (φ(r) = r·ln(120/r); a quintuple is never net-positive) and packs them greedily with shared donors on nested chains. On the bare gen4 word this gives a 695-entry selection (416 pairs, 24 twin pairs entering on 16-dimensional intersections, 255 multi-donor families; model κ 7.24029·10⁻⁴ before any retiming), whose helpers largely coincide with #283's 741-pivot selection. On helpers that no part of #283 touches (its 1,612 kernel roles, 16 sinks, 60 transported candidates), 91 families remain net-positive — 87 quadruples and 4 triples: (3, 3, 3, 3) with e = 2 × 49, (5, 5, 5, 5) with e = 4 × 10, families of first dimension 10–17 entering on e = r − 1 or r − 2; entrance rank 456, 14 donors shared. All are legal at #283's global cut 520,031, so they are appended to inputs/kernel-original.json in #283's schema and go through its sink remap and kernel transform unchanged.

	#283	this package
kernel pivots / entrance rank (with transports)	741 / 2,074	832 / 2,530
one-stage paid rank mass	425,358 → 422,844	425,358 → 422,388
literal stock (120 replicas) / normalized	2,551,580 / 510,316	2,549,300 / 509,860
κ (fixed-prime price)	7.26111344123733·10⁻⁴	7.26291692225845·10⁻⁴

The ledger model predicted #283's price → 7.26291688·10⁻⁴; the package's pricers give exactly that before the fixed-prime refinement.

The one code change (PROOF.md, section 3)

#283's code/banks.cpp tiles a residual-r role with four r-blocks and (30 − r) rank-4 fillers per bank; the 91 families need 30,060 rank-4 filler copies where #283's word leaves 10,380. The allocator now lets a residual group that would exhaust the rank-4 pool use rank-3 fillers instead (4r + 4b + 12k = 120 with b = (30 − r) mod 3), both remainders and the pool asserted, every bank width 120 and full; when the pool suffices the allocation is byte for byte #283's. Here only #283's own 29 rank-19 transported entrances switch (pattern [6 × 4, 3 × 32] in 870 banks; 3⁴⁰ banks 2,118 → 1,422). Without the change, 25 of the families fit the pool and give 7.26226544·10⁻⁴ (rows kept in discovery/).

Checks

#283's verifier, otherwise unchanged, end to end: the nine #276 stages on the vendored immutable package; sink transform and audit; kernel transform (832 pivots, 13,136 compensation reads removed, 3,462 Q/Q⁻¹ gates, all 19,914 columns forward and inverse, omitted-operation controls non-vacuous); #279's retiming (880 gates), #280's transports (29) and restorations (440), the 220 target squares; independent legality and prefix audits (1,464,320 kernel equalities on 1,956 helpers); 23,434 global columns; bank review with the new allocation (4,923,000 assignments, chart bound unchanged, max 312 factors); price, fixed-prime price (47 constraints, adjacent grid rejected) and finite invoice; every receipt regenerated and pinned. The families were also replayed on the bare gen4 word with per-family omitted-setup and omitted-restore controls (discovery/replay_families.py).

Verify

sudo apt-get install -y g++ libboost-dev
python -m pip install -r research/gen4-collective-families-283/requirements.txt    # sympy 1.14.0
python3 -B research/gen4-collective-families-283/verify.py --output /tmp/gen4-collective-replay   # about 16 minutes locally; fresh directory outside the package

Prints the certificate PASS_SOURCE_REGENERATED_COMPOSITION with κ. The workflow runs it on Ubuntu with Python 3.11 and 3.13.

Scope and limits

• Same conditional status as Conditional κ = 7.22869827347495e-4: a new paired-cube bit word (arc-aware local circuit, centre-sharing pair module) in the five-stage completed banks #276/Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283 (finite construction under the retained public all-size hypotheses; no Lean certificate); Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283's own correctness claims are inherited, not re-proved.
• The 24 e = 16 twin pairs and most multi-donor families of the bare-word selection sit on helpers Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283 already uses and were not re-derived on its word; per-family cuts were not needed (all 91 are legal at the global cut).
• Diff against Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283's package (discovery/README.md): inputs/kernel-original.json (+91 rows), inputs/kernel-remapped.json, code/banks.cpp (filler block), one sentence in PROOF-PR283.md, regenerated expected/ and RESULT.json, MANIFEST.json, this package's notes and discovery/.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The composed word, the donor-first kernel selection, sinks, audits, pricers and the bank allocator extended here are Dugongue's (#283, on #254/#259/#272); the gen4 word is DreamingOfClouds' (#276); the retimings and target squares are rohanarun's (#279); the transports and restorations are utcorvusvolat-dotcom's (#280); the target-prefix mechanism is eumemic's (#268) on its source527 five-stage lineage; the five-stage layout is hcg890's (#234) with Jacob Sussman's construction. This package contributes the collective-kernel census, the 91 families, the rank-3 filler block, the receipts 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: 91 additional collective families on the gen4 composition.

Review and limits: Authenticated PR286 Linux CI explicitly fetched this exact PR284 head, checked the identical manifest and passed its entire native parent verifier (601.8s). Redundant local rerun stopped after establishing that binding; no fresh complete local pass claimed.

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: 2c991a3ac400057059145c837e8b32bba9128346. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.