Conditional κ = 7.24733001963261·10⁻⁴: 695 collective response-kernel entries (per-entry cuts, entrances of any dimension) on #276's gen4 word, composed with #279's descent retiming (+0.258% over #276)

[chafreaky]

Conditional κ = 7.24733001963261·10⁻⁴: 695 collective response-kernel entries with per-entry cuts on #276's gen4 word, composed with #279's 880-gate descent retiming (+0.258% over #276, +0.161% over #279)

One self-contained package, research/five-stage-gen4-collective-kernels/ (renamed from the first version of this PR, which carried 424 line pairs at 7.23943728615132·10⁻⁴), with its own workflow. No existing file is changed.

κ = 724733001963261/10¹⁸ = 7.24733001963261·10⁻⁴, the bit side binding (bit coarse 7.25259·10⁻⁴, complex 7.47455·10⁻⁴). This is the standalone construction in #276's own Python pipeline; #283/#284 compose further levers (terminal sinks, transported entrances, early restorations, target squares) in Dugongue's native pipeline and stand above it — the contribution here is the collective families themselves, verified in #276's chain.

Result

#276 (DreamingOfClouds) introduced the gen4 bit word; none of the entrance levers developed on the #249 word (#254–#275) had been carried to it. This package carries response kernels there, generalised: as a stage of #276's own six-int transcript pipeline, after #279's descent stage (vendored unchanged), 695 entries — 416 twin pairs on lines (e = 1, #254/#268's mechanism), 24 twin pairs entering on their full 16-dimensional first-frame intersection (each worth more than ten line pairs), and 255 multi-donor families (208 quadruples, 47 triples: the pivot's response is the XOR of its donors', all members share a nondegenerate common entrance of dimension 1–18, 59 donors serve several entries along nested chains; #259/#272's collective kernels).

one invocation, m = 120	#276	#279	this package
entrances (ranks)	2,466 (20, 21)	2,466	3,161 (1 × 476, 2 × 93, 4 × 29, 5–18 × 97, 20 × 2,200, 21 × 266)
one-stage paid rank mass	425,742	—	423,714
literal stock, 60 replicas / banks per stage	1,283,035 / 172,127	—	1,277,965 / 171,113
κ	7.22869827347495·10⁻⁴	7.23571590007464·10⁻⁴	7.24733001963261·10⁻⁴

• Per-entry cuts. On gen4 the compensation reads are chronological: each plain dirty helper's zero-frame reads precede its first other use (by at least 41,247 records), so there is no single prefix cut; each entry is cut at the later of its members' last zero-frame reads, bound by content. The twin census of the 13,944 plain helpers finds 1,568 twin classes and 568 pairs with a nondegenerate common line; the collective census enumerates the low-weight F₂ circuits of the 12,352 distinct responses (3,158 triangles, 23,421 quadruples), keeps those with a nondegenerate common entrance in all first frames, prices them under the deficit-fixed ledger (φ(r) = r·ln(120/r); quintuples are never net-positive) and packs them greedily with shared-donor chains. Total entrance rank 2,028.
• Rewrite (eumemic's κ = 7.11191385398487e-4: compose kernel entrances with 217 target-prefix groups #268 kernel_transform.py, generalised): the pivot's compensation reads are deleted (12,520 in all), it enters at E, each donor pays its shear at E after the cut and the inverse at the full frame (2,316 ADDs); one-stage paid rank mass 425,742 → 423,714.
• Checks. kernel_transform.replay runs every formal column (19,930) forward and inverse on the emitted word, rejects both omitted-gate controls (2,713 / 1,061 wrong rows), checks the prefix relation (pivot response = XOR of the donors' responses) on the literal prefix for every entry, rebuilds the stream frame paths independently of the MOVEs and checks the nested donor chains; then every 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 stage runs fresh — virtual, raw, bit, scalar (eleven controls), primes (every entrance basis factored; 664 charts ≤ 576 factors, normalizer bound 815), banks (every residual width r tiled by (r⁴, 4³⁰⁻ʳ) banks, 20 pattern kinds, every bank width 120 and full; selector calls 963,705,578,400 < 2⁴⁰), complex, math, finite — with ten omitted-stage negative controls, and the geometry stage samples an actual entrance basis for every new rank. Every 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 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 all of them).

Verify

python -m pip install -r research/five-stage-gen4-collective-kernels/requirements.txt    # sympy 1.14.0
python3 -B research/five-stage-gen4-collective-kernels/verify.py --output /tmp/gen4-collective-kernels-verification   # about 8 minutes; not under -I
python3 -B research/five-stage-gen4-collective-kernels/gen4bit/producer/regenerate.py --work /tmp/gen4-regeneration

verify.py is #276's with the kernel stage required and κ pinned (PASS_IMMUTABLE_GEN4_KERNEL_PAIRS_FIVE_STAGE_BANKED_CONSTRUCTION). The workflow runs both commands 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 and the source527 lineage (finite construction under the retained public all-size hypotheses; no Lean certificate). Conditional κ = 7.24098741932703e-4: 880-gate concave-descent retiming and 220 target-prefix squares on #276's gen4 five-stage banks #279 is a draft at the time of writing: its stage is vendored unchanged and its correctness claims are inherited, not re-proved. One composition detail: the used-frame inventory passed to the prime check keeps the inherited producer inventory (a retimed gate's former frame is still a required base of the virtual word), a superset; Conditional κ = 7.24098741932703e-4: 880-gate concave-descent retiming and 220 target-prefix squares on #276's gen4 five-stage banks #279's own package builds that inventory from actual uses only, which may be why its full replay is still pending.
• Not composed here: terminal sinks, transported entrances, early restorations and target squares (Conditional κ = 0.000724189082265949: gen4 retiming, 440 early restorations and 60 dirty entrances #280/Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283); Conditional κ = 7.26291692225845·10⁻⁴: 91 collective response-kernel families on helpers untouched by #283, at its own cut (+0.0248% over #283) #284 adds 91 of these families on Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283's word. Not exploited: cleanup sandwiches (gen4 helpers are compute/read/uncompute nodes with mirrored cleanup, so none exists), residual widths 3 and 4, the admissible pairs that are net-negative under φ.
• Diff against 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's package: kernel_transform.py, kernel-selection.json, pins.py, expected/kernel-pins.json, KERNEL-PROOF.md, descent_transform.py and descent-selection.json (Conditional κ = 7.24098741932703e-4: 880-gate concave-descent retiming and 220 target-prefix squares on #276's gen4 five-stage banks #279), modifications to portable_bit.py, raw_ledger.py, code/global_lowering.py, code/geometry527.py, bank_template.py (the (r⁴, 4³⁰⁻ʳ) patterns), bank_check.py, scalar_check.py, prime_check.py, math_check.py, finite_check.py, verify.py; discovery/; this package's notes (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's kept as README-PR276.md, PROOF-PR276.md).

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The gen4 word and its package are DreamingOfClouds' (#276); the descent retiming stage is rohanarun's (#279); the kernel stage adapts eumemic's #268 (its mechanism is Dugongue's #254/#259/#272, the tiling rule #283's); the five-stage completed-bank package is eumemic's source527 lineage (#251/#249/#244/#210) on hcg890's #234 layout, with Jacob Sussman's five-stage construction, Evan McKinney's and rohanarun's banks, ikeboy's complex helper and icekylinx's paired cubes. This package contributes the per-entry-cut censuses and selection, the generalised stage (any entrance dimension, multi-donor families, shared donors), the bank tiling for every residual width at 60 replicas, the per-rank geometry sample, the re-pinned checks, the composition with #279, 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: Collective response-kernel entrances on gen4.

Review and limits: Source-bound full Linux pipeline and independent producer regeneration. Multi-donor response relations and common nondegenerate entrances are charged explicitly.

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: 841a99b8cf0609ce461b208ed6367463459dd7f5. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.