Conditional κ = 7.45513573133379·10⁻⁴: early restorations and terminal sinks on the gen5 kernel word, with #233's complex word as supplier (+0.107% over #290)

[chafreaky]

Conditional κ = 7.45513573133379·10⁻⁴: early restorations and terminal sinks on the gen5 kernel word, with #233's complex word as supplier (+0.107% over #290)

One self-contained package, research/five-stage-gen5-restorations-sinks/, with its own workflow. No existing file is changed.

κ = 745513573133379/10¹⁸ = 7.45513573133379·10⁻⁴: +0.1073% over #290's 7.44714499357628·10⁻⁴, above the rungs stacked on #290 (#291–#293, best 7.44729·10⁻⁴), and +0.187% over #287. Bit coarse 7.46070·10⁻⁴, complex coarse 7.54736·10⁻⁴; the bit side binds with a 1.16% margin.

What it is

The #290 package (#285's gen5 word, #287's 904-gate descent and 220 target squares, our 450 collective kernel entries) with three changes, each a stage or input of the same Python pipeline, each priced on the whole word (the φ-ledger model predicted every κ below exactly before the stage was built):

step	mechanism (credited)	census on gen5	κ
#290	—	—	7.44714499357628·10⁻⁴
complex supplier → #233's word	#233/#256 (ours), Lean-checked in jacobalansussman/wht-power-saving-lean#2	gcert/1 certificate replaces #193's	unchanged (bit binds), complex cap 7.47455 → 7.54736·10⁻⁴
440 early helper restorations	#280 (utcorvusvolat-dotcom), endpoint-aware banking as in #283 (Dugongue)	at the first cleanup ADD (record 601,737): 440 rank-20-entrance helpers whose single remaining incidence is a −= b, with entrance, both frames and a nondegenerate rank-23 join	7.45272862652714·10⁻⁴ (+0.075%)
8 terminal sinks	#283 (Dugongue; source527 lineage)	16 match the shape; 8 whose targets are not read by #287's squares are legal (the other 8 would need a downward move)	7.45513573133379·10⁻⁴ (+0.032%)

The complex swap does not change κ by itself; it keeps the restorations' and sinks' gains (+0.107% together) from being capped by #193's complex supplier, which sits only 0.29% above #290's bit side.

Checks

python3 -B verify.py runs twelve fresh stages — virtual, raw, bit (with #287's stages), kernel, restore, sink, scalar, primes, banks, complex, math, finite — with twelve omitted-stage negative controls:

• restore: 1,768,348 crossed-gate commutation checks; all 19,406 columns forward and inverse; omission control rejected (440 wrong rows); both reflected ledgers; completion P_σ + I − P_E, and the geometry stage rejects the old I − P_σ residual;
• sink: all 19,398 remaining columns forward and inverse; three omission controls rejected (32 / 24 / 24 wrong rows); each removed helper frees exactly 24 rank; the 4,400 deficit re-checked at R = 15,878;
• banks: charts built on (E ∩ σ⊥ | σ | E⊥), 985 charts (440 new), max 576 factors; literal stock 1,244,515 → 1,242,935; role census checked against the removed roles;
• complex: 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 gcert/1 program through the package's complex stage unchanged in logic (labels, splice, scalars, precision guard below 2⁴⁸; all complex mutation controls rejected), complex coarse 754736418878859/10¹⁸ pinned at the 10⁻¹⁸ grid;
• every changed literal is a recomputed pin in expected/kernel-pins.json (pins.py now also refuses one name with two values).

Verify

python -m pip install -r research/five-stage-gen5-restorations-sinks/requirements.txt    # sympy 1.14.0
python3 -B research/five-stage-gen5-restorations-sinks/verify.py --output /tmp/gen5-rs-verification   # about 5 minutes; not under -I

Prints PASS_IMMUTABLE_GEN5_KERNEL_RESTORE_SINK_FIVE_STAGE_BANKED_CONSTRUCTION. The workflow runs it and #285's generator check on Ubuntu.

Scope and limits

• Conditional finite construction under the retained public all-size hypotheses (no Lean certificate for the bit side). 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, 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 and 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) #290 are open; their word and stages are inherited.
• Not composed: Conditional κ = 0.000724189082265949: gen4 retiming, 440 early restorations and 60 dirty entrances #280's transported dirty entrances (on gen5, 1,911 of the 1,967 untouched plain helpers have a target read by a square, leaving ≈ +0.003%); 354 restoration candidates whose join is the full frame; single-target sinks (geometrically illegal on this word); the retiming rungs Conditional κ = 7.44728650221677e-4: second concave-descent retiming (89 gates, constructed frames) after #290's kernel entries on the gen5 word #291/Conditional κ = 0.000744719882668682: 65 compatible retimings on PR290 kernels #292/Fixed-prime eight-level gen5 descent: conditional κ = 744728654228590183370616/10^27 #293 on 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) #290 (compatible in principle, not rebound here).

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The early-restoration mechanism is utcorvusvolat-dotcom's (#280); the terminal sinks and endpoint-aware banking are Dugongue's (#283); the gen5 word is DreamingOfClouds' (#285); the descent and target stages are rohanarun's (#287); the kernel stage and the complex word are ours (#290, #233/#256), on Dugongue's (#254/#259/#272) and eumemic's (#268) mechanisms and the source527 five-stage lineage, hcg890's #234 layout and Jacob Sussman's construction.

[Mister-Meeseeks]

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

Contribution: Early restoration and terminal sinks on collective gen5.

Review and limits: Source-bound full Linux word and independent producer replay. Validity rests on actual writer/endpoint conditions, not generic scratch reuse.

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: a41781bb9bb45c368046e4e387c6ffeb041c8208. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.