Conditional κ = 7.08582410781108e-4: completed width-120 entrance banks on #234's five-stage bit supplier

[rohanarun]

Conditional κ = 177145602695277/250000000000000000 = 0.000708582410781108, +0.694% over #234 (0.000703701743496697) and +1.99% over #230. Full local run passed (--full re-runs #234's own six-stage verifier first).

Idea

#234's five-stage bit supplier completes each of its 2,279 genuine independent entrances with a bundled rank-5a exterior child (2,200 of rank 100, plus 18 of rank 60, 48 of rank 65 and 13 of rank 90). Its ledger notes these are "not banked".

This package applies PR197/#205's completed-bank construction to the five-stage cover at width m = 120: route each entrance residual I − P_σ through an exact chart into full-width banks whose disjoint residual involutions multiply to the full endpoint, and drop those exterior children.

• Charts: 231 distinct entrance frames (220 PR200 rank-20 + 11 PR210 joint frames), G-orthogonal complement 15a−(Σa)1 for G=9I−J plus exact kernel basis; fraction-free inverse and factor replay (≤221 factors, all integers < 2^80).
• Banks: 12 data copies → 60 residual occurrences per chain (5 stages × 12 copies). Residual ranks 4/12/11/6 and 24 fill exactly 177,179 banks of width 120 (360×(8·11+8·4), 108×(10·12), 39×(20·6), 4,304×(30·4), 172,368×(5·24)).
• Colouring: proper 60-edge-colouring of the 998,580-incidence chain/bank multigraph (colour = stage + 5·copy), checked at both ends; a duplicated colour is rejected.
• Profile: W = 12·4v + 177,179 = 261,659, m = 120, deficit 52,800 = 12·4,400 (unchanged per copy), largest child 50. Only the 2,279 exteriors are removed.
• Arithmetic: Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234's own interval moment engine (with the 10⁻¹⁶ bad-class envelope), three finite ordinary levels and Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234's unchanged outer-47 assembly (η=10⁻¹², β=10⁻⁹). Coarse bit saving 7.090849e-4 (was 7.041973e-4); 47 strict inequalities and 7 margins pass; the next 10⁻¹⁸ grid point is rejected. The bit side still binds (complex 7.4745e-4).

	bit coarse	κ
#234	7.041973e-4	0.000703701743496697
this	7.090849e-4	0.000708582410781108

Reproduce

git fetch https://github.com/CrocSwap/integer-mult-bounds pull/234/head
git worktree add --detach ../pr234 af3fe331ca60c936229e060681ffccbc1c208678
python3 -m pip install sympy==1.14.0
python3 -B research/five-stage-banks/verify.py --pr234-root ../pr234 --full

About 4 minutes (3.5 of them are #234's own verifier). A workflow is included.

Scope

Conditional finite composition. All #234 dependencies are retained (five-stage physical/scalar/prime checks, complex symbolic theorems, weighted q-local compiler, complete-stream movement, copied-centre/restored-row, ordinary leaves, prime supply, precision, all-size reduction). The bank endpoint argument is PR197's, applied with five stage windows; the extra routing bill 12,227,329,044 < 2^40 is absorbed by the inherited ordinary toll as in PR197. Source-owned PR210 gauges are not banked.

Credits: hcg890 (#234) and Jacob Sussman (five-stage layout); Evan McKinney (PR197 banks); eumemic (#210 helper); Chafik Boukhalfa (#200); Avi Eisenberg (#193); all contributors credited by those packages.

Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[Mister-Meeseeks]

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

Contribution: Completed width120 entrance banks.

Review and limits: Prior exact-head original changed-bank verifier passed; source-bound Linux full dependency replay retained.

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: a8b0ce212bc9f09353a8161bf8a5cee9cefd600c. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.