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

[DreamingOfClouds]

κ = 148557132624107/(2·10¹⁷) = 7.42785663120535e-4, conditional, with the bit supplier binding.

compared with	κ	gain
highest open claim when I wrote this (#284, 07:10 UTC Oct 10)	7.26291692225845e-4	+2.27%
#276's gen4 package, which this extends	7.22869827347495e-4	+2.76%

This is a second structural change to the bit word. The gain does not come from frame or entrance tuning.
#276's gen4 word keeps its arc-aware local circuit. Its two module circuits are replaced by modules
annealed against the exact one-instance compile cost:

Module	gen4	gen5
pair + copied centre	448 additions, debt 184	463 additions, 299 arcs, debt 164
all-but-one	34 additions, debt 12	32 additions, 21 arcs, debt 11

The virtual word needs 17,292 auxiliary roles: 612 fewer than gen4 and 1,616 fewer than PR200. The
generator is byte-identical to #276's; only its module data and frozen arcs change.

cd research/five-stage-gen5-banks
python3 -m pip install -r requirements.txt     # sympy==1.14.0
python3 -B verify.py --output /tmp/gen5-five-stage-verification     # about 5 minutes, no network

What changed

There is one new self-contained package, research/five-stage-gen5-banks/, plus its own path-filtered
workflow. No existing file changes. The package is #276's package with the bit word swapped and the
constants re-pinned.

one invocation, m = 120	gen4 (#276)	gen5 (this PR)
virtual auxiliary roles	17,904	17,292
compensated reuse pairs	1,494	1,406
independent dirty registers R	16,410	15,886
independent entrances	2,466 (ranks 20, 21)	2,554 (ranks 17, 18, 20, 21)
literal stock (60 replicas)	1,283,035	1,247,070
normalized stock / deficit	256,607 / 52,800	249,414 / 52,800
bit coarse saving	7.23392746398452e-4	7.43337804075788e-4
κ	7.22869827347495e-4	7.42785663120535e-4

The complex supplier is unchanged (PR234/PR193, coarse 7.47454944651775e-4). The bit side still binds, now
0.55% below it, so the complex supplier, for example #233's word (7.547e-4 in this layout per #256),
is about to become the lever.

Why compile cost and not debt

Minimizing additions − carrier arcs alone is the wrong objective. A cyclic-interval all-but-one module
reaches the proven floor of debt 10, yet costs the whole word 4.5%, because it cuts roles into many
small rank steps.

So both modules were annealed on the exact one-instance compile cost: a single module instance compiled
in its real geometry with the generator's own compiler. Closed-form node-rank evaluators reproduce that
compile up to a constant offset, which allowed long annealing runs (10⁷ or more iterations each).

Region	Debt change	Raw a_b change (whole word)
pair module	184 → 164 per instance	+1.13%
all-but-one	12 → 11	+0.83%
together		+2.76%

The predicted and actual gains agree to 0.01%. gen5bit/README.md has the details.

The rank-17 entrances need width-7 bank blocks, so 10 mixed (7¹² 4⁹) banks are used per stage, as
source527 used mixed 11/4 banks. The total residual width is even, so the 60-replica tiling is exact.

What verify.py checks (9 mandatory stages, same as #276)

• virtual: the PR168-v4 checker on the virtual word, with all 5 controls rejected; then PR200's
  physical-word proof (1,710 changed frames, 1,406 handoffs, the physical row, the physical word on all
  19,406 columns over F₂ and the integer decoder), with 4 adverse controls rejected.
• raw / bit: the census of every physical path (both reflections), the event emitter, parity fusion
  (1,500,000 even payload ADDs elided), the five-stage global lowering (479,144 paid calls, deficit
  4,400), and exact h=24/m=120 geometry, including one actual entrance basis of each rank 17, 18, 20
  and 21.
• scalar: the retained source527 scalar program on all 19,406 F₂ columns and the signed integer
  decoder in both directions, with 11 actual-context corruptions rejected. The controls now always pick
  odd-coefficient responses, since even ones vanish over F₂.
• primes: 25,802 consumed bases, at most 118 bits, every residual below 2⁸⁰.
• banks: 411 exact charts and all 4,765,800 assignments; the 2,554 rank-5a completions are removed.
• complex, math, finite: both moment engines with the full 10⁻¹⁶ fallback, all 47 strict
  inequalities and 7 margins (the adjacent κ grid point is rejected), and the literal finite bill with
  757,938,545,400 selector calls.

gen5bit/producer/regenerate.py rebuilds the five pinned bit-word files byte for byte, in about 2
minutes. I replayed the package in my working environment, in a clean Python 3.12 + sympy 1.14.0 venv
from another directory, and from a fresh git checkout. All three give the identical certificate.

Outside the package, a read-only audit:

• reproduced the coarse value and κ with its own 400-bit ball arithmetic;
• re-derived every frame, all 1,406 aliases, every target chronology, the F₂ replay on all 19,406 columns
  and the helper ledger with exact arithmetic;
• confirmed that no check was weakened relative to 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.

Status

This is a conditional finite construction with exactly the retained interfaces of #276, #210 and #234.
There is no new Lean and no multiplier benchmark. The composable gen4 mechanisms (#279–#283: response
kernels, retimings, target-prefix squares, early restorations) should carry over to gen5 with the
constants re-pinned. I have not done that here.

Attribution

Everything credited in #276 is retained:

• Henry Grant (hcg890)'s Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234 and Jacob Sussman's five-stage construction;
• eumemic's source527 package and Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 generator and checker;
• Chafik Boukhalfa's A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 physical-word classes;
• Evan McKinney's Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197 and Rohan Arun's Conditional κ = 7.08582410781108e-4: completed width-120 entrance banks on #234's five-stage bit supplier #237 banks;
• Avi Eisenberg's Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 complex helper;
• icekylinx's Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 framework;
• the PR207, PR216, PR211, PR230 and PR210 notices.

The gen5 modules (the module search with exact compile-cost evaluators and annealing, run by Claude
Opus 5.5 agents) and the integration were prepared with Claude Opus 5.5 (Anthropic), via Claude Code. See NOTICE.md.

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: Compile-cost-optimized pair and all-but-one modules for gen5.

Review and limits: Source-bound full Linux pipeline and independent producer regeneration. Objective is the compiled moment, not additions alone.

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: b40c1a3d8f40ee3764b9f13fe850f663f2d8c834. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.