Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314)

[DreamingOfClouds]

κ = 152846172264049/(2·10¹⁷) = 7.642308613202450e-4, conditional. The bit supplier binds.

compared with	κ	gain
#314, the highest open claim when I wrote this (15:55 UTC Oct 10)	7.55120411314104e-4	+1.207%
#285 (gen5 five-stage package, p = 12)	7.42785663120535e-4	+2.887%

The open frontier (#312, #314) is the p = 12 w3 word priced against #304's complex supplier (7.6359e-4), with the
bit side binding. This package changes both suppliers:

1. The bit word at cube size p = 10. The paired-cube bit word (PR168 → 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 gen4 → 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 gen5) uses
   h = 20 coordinates and v = 960 sources instead of p = 12, with h = 24 and v = 1,760. The five-stage width becomes
   m = 5h = 100. Its certified bit root is 7.64815357e-4; gen5's was 7.43338e-4 and gen5b's 7.47268e-4.
2. A stronger complex supplier: a centre-sharing complex program, 7727/10⁷. The source-assisted complex word
   (Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184/Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193/Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194) is built with A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's recipe, but its pair module also supplies the copied star centre.
   • Its coarse saving with the 10⁻¹⁶ fallback is b = 7.72714351e-4: A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's word gives 7.6359e-4 and 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
     7.5474e-4 (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 figure uses the inherited no-fallback convention).
   • Every run checks the program with Sussman's gx.check1 (exact scalar identity) and his gxcore mirror.
   • The same gcert bytes were also kernel-checked in Lean with Sussman's unchanged generator. That run is
     local and not pinned in the package; details below.

cd research/five-stage-p10-banks
python3 -m pip install -r requirements.txt          # sympy==1.14.0
python3 -B verify.py --output /tmp/p10-five-stage-verification     # about 3 minutes, no network

Why p = 10

Under the five-stage price, the saving is about a ≈ D/L:

• D = 4v − 5h(h−2) is the deficit;
• L = Σ r·n·ln(m/r) is the ledger;
• m = 5h.

For the paired cubes, D/v = 4 − 15/(p − 2) saturates as p grows, while m grows linearly in p. PR168 chose p = 12
under the three-stage price, and its p = 11 and p = 13 entries used weaker modules. Jacob Sussman's open-directions
note lists "neighbouring sizes … not priced in the five-stage word" as an open lead.

I priced each cube size with one recipe for all of them: witness-cut pair module plus centre, nested-prefix
all-but-one, gen5's local design and the same compiler. The table gives the five-stage virtual price. These are
measurements on this pipeline, not theorems.

p	9	10	11	12	13
five-stage virtual a (·10⁻⁴)	6.480	6.638	6.541	6.424	6.217

Through the physical layer, the p = 10 cut word prices 3.5% above the p = 12 cut word (unbanked). With annealed
modules the p = 10 word reaches 7.4425e-4 virtual, against gen5's 7.2824e-4. The complex word shows the same shape
on an ensemble of library-module draws:

p	9	10	11	12
complex five-stage price (·10⁻⁴)	6.48 ± 0.06	6.45 ± 0.04	6.30 ± 0.05	6.08 ± 0.05

The complex supplier in this package is still the p = 11 word. It does not bind, so the bit side is the one that
matters here.

The p = 10 bit word

one invocation	gen5 (#285), p = 12, m = 120	this package, p = 10, m = 100
virtual auxiliary roles	17,292	9,120
compensated reuse pairs	1,406	890
independent dirty registers R	15,886	8,230
formal F₂ columns 2v + R	19,406	10,150
independent entrances (ranks)	2,554 (17, 18, 20, 21)	1,270 (16, 17)
normalized stock / deficit	249,414 / 52,800	132,606 / 24,480
bit coarse saving (certified root)	7.43337804075788e-4	7.64815357146385e-4
κ	7.42785663120535e-4	7.64230861320245e-4

The generator is #285's (eumemic's PR168 generator with #276's local-design and centre-sharing changes). It takes
three p = 10 data files:

• Pair module (pm_p10.json): 36 inputs plus the centre root, 273 additions, debt 95. It is a restriction of
  gen5's pair module to 9 points, annealed against the exact one-instance compile cost. The closed-form rank
  evaluators were re-validated at h = 20.
• All-but-one module (qmod_p10.json): 24 additions, debt 8. That is the floor, because the eight size-7
  roots cannot donate.
• Local design (local_design_p10.json): the cube-local recipe trees of a pair-aware local-design search run
  on the p = 12 word. It is cube-local, so it is reused unchanged.

The physical layer follows PR200's rules: 890 compensated reuse pairs with rank h − 3 = 17 recipients, and 998
descended frames. Two of the 892 matched pairs are dropped, the minimum that makes the 60-replica bank width
divisible by m = 100. That costs about 1.5·10⁻⁸ in the unbanked price.

What changed in the package

There is one new self-contained package, research/five-stage-p10-banks/, plus its own path-filtered workflow.
No existing file changes.

The package is #285's with every p = 12 literal replaced. Each became either a cube-size formula, or a value
pinned in word-pins.json and checked by strict equality. discovery/repin.py records a new word's pins;
verify.py itself only runs strict. Specifically:

• Five-stage layout:
  • m = 5h;
  • idle ranks 2h−2, h−1, 2h+2 and 4 (38/19/42/4 here; the complex side's are 42/21/46/4 at h = 22);
  • the deficit 4v − 5h(h−2) = 2,040, asserted from the actual paid histogram.
• code/geometry527.py: builds the exact 100×100 rational routes and checks the eight idle boundary
  projectors, the completion projectors (trace 5a) and the five disjoint windows. It does so on two actual rank-16
  entrance bases and one rank-17 basis.
• Banks: an exact tiler replaces gen5's fixed patterns and asserts zero padding: (3²⁰ 20²) × 210,
  (4²⁵) × 2,880 and (20⁵) × 83,436 banks per stage. bank_check.py enumerates all 2,469,000 assignments.
• Primes: the stripped primes are {2, 3, 11} (det 9G = −11·9¹⁹ at h = 20), with every residual below 2⁸⁰.
• Finite bill: it pays h = 20 copied-centre temporaries and 5h = 100 copy/erase episodes per replica. That is
  my reading of the inherited "24 temporary roles" and "120 fresh copies". With the original constants 24 and 120,
  the bill also passes with the same cutoff.
• Complex stage: the complex chain comes from a Conditional κ = 7.46709720962478·10⁻⁴: gen5b — exactly optimal local design and a debt-10 all-but-one module on #285's word, #287's stages re-derived, #233's complex supplier (+0.348% over #287) #296-derived package, re-pinned for the new gcert.
  code/complex_gx.py is new and runs gx.check1(scalar=True) and gxcore.mirror on the pinned program, with a
  flipped-sign control. The complex coarse b is certified with the fallback, and the adjacent grid point is
  rejected.
• math_check.py: it can also certify a complex-bound package by feeding the largest grid point at or below
  (1 − β)·b to the bootstrap. That path is not exercised here: the bit binds, so the assembly is 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's.

Verification

• Replays:
  • The package: Python 3.12 and 3.14 (the versions in the workflow matrix), from fresh copies and from
    git archive of this commit, are byte-identical apart from timings. verify.py takes about 3 minutes.
  • regenerate.py reproduces the five pinned bit-word files byte for byte (about 45 s).
• Two independent read-only audits by separate Claude Opus 5.5 agents that I directed, with no blockers. They used no
  package code for their own arithmetic:
  • Bit side: a diff of every check against 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, with no weakened check. The only loosened line,
    fallback == 346314 < 460800, became the load-bearing fallback < 32m². Mutation tests of geometry527
    reject wrong idle ranks and a truncated completion projector.
  • Bit root: an own 400-bit interval recomputation gives c as the exact fallback-inclusive floor. It also
    re-derived the tiling, the prime exclusions and the finite bill, and gave an independent decoder check at
    p = 10.
  • Assembly and complex:
    • κ recomputed exactly in its own rational arithmetic;
    • the complex transplant diffed file by file against its verified source;
    • gx.py and gxcore.py checked byte-identical to Sussman's f010392c;
    • three extra program mutations, all rejected by both checkers;
    • the complex price recomputed independently.
• The complex program alone: an earlier audit, also by a separate Claude Opus 5.5 agent, re-derived its decoder over all 1,320 × 1,320 (target, source)
  pairs and re-applied the Graph.finish patch byte for byte. It replayed the scalar identity independently and
  confirmed 7727 is certified and 7728 rejected, with and without the fallback.
• Lean (local, not pinned):
  • Setup: Sussman's wht-power-saving-lean at f010392c (Lean v4.34.1, Mathlib d13f23b), with his unchanged
    tools/gx/regen_check.py.
  • What was checked: the exact gcert pinned here (sha256 bb0499ab…4d34) compiled and kernel-checked as
    OAI.PowerSaving.WHT.wht_main_block_B2Gcc27x, exponent 1 − 7727140/10¹⁰, 34/34 modules. tools/Compare.lean
    passed against a challenge that differs from his ChallengeB2Gp193x only in exponent and names. The axioms are
    propext, Classical.choice and Quot.sound.
  • Not run: the official comparator and a clean build from nothing.
  • Scope: this covers the complex program's Walsh–Hadamard saving only. Because the bit binds, κ is unchanged
    if b is replaced by the Lean exponent.
  • A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's program too: its 7635 program was checked the same way (wht_main_block_B2Gcm35x,
    1 − 7635875/10¹⁰).

Status, honestly

• Conditional: this is a conditional finite construction with 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's retained interfaces: the all-size
  compiler, common weighted charts, restored rows, selectors, routing, prime supply, precision/recovery, complex
  symbolic correctness and analytic transfer. There is no Lean for the bit word, the bootstrap or the assembly, and
  no multiplier benchmark.
• Inherited texts: proof/, UPSTREAM-PR234-PROOF.md and the notices are inherited verbatim. They state
  p = 12 / source527 numbers and describe PR193's complex program. PROOF.md says so and gives the p = 10 values.
• The complex program is new and unpublished elsewhere.
  • It was built with A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's recipe from a centre-sharing pair module, the earlier re-annealed triple module
    (tmod_final, v1; not A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's v2) and A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's all-but-one module.
  • It needs a 10-line patch to PR Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's Graph.finish and a 46-root loader. The patch and modules are shipped as
    provenance in inputs/complex/centre-mw/, but the loader is not.
  • The gcert bytes are kept as emitted, with non-canonical key order, because the Lean run checked exactly those
    bytes.
• Composable follow-ups: the community stages on gen5b and the w3 word should carry over to the p = 10 word
  with the pins re-recorded. These include kernels, restorations, sinks, target squares, reorderings and
  dirty-response transport. A p = 10 complex word (7737/10⁷ with A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's recipe, 7795/10⁷ centre-sharing) exists
  in my research tree but is not needed while the bit binds. I have not done either here.

CI note. This branch also carries Rohan Arun's #318 commit, cherry-picked unchanged (it re-pins the root
Makefile and README.md hashes in research/joint-dual/SOURCE.json). Without it the repo-wide verify (…, joint)
job fails on main and on every open PR. The commit is identical to #318's and drops out once #318 is merged.

Attribution

• Five-stage layout and gcert/gx: 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,
  gcert/1, gx.check1 and gxcore.
• Generator, checker and package: eumemic's Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 generator and checker, and the source527 package with its
  lineage. The notices of PR207, PR216, PR211, PR230 and PR210 are retained.
• Physical layer and complex re-pin: Chafik Boukhalfa (chafreaky)'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. His 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
  reuse pairing, The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256 gcert emitter and Conditional κ = 7.46709720962478·10⁻⁴: gen5b — exactly optimal local design and a debt-10 all-but-one module on #285's word, #287's stages re-derived, #233's complex supplier (+0.348% over #287) #296 re-pinning of the complex checks for a new gcert are also used.
• Banks: Evan McKinney's Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197 completed-bank method and Rohan Arun's Conditional κ = 7.08582410781108e-4: completed width-120 entrance banks on #234's five-stage bit supplier #237 width-120 banks, here at width 100.
• Complex word: Avi Eisenberg (ikeboy)'s Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193/Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194 complex helper, and icekylinx's Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 framework and Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184
  frame flow.
• Earlier credits: everything credited in 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 = 7.42785663120535e-4: gen5 — compile-cost-annealed pair and all-but-one modules on #276's gen4 bit word, five-stage completed banks #285 and A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304 is retained. See NOTICE.md.

The cube-size scan, the p = 10 module annealing, the centre-sharing complex program, the package generalization,
the audits and the Lean run were prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code (Claude Opus 5.5 agents).

🤖 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: p=10 paired-cube bit word and centre-sharing complex supplier.

Review and limits: A construction change in both suppliers; includes physical, scalar, determinant, bank, recurrence and finite-invoice checks plus producer regeneration.

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: 64bccfb67494f7c4549bfb2f186b400a3cfefb42. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.