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

[DreamingOfClouds]

κ = 144573965469499/(2·10¹⁷) = 7.22869827347495e-4, conditional, with the bit supplier binding.

compared with	κ	gain
highest open claim when I wrote this (#275, 04:17 UTC Oct 10)	7.13690108083058e-4	+1.29%
the source527 package (#210 head), whose machinery this reuses	7.10069340338651e-4	+1.80%

The gain does not come from frame or entrance tuning. It comes from a new paired-cube bit word (gen4) whose virtual circuit needs 1,004 fewer
auxiliary roles than PR200's. Here that word is carried through the complete five-stage completed-bank
package.

cd research/five-stage-gen4-banks
python3 -m pip install -r requirements.txt     # sympy==1.14.0
python3 -B verify.py --output /tmp/gen4-five-stage-verification     # about 5 minutes, no network

What changed

There is one new self-contained package, research/five-stage-gen4-banks/, plus its own path-filtered
workflow .github/workflows/five-stage-gen4-banks.yml, which runs verify.py and the regeneration on
Python 3.11 and 3.13. No existing file changes.
It is eumemic's source527 five-stage package (PR234 layout, stage-private width-120 completed banks,
parity fusion, exact geometry, primes, finite bill and outer assembly) with the bit helper replaced.

one invocation, m = 120	source527 (#210 head)	gen4 (this PR)
virtual auxiliary roles	18,908 (PR200 word)	17,904
compensated reuse pairs	1,760	1,494
source loans / terminal sinks	527 / 34	0 / 0
independent dirty registers R	16,587	16,410
independent entrances	2,279	2,466 (2,200 rank 20, 266 rank 21)
literal stock (60 replicas)	1,304,935	1,283,035
normalized stock / deficit	260,987 / 52,800	256,607 / 52,800
bit coarse saving	7.10573897320284e-4	7.23392746398452e-4
κ	7.10069340338651e-4	7.22869827347495e-4

The complex supplier is unchanged: PR234/PR193, coarse 7.47454944651775e-4. The bit side still binds, so
the remaining headroom on this layout is about 3.3%.

The gen4 bit word

PR168's p = 12 word is kept: the same roots, decoder and partner-pair mixing. Three parts of the circuit
change, and R = c + q − |M| falls from 19,788 (main) or 18,908 (PR200) to 17,904.

region (additions − carrier arcs)	PR168 L1	gen4
per-cube local channels (220 cubes)	22 each	16 each
pair module + copied centre (24)	191 each	184 each
all-but-one module (132)	15 each	12 each

1. Arc-aware local design. Every plane channel is any recipe tree of support-disjoint additions, not
   only PR144/189/200's edge, face-diagonal and fixed-coordinate options. Each cube uses 29 additions,
   but they create 13 in-place carrier arcs instead of 4: five partner-root arcs from the flip01
   face-diagonal pairs, and eight chain arcs. A CP-SAT model over all 15 recipe trees per plane proves
   that debt 16 per cube is optimal.
2. Centre-sharing pair module. The pair module's 56th root is star(2i+bit), the sum of all 55 inputs.
   It costs 10 extra additions on module nodes, replacing a separate 34-addition centre tree.
3. Re-annealed all-but-one module. It has 34 additions and 22 arcs.

Its physical layer applies PR200's rules with new producers:

• 1,494 compensated late-read reuse pairs, from a maximum Hopcroft–Karp matching;
• 1,802 descended operation frames: equal-frame groups moved to the join of their inputs or the meet of
  their outputs, accepted only when the five-stage cost falls.

What verify.py checks (9 mandatory stages)

• virtual: eumemic's PR168-v4 package checker on the virtual word, unmodified. It covers exact
  frames, the mod-2 decoder, geometry, chains, G-nondegeneracy, the F₂ replay and the ledger recount,
  with all five of its mutation controls rejected. The stage then runs PR200's physical-word proof,
  unmodified and exactly as PR200's prove.py runs it:
  • all 1,802 changed frames and 1,494 handoffs;
  • the physical row: phase one, actual operands, nested chains, deficit;
  • the physical word on all 19,930 columns over F₂ and the integer decoder;
  • four adverse controls rejected.
• raw: a fresh census of every physical source, internal and target path, with both reflected
  ledgers.
• bit: the source527 event emitter and parity fusion (1,510,656 even payload ADDs elided, no MOVE
  merged), then PR234's five-stage global lowering (487,546 paid calls, deficit 4,400). Exact rational
  h=24/m=120 geometry covers two actual ports, two rank-20 entrances and, newly, one rank-21 entrance.
• scalar: the source527 scalar program, byte-identical with all its selections empty. It runs on all
  19,930 formal F₂ columns and on the signed integer decoder in both directions. The emitted
  parity-filtered word and its literal inverse are checked independently on every column. Eleven
  actual-context corruptions are rejected; they replace the 25 source527 mutations, whose features gen4
  does not have.
• primes: cleared-Gram determinants of all 26,249 consumed bases, at most 118 bits; every residual is
  below 2⁸⁰.
• banks: 370 exact charts, with patterns (3⁴⁰) × 399, (4³⁰) × 4,400 and (24⁵) × 167,328. All
  4,923,000 role/replica/stage assignments are enumerated, and exactly the 2,466 rank-5a completions are
  removed.
• complex, math, finite: the retained complex supplier, two exact moment engines with the complete
  10⁻¹⁶ fallback (the adjacent bit grid point is rejected), all 47 strict outer inequalities and seven
  margins (the adjacent κ grid point is rejected), and the literal 60-replica finite bill with
  502,915,820,400 bank selector calls.

gen4bit/producer/regenerate.py rebuilds the five pinned bit-word files from the generator, the four
p = 12 data files and both producers, byte for byte, in about 2 minutes. This is provenance only:
verify.py never runs a producer.

I also replayed the package independently: a fresh copy, run from another directory under a clean
Python 3.12 + sympy 1.14.0 venv, gives the identical certificate.

Outside the package, two read-only audits imported no package code and used their own exact arithmetic
(FLINT and Bareiss). They reproduced:

• all 26,058 frames and the frame legality of every operation, including that labels equal actual values;
• all 1,494 aliases and every target chronology;
• an independent F₂ replay on all 19,930 columns;
• the helper ledger;
• the coarse saving and κ.

Status

This is a conditional finite construction with the same retained interfaces as #210 and #234:

• the all-size weighted compiler and common ancestor charts;
• restored rows and completed ordinary leaves;
• selectors, exact-tape routing and prime supply;
• precision/recovery and retained complex symbolic correctness;
• analytic transfer.

There is no new Lean and no multiplier benchmark.

Composes with

None of these is included, and each would re-pin the package's constants:

• the five-stage entrance and retiming work (κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249–Conditional κ = 0.000711457853376510: 887 entrances and connected-block retiming #270);
• PR210-style source loans;
• terminal sinks;
• more reuse pairs.

The bit word and these mechanisms act on different parts of the ledger.

Attribution

• The five-stage layout is Henry Grant (hcg890)'s Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234 with Jacob Sussman's construction.
• The package machinery is eumemic's source527 package, with OpenAI Codex assistance, rebound to the
  new helper.
• The banks come from 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.
• The physical-word classes are 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, vendored unchanged.
• The checker and generator are eumemic's Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168, on icekylinx's Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 framework.
• The complex supplier is Avi Eisenberg's Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193.
• Every upstream notice and AI-assistance disclosure is retained.

The gen4 circuit, both physical-layer producers and the integration were prepared with Anthropic Claude
assistance. The circuit and module search was run by Claude agents using annealing, CP-SAT and exact
one-instance compiles, and every number was re-run with the exact generator. See NOTICE.md.

🤖 Generated with Claude Code

[chafreaky]

Built on this: #282 carries the first of the #249-word entrance levers to the gen4 word — 424 response-kernel twin pairs (#254/#268's mechanism) with per-pair cuts, since gen4's compensation reads are chronological (each helper's zero-frame reads precede its first other use, so there is no single prefix cut), tiled with (23⁴, 4⁷) banks at 60 replicas — giving 7.23241·10⁻⁴ on your word alone (+0.051%) and 7.23944·10⁻⁴ composed with #279's descent stage. Your package's stages run unchanged with the changed literals re-pinned (expected/kernel-pins.json); the generator check passes as well. The plain helpers' response matrix has F₂ nullity 12,184 on gen4, so there is a lot of collective-kernel room left.