Searched pair module in the paired-cube bit word at p = 11: conditional κ = 5.616446e-4

[GamingPuzzled]

Draft: locally make paired-cube-verify and the paired-cube tests pass on Python 3.11. Fork CI is running.

Under #157's retained interfaces,

κ = 2808223/(5·10⁹) = 5.616446×10⁻⁴, +1.106% over #157 (5.555021e-4) and +1.06% over #158 (5.557315e-4). The complex side now binds.

Change. Only the bit supplier changes. #157's complex word is unchanged and its certified fields are identical.

#157's paired-cube bit word uses p = 12 with a pair-disjoint module derived by a witness cut. #157's complex word uses a searched pair-disjoint module (pmod_sa1, eumemic) on 10 points. That module satisfies exactly the same pair-disjoint contract, so the bit word at p = 11 can use it as is.

• The module is copied unchanged as data/pair_module_p11_annealed.json.
• The carrier matching is re-solved and frozen as data/arcs_p11_annealed.json.
• The derived modules and their pins are untouched.

bit word	pair module	R	W	gauges	coarse a0
#157: p = 12	derived	25,624	29,144	5,720	5566382/10¹⁰
p = 11	derived	18,964	21,604	4,290	5502556/10¹⁰
this: p = 11	searched (#157)	18,106	20,746	4,290	5641569/10¹⁰

The bit coarse saving rises above the complex saving (5622769/10¹⁰), so κ is now complex-bound.

Checks

• paired_cube_bit_word.py --p 11 --check regenerates the word and profile byte-identically from the frozen arcs.
• check_paired_cube_bit.py --dir out --p 11, the independent checker, passes: 15,280 exact frames, the decoder identity, all chains, partner-pair chronology, nondegeneracy, the literal F₂ replay and the ledger. All 5 mutation controls are rejected.
• moment_certificate.py gives coarse 5641569/10¹⁰, with the next grid point rejected.
• derive_pair_modules.py --check still passes (derived pins unchanged), and the p = 12 word still checks.
• make paired-cube-verify passes: the complex producer and all its independent checks, then the network certificate with both moments, the finite router and all 47 strict constraints and 7 margins.
• The network changes are confined to the bit row loader (p = 11), its dimension pin and the constants COARSE and KAPPA.
• certificates/paired-cube-input.json is regenerated from the certificate computation. Every complex field is unchanged; only the bit fields and κ change.

Scope. Every interface #157 retains stays an assumption. This is a finite conditional witness.

Credits:

• eumemic: the paired-cube bit word, the searched pair module, and Paired cubes at p = 11 with a searched pair module, physical frames and a paired-cube bit word: conditional κ = 5.555021e-4 #157's complex word with its physical frames and reuse.
• icekylinx: paired cubes (Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144, Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130).
• an664: completed-core sharing (Completed-core sharing: conditional kappa > 2^-13 #128).
• Zhihao Chen and Swapnil Jain: bit lineage.
• All inherited contributors.

Composition prepared by Joel Pulikkan with Anthropic Claude assistance.

🤖 Generated with Claude Code

[RobGruhl]

Upstream asks for independent replays, so I replayed this PR's checks on another machine and did one extra thing
with research/paired-cube-bit/check_paired_cube_bit.py: I re-ran it with its three replay seeds
(random.Random(1000 + seed)) replaced by recorded random ones, to see whether the result depends on the scratch.
206 runs in two batches, across the 18 open PRs that carry this same checker (#160, #163, #165, #167, #168, #169,
#173, #178, #179, #181, #182, #185, #187, #189, #191, #193, #194, #202).

The positive claim held everywhere. All 206 main replays passed (PASS p=12, scratch and sources restored) on
input vectors the checker had never been run with. That is evidence the authors' own run cannot provide, and it is
in your favour. The frame-dimension mutation control I ran alongside (one dim 24 -> 25, manifests re-pinned) was
rejected on all 28 PR heads tried.

One negative control depends on the scratch. broken_mix was accepted in 24 of the 206 runs (7 of 80, then
17 of 126), so the script exited FAIL: control broken_mix was accepted on a correct word. Reading the checker, the
reason is exact: broken_mix sets skip_mix on entries[0], and the only check that sees it is replay(), where
the receivers get X[carrier] instead of X[carrier] ^ X[passive]. The control is rejected iff x[passive] is 1
in at least one of the three random input vectors. With passive = 3 that is one fair bit per seed, so a correct
word is reported FAIL with probability 1/8 under fresh scratch (observed 11.7%). The hard-coded seeds give
x[3] = 1, 1, 1, so the control works exactly as designed for them. From the recorded seeds I predicted which runs
would miss: 22 predicted, the same 22 observed, over the 132 runs whose data files I had fetched, with no false
positives.

The other four controls are caught by the exact checks before the replay runs, which is why only this one depends
on luck. None of this touches the word, the ledger recount, or kappa.

Suggested hardening, one line in replay(): add a deterministic input vector with every source bit set
(x = [1] * v, random y0, z0) alongside the random ones. It exposes a skipped mix on any entry with certainty
and leaves the other controls unchanged. Happy to send it as a PR against whichever of these branches you prefer,
and to share the seed maps and logs.

(Replays ran on EC2 Graviton. I used Claude Code for the analysis and reviewed it before posting.)

[RobGruhl]

Correction to the numbers above, after a second review of my records:

• The 206 runs included 32 runs of four other verifiers from the same lineage (paired-cube-diagonal-*, paired-cube-local-bit-168, paired-cube-twin-local-168), which do not have the broken_mix control. The right denominator for this checker is 174 direct runs (48 in the first batch, 126 in the second), all 174 printing PASS p=12, with broken_mix accepted in 24 of 174 (13.8%), against the expected 12.5%.
• The seed prediction covers all 174 runs: 24 of 24 misses and 150 of 150 non-misses predicted from the recorded seeds.
• The frame-dimension mutation control ran 28 times across 20 PR heads, 24 of them on this checker across the 18 heads listed; all 28 were rejected, and the 18 logged runs name dimensions of frame 0, not "28 PR heads" as I wrote.

The mechanism and the suggested one-line hardening are unchanged. Apologies for the loose counting.

[Mister-Meeseeks]

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

Contribution: Searched partner-pair module in the p11 bit word.

Review and limits: Historical construction; distinguish the changed bit module from the unchanged complex supplier and inherited all-size reduction.

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: da3385314d935e57f8129c8734c27104b125e572. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.