Mixed-gap bit all-but-one module on #168: conditional κ = 6.129700e-4

[GamingPuzzled]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{6129700}{10^{10}}=6.129700\times10^{-4}.$$

+0.547% over #168 (6.096379e-4) and +0.31% over #171 (6.110684e-4). This stacks on #169, which is on #168 at 98c115b53742b6613ad630de4d493f37b0119da7. Their words, modules, producer and assembly are prerequisites, not contributions of this PR.

Change: a mixed-gap all-but-one module for the bit word

The bit supplier binds in #168, so only the p = 12 bit word changes. The complex supplier and the assembly are #168's, unchanged.

#168's nested-prefix module (n = 10) builds prefixes p_{k+1} = p_k + x_k and suffixes s_k = x_k + s_{k+1}. It forms each middle output y_j (all inputs except x_j) in the left-gap form L: p_{j-1} + (x_{j-1} + s_{j+1}). Its mirror is the right-gap form R: (p_j + x_{j+1}) + s_{j+2}. Choosing the form per output and searching the patterns selected RLLLRRRL; no single flip of it scores higher. All-L reproduces #168 exactly.

The selected module has one more addition (32 vs 31), but the re-solved carrier matching gains more links than that costs. Net, the bit word has 132 fewer roles per vertex. Its operation frames then get #169's descent: 1,320 frames, 264 of them leaving their original frame.

Bit supplier, p = 12	R	W	Replaced frames	Coarse saving
#168	22,252	25,772	0	6105820/10¹⁰
#169	22,252	25,772	1,716	6115023/10¹⁰
this	22,120	25,640	1,320	6139215/10¹⁰

Verification

python3 -B research/paired-cube-bit-rl-module/prove.py        # about 3 minutes, standard library

0. Rebuild. build.py rebuilds out/ byte-identically from the frozen arcs_p12.json through Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's unchanged producer; only its p = 12 module is swapped. Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's standalone bit checker (Checker.run) then recomputes the exact frames, the decoder identity, every chain, the partner-pair chronology, nondegeneracy, an F2 replay and the ledger.
1. Replaced frames. Certify extended paired cubes and bit frames (κ=5.94272360424954e-4) #165's checker applies the replacement frames and checks each one: value span inside, exact nondegeneracy, nested physical role chains, and preserved rank mass with a full recount. It then checks the complete F2 identity and the integer decoder over every source, target and dirty column. Omitted compensation, a missing partner delivery and a zero frame are rejected.
2. Prime witnesses. Every new Gram determinant factors as powers of 2, 3 and 5 times a residual below 2^80.
3. Assembly. Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's own scripts/paired_cube_network.py functions, unchanged, certify the bit moment with its contaminated fallback, the complex supplier, the finite bridge, and all 47 strict constraints and 7 margins. The next 10⁻¹⁰ coarse and κ grid points are rejected, and so is the original-prefix control. certificate.json and prime-witnesses.json reproduce byte-identically.

No existing file changes; a new workflow runs prove.py. Locally, #169's base passes 12 of the 14 make verify-* groups with no tracked-file changes. The other two need OpenMP and Boost, which this Mac lacks.

Scope and credits

All of #168's conditional interfaces remain assumptions. This is not an unconditional multiplication theorem or a runtime claim, and no global optimum is claimed.

• @eumemic: Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's nested-prefix module, word, producer and assembly.
• @chafreaky: Certify extended paired cubes and bit frames (κ=5.94272360424954e-4) #165's bit checker and prime witnesses.
• Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130/Physical frames and scratch reuse on Cayley covers: conditional κ = 3.32205398986e-4 #131: the frame lemma.
• @jon314159: Causal bit-frame refinement of #169: conditional kappa = 6.11068410271618e-4 #171's causal frame refinement is independent of the module and could stack on this word.

Prepared by Joel Pulikkan with Anthropic Claude assistance; Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: Mixed-gap all-but-one module, rematching and frame descent.

Review and limits: Fresh full prove.py passed. More scalar additions can reduce the compiled role/moment cost; no global module optimum is claimed.

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: 55d19073972c9a334203228e3af8f84997a1c2ad. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.