Causal bit-frame refinement of #169: conditional kappa = 6.11068410271618e-4

[jon314159]

PR168's bit supplier limits its conditional multiplication bound. This refines PR169's credited frame plan with causal prefix/suffix moves and certifies

kappa = 305534205135809 / 500000000000000000 = 0.000611068410271618.

The saving is 0.23465% above PR168 and 0.08389% above PR169's published value. These compare asymptotic exponent savings, not measured runtimes.

Stacked on #168 at 98c115b53742b6613ad630de4d493f37b0119da7, with #169 at 4895ba1104cedc7908b6d6ba64fdd6290e0ed7ae supplying the exact initial bit-frame plan. Their modules, arcs, scalar words, complex physical layer and balanced layout are prerequisites, not contributions of this change. The incremental changes over #168 are additive research packages and one workflow.

Construction and matched comparison

The selected search accepts 240 downward causal-prefix moves and 120 upward causal-suffix moves, ending with 2,196 replacement operation frames. Every accepted move has a strictly improving complete boundary moment proved by rational intervals, including its fallback edge-count change. Scalar operations, selected gauges, register stock W=25,772 and rank deficit 1,936 stay fixed.

A further search from #169 improves the earlier 2,172-frame result. Compatible pointwise-intersection and span starting plans were also checked: the span search reaches the same final frame records as the #169 start, while the intersection search retains the earlier paid profile. The simpler #169 start is selected and reproduces deterministically.

Witness	Conditional kappa
#168 published	0.0006096379
#169 published	0.0006105562
#169 with the same refinement rules used here	0.000610778813757499
This checked plan	0.000611068410271618

Thus the final plan remains 0.04741% stronger under identical certification settings. compare_pr169.py independently checks #169's exact geometry, full child profile, determinant factors and every formal bit column before comparing both profiles. The same least-paid-atom rule is used; each profile pays its own required atom exponent.

Verification improvements

• Exact support containment, all physical and target chains, nondegeneracy and complete paid child recount. A replacement may enlarge its original frame only when the actual physical constraints pass.
• Every bit source, target and arbitrary-dirty column over F2 and against the defining integer decoder; every complex column over the rationals in both shear directions. The complex replay retains PR163's injective packing proof, with its coefficient and final-residual bounds recomputed for Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168.
• Exact Gram determinants and factor/residual witnesses preserve every retained prime q > 2^80. Negative controls distinguish a large determinant made of small factors from an unproved large residual.
• Source-derived bad-fraction bound 2*m^3/2^80 and the complete direct fallback bill 6*N0*(N0-1)+3*N0+6*(N0-1), N0=2m. No fallback edge, reversal or dirty correction is omitted.
• Exact coarse-root refinement, the least strictly paid atom on the stated grid, a refined complex supplier, and all 47 strict assembly constraints and seven margins. The full finite group, logical scalar inventory, router, semantic guard and restored-row charges remain present. The next coarse/final grid points and preceding unpaid atom are rejected.
• A separate source-bound obstruction proves the fixed selected bit gauges admit zero donor handoffs in both Connected bit-frame descents on #163: conditional κ = 5.94035602295411e-4 #164 and Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168. It does not rule out different gauges or compilers.
• Frozen source closure, strict frame-plan schema validation, canonical LF output, and certificate-generated numerical prose avoid stale counts, unsupported supplier substitutions and platform-dependent hashes.

Reproduction

Standard-library Python 3.11+; run without -O from the repository root:

python3 -B research/paired-cube-bit-descent/search.py
python3 -B research/paired-cube-bit-descent/verify.py
python3 -B research/paired-cube-refinement/test_arithmetic.py
python3 -B research/paired-cube-bit-reuse/check_obstruction.py --check
python3 -B research/paired-cube-refinement/verify.py
python3 -B research/paired-cube-refinement/compare_pr169.py

The selected finite word, deterministic search, exact arithmetic and read-only certificate reproduction pass. #164's full package verifier and #168's published package gates also passed. This does not claim that all fourteen unrelated historical repository verification groups were rerun. The new workflow runs the search, controls, complete successor certificate and matched comparison. It also passed from a clean GitHub-hosted Linux checkout at commit 89d0c75f8bf97131db21bc610e7546b699afcb9a in 4m45s.

See research/paired-cube-refinement/RESULTS.md, PROOF.md, REVIEW.md and the exact JSON certificates.

Scope and credits

This remains conditional on the inherited all-size Clifford/tensor, uniform weighted bit compiler, internally borrowed and restored rows, balanced positional layout, analytic recovery and fixed-tape interfaces. It is not an unconditional multiplication theorem, a practical runtime claim, or a global optimality result.

• @icekylinx and the repository contributors: paired cubes, shared-core accounting, bit and finite-bridge lineage.
• @eumemic: Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's modules and physical layer; Connected bit-frame descents on #163: conditional κ = 5.94035602295411e-4 #164's connected descent and prime-factor work, with their disclosed assistance and credited predecessors.
• @DaysSky: closure compiler and pairing work adopted by Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168.
• @chafreaky: Fused paired cubes and paid balanced transfer: conditional κ = 5.93970203079492e-4 #163's formal-column checker, exact intervals, paid atom and fine transfer; the Certify extended paired cubes and bit frames (κ=5.94272360424954e-4) #165 replacement lineage credited by Bit operation-frame descent on #168's bit word: conditional κ = 6.105562e-4 #169.
• @GamingPuzzled (Joel Pulikkan): Bit operation-frame descent on #168's bit word: conditional κ = 6.105562e-4 #169's exact initial plan and its lower/upper operation moves, prepared with Anthropic Claude assistance. Generic ascent/descent and that plan are not claimed as new here.
• All inherited frame, reuse, bit, atom-toll and balanced-layout notices remain intact.

This refinement was prepared with substantial OpenAI Codex assistance; Apache-2.0.

[Mister-Meeseeks]

Thank you @jon314159. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Causal bit-frame refinement, matched repricing and fixed-gauge reuse obstruction.

Review and limits: Fresh full certificate, exact reuse obstruction and PR169 comparison passed. Empty compatibility is proved only for the two named gauge snapshots, not all reuse mechanisms.

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: 89d0c75f8bf97131db21bc610e7546b699afcb9a. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.