Bit operation-frame descent on #168's bit word: conditional κ = 6.105562e-4

[GamingPuzzled]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{6105562}{10^{10}}=6.105562\times10^{-4}.$$

+0.151% over #168 (6.096379e-4). Stacked on #168 at 98c115b53742b6613ad630de4d493f37b0119da7; its word, modules, arcs, complex physical layer and assembly are prerequisites, not contributions of this PR.

Change

The bit supplier binds in #168, and #168 ships its p = 12 bit word with no operation-frame descent. This PR descends 1,716 operation frames of that word, the step #163–#165 applied to #161's bit word. 264 of the frames leave their original frame, as in #165. The scalar word, gauges, persistent roles and complex supplier are unchanged.

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

Verification

python3 -B research/paired-cube-bit-descent-168/prove.py      # about 1 minute, standard library

• Bit word. Certify extended paired cubes and bit frames (κ=5.94272360424954e-4) #165's bit checker (word.py, unchanged except for paths) applies replacement-frames.json to Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's committed bit word. For every replaced frame it checks:

  • the value span lies inside the frame;
  • the frame is exactly nondegenerate;
  • every physical role chain stays nested;
  • rank mass is preserved, with the complete child histogram recounted.

  It then checks the complete F2 identity and the defining integer decoder over all source, target and dirty columns. Omitted compensation, a missing partner delivery and a zero frame are rejected.

• Prime witnesses. All 1,536 distinct new Gram determinants factor into powers of 2, 3 and 5 times a residual below 2^80 (Certify extended paired cubes and bit frames (κ=5.94272360424954e-4) #165's script). Every retained prime q > 2^80 stays valid.

• 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 of the balanced-prefix assembly. Only the bit row and the two grid points are substituted. The next 10⁻¹⁰ coarse point and the next κ point are rejected; the original-prefix control is rejected.

• certificate.json and prime-witnesses.json reproduce byte-identically.

• Generator. generate.py regenerates replacement-frames.json byte-identically in about 70 s. It runs alternating passes; each operation's frame moves to the lower bound (span of both roles' previous frames and the value span) or the upper bound (intersection of their next frames) when nested, nondegenerate, prime-witnessed and cheaper.

• A new workflow runs prove.py. No existing file changes.

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 and its full lineage (@icekylinx's 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, and others).
• @chafreaky: Certify extended paired cubes and bit frames (κ=5.94272360424954e-4) #165 bit-frame replacement checker and prime witnesses, and Fused paired cubes and paid balanced transfer: conditional κ = 5.93970203079492e-4 #163.
• Connected bit-frame descents on #163: conditional κ = 5.94035602295411e-4 #164: connected bit-frame descents.
• 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.

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: 1716 operation-frame descents on the PR168 bit word.

Review and limits: Fresh prove.py passed with complete finite word/profile checks and rejected controls; fixed scalar word and endpoint contract.

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: 4895ba1104cedc7908b6d6ba64fdd6290e0ed7ae. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.