Conditional kappa 0.001584931113695072: literal bit bad-class charge on PR535

[rohanarun]

Conditional κ = 0.001584931113695072, versus #535's 0.001584931111785787.

This is a tiny arithmetic refinement: absolute gain 1.909285e-12, approximately 0.0000001205%. It leaves #510's R480 bit word, #535's complex word, both physical assignments and histograms unchanged.

The source theorem in notes/three-stage-cover-bit.tex bounds each bad edge using bad fraction 2m^3 / 2^80 and fallback charge 6N(N-1)+3N+6(N-1), N=2m. At bit dimension m=64 these are 1/2^61 and 98682, below the rounded 10^-16 and 32m^2 still used by #510. This transfers the refinement from #461/#464/#465 to the now bit-bound #535 composition; it is not a new fallback theorem or circuit.

New bit c: 0.001587447108007899663966. Complex b remains 0.001589991533606044174031; the bit side binds. The old finite invoice allowances remain conservative because the fallback operation count only decreases, with the prime threshold unchanged.

The packaged verifier now passes on Python 3.11.15 and 3.14.8 with identical mathematical output. Published evidence includes both receipts and the tested manifest. Both rational moment engines, all 47 strict outer inequalities, six bootstrap cutoffs and six negative controls pass; the old endpoints and kappa are also reproduced. The draft was submitted before these cross-version checks. GitHub CI remains separate and pending.

Reproduce: python -B research/r480-sharp-charge535/verify.py /absolute/new/output (Python 3.11+, standard library, assertions enabled).

This is arithmetic repricing of cited suppliers, not a new physical-operator admission. PR510/PR535's all-size compilation, mutable-source/common-label realization, normalized wrappers, whole-call hosting, restored-work, precision/tape and analytic assumptions remain. Portability of the source bad-class theorem to the compressed cover is retained conditionally, not newly proved. No unconditional theorem, Lean completion or measured runtime gain is claimed.

Prepared for rohanarun with OpenAI Codex assistance. Credit: Srikrishnabalaji (#461) and subsequent #464/#465 applications for the sharper charge; icekylinx for the source theorem and #535; DreamingOfClouds and disclosed Claude assistance for #510; eumemic and all preserved predecessor notices for the bit construction. Original licenses and attribution are retained.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution at 915ca69db7f98c70b97ce96e1fcc96818517584c is incorporated in the reviewed family through #588, published in 9594b61.

For m=64 the literal fallback charge uses F=98682 and bad fraction bounded by 1/2305843009213693952; good classes retain full weight and bad edges are bounded separately. Both old endpoints and the new exact rational endpoint are reproduced. This is arithmetic-only under the compressed-cover transport premise, not a new proof of that transport or a new scalar circuit.

Preserved implementation and research:

• research/r480-sharp-charge535

The family review and individual disposition record the scope, evidence and attribution. Parallel and intermediate contributions remain credited even where a later composition has a stronger bound. The published multiplication bound remains conditional on the documented compiler, realization, hosting and analytic assumptions; these finite checks are not a full Lean multiplication proof.