Conditional κ = 6.84696826673891e-4: paid XOR butterflies with cascade-optimized frames

[hcg890]

Result

Conditional κ = 0.000684696826673891

Exact value: 684696826673891 / 10^18

Improvement over PR211: 0.1241%

This combines a new 440-butterfly XOR rewrite with PR211’s cascade-optimized frames, using the existing PR200 bit construction, PR197 completed-residual packing and retained PR193 complex supplier.

Quantity	PR211 at c63e50a	This PR
Assembled κ	0.000683847872495777	0.000684696826673891
Normalized stock W	56,402	56,402

What contributes

1. New scalar rewrite: 440 XOR butterflies. Each replaces (a+d)+(b+c) with (a+c)+(b+d) at two single-use pair nodes feeding the same four-sum. This changes 880 scalar controls and 880 graph nodes while preserving the carried value. The unused pair has no later consumer, and reverse execution restores every temporary.

2. Inherited frame optimization: PR211’s 6,191 entries. Three entries, at operations 15154, 16230 and 16881, overlap the rewritten pair operations. They are explicitly replaced by the admitted butterfly pair frames, and the complete combined chronology and geometry are checked.

3. Inherited terminal elimination, packing and complex supplier. The existing 34-sink transformation and completed-residual packing are retained and repriced on the combined word. Their savings are not claimed as new butterfly improvements.

The measured κ improvement is for this complete composition versus PR211. Separate mechanism gains are not added together.

Combined cost and validation

The final paid profile has rank mass 4,055,136, deficit 5,808 and largest child 22. Nine complete core invocations retain 45,842 banks and 154,026 incidences, charging 7,093,890 literal XORs and 5,701,209,426 selector calls. Assembly uses three finite bootstrap levels.

The complete replay and profile/price binding pass:

• All 18,908 integer dirty-response rows are preserved.
• Baseline F2/defining-Z checks and all 20,634 terminal columns pass, including both signed defining-Z directions, restoration and hostile controls.
• Prime witnesses cover 25,723 frame IDs in 24,611 distinct bases.
• Two outward-rational paid-moment engines and all 47 strict assembly inequalities pass.
• A clean public-source reproduction passes all 6 retained complex stages and 18 changed-bit stages, including persisted-artifact and prime-archive verification.

Scope

The proposed consequence is conditional:

T(n) = O(n (log n)^(1−κ))

Weighted compilation, separated completed-core residuals, restored-row and ordinary-selector contracts, routing, prime supply, precision/recovery and fixed-tape analytic transfer remain inherited hypotheses. These finite certificates do not establish an unconditional theorem or practical multiplication speed. SOURCES.json, README.md and PROOF.md provide exact dependencies, reproduction instructions and the argument.

Lineage and credits

Rohan Arun (@rohanarun): PR211 cascade frames and PR185/187 predecessors. Dugongue: PR207 frame refinement and PR186 banks. Chafik Boukhalfa (@chafreaky): PR200/202. Evan McKinney (@evmckinney9): PR197 packing. Avi Eisenberg (@ikeboy): PR193 adaptation. @icekylinx and @eumemic: flow and paired-cube antecedents. James Chang (@jamesyc), Jean-Guillaume Dumas and Bruno Grenet: accumulation antecedents. Maxime Fleury (@maxime-fleury): PR204’s independent combination model. Original notices and earlier credits are retained.

Assisted with ChatGPT.

[Mister-Meeseeks]

Thank you @hcg890. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Paid XOR butterflies with cascade-optimized frames.

Review and limits: Prior exact-head changed bit construction, strict complex adapter, packing and assembly replay. Original dependency-version gate and adapted environment are explicitly distinguished.

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: b739fc226a54a4a507989aa1bef092d491287451. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.