Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers

[hcg890]

This source-bound five-stage composition gives the conditional integer-multiplication bound T(n) = O(n (log n)^(1−κ)), with

κ = 703701743496697/10^18 = 0.000703701743496697 > 2⁻¹¹.

It combines a new F₂ bit construction with Sussman's published five-stage complex baseline. The bit side binds.

Paid quantity, per cover vertex	F₂ bit	Complex
Dimension m	120	110
Live stock W	23,683	14,692
Positive-rank calls	495,304	361,405
Rank mass	2,837,560	1,613,040
Deficit mW − rank mass	4,400	3,080
Largest child	100	46
Paid-moment root, approximately	0.000704197288591	0.000747454944652

The bit source is pinned to PR210 at df95878. For comparison, PR210's newer c4c21244 reports κ = 0.0006956735022133; the assembled κ here is approximately 1.154% higher. That newer head is not incorporated here; all 57 of its verification checks passed at the comparison check on 9 October.

Each independent bit helper crosses five disjoint address windows before one paid exterior completion at five times its entrance rank. All 2,279 completions and 120 copied-center calls remain charged. Source ownership, full forward/inverse words and rational partial-swap identities bind the construction to the actual helper. The rank-100 children require four recursion levels to halve the width.

The fresh source-to-certificate verifier passes all six mandatory stages: all 20,163 bit columns in both directions, physical chronology and copied-center lifetimes, 26,380 exact basis determinants, the complex scalar/label splice, finite accounting, three ordinary-bootstrap steps and all 47 strict outer constraints. Inputs remain unchanged.

Run the complete verifier:

python3 -m pip install -r research/five-stage-source-bound-v8/requirements.txt
python3 -B research/five-stage-source-bound-v8/verify.py --output /tmp/five-stage-v8-verification

The result retains the symbolic complex correctness, weighted compiler, complete-stream copying, restored-row, prime-supply, routing, precision/recovery, ordinary-leaf and all-size analytic hypotheses. No new Lean kernel build or practical multiplication benchmark is claimed.

Approaches tested

• Completing helpers after every invocation gives a bit moment root near 0.000665572. Bundling each helper's five windows retains the paid corrections and improves that root to 0.000704197.
• The old fixed integer endpoint is unnecessarily restrictive; this construction proves its F₂ endpoint and bounds its own signed integer lift.
• The current four-bank/common-anchor interface with rank-one cross-bank shears needs five helper calls. A paid four-call workaround reaches only about 0.000305. Tested cross-port variants give relabelings or costly corrections; the tested three-bank model has an optimistic six-call root near 0.000524.

These are supplier moment results, not final multiplication exponents or global impossibility claims. Different helper actions and boundary geometry remain open.

Lineage and credits

Jacob Sussman: f010392, introduced in PR209, supplies the five-stage layout and complex baseline; Apache-2.0, NOTICE and the original Claude attribution are retained. Avi Eisenberg (@ikeboy): PR193 at 187e101, the retained complex helper. @eumemic: PR210 at df95878, the pinned bit helper. Earlier contributor credits and assistance disclosures remain in the original source notices.

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: Five-stage bit and complex suppliers.

Review and limits: Prior exact-head complete six-stage finite replay, supplemented by source-bound CI. Complex symbolic, all-size and formal dependencies remain separate.

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: af3fe331ca60c936229e060681ffccbc1c208678. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.