Fixed-prime refinement of PR259 (kappa = 0.000711029389987426878345881)

[sennemmi]

Research closeout — Draft retained; no current-record claim. This is an exact fixed-prime arithmetic refinement of PR #259. It inherits #259's finite physical construction and changes the conditional bit-supplier calculation only.

Result and contribution boundary

PR #259: 142205877210807 / 200000000000000000
PR #261: 711029389987426878345881 / 10^27
Exact gain: 3933391878345881 / 10^27

At m = 120, the certificate fixes q = 2^127 - 1, uses rare-class density rho = 3456000/q < 10^-16, and charges the complete 32m^2 rare-child fallback. It retains #259's 518 pivot entrances, four cuts, physical frames, routes, roles, replicas and lifetimes unchanged. It makes no new physical-DAG saving.

The finite arithmetic replay regenerates the cohort profile and invoice, uses two rational interval engines for the coarse endpoint, checks eight finite levels, 47 strict outer constraints and seven margins, and rejects the adjacent 10^-27 grid point. The fresh finite coefficient is 62639910650427201 < 2^80 < q; selector, chart, normalizer, signed-prefix and inherited physical costs remain charged.

Attribution: Dugongue's #259 provides the inherited multi-cut physical witness and invoice; its pinned source lineage comes from #249. The fixed-prime approach credits Nespoli #235 and Arun #252, with #255/#258 refinements acknowledged in the certificate history. The exact recomputation, certificate, documentation and workflow integration in this PR were prepared by sennemmi with substantial OpenAI Codex assistance. Upstream authorship, source notices and license files are retained; the added package carries its stated Apache-2.0 license.

Incremental review path

The full PR comparison against main is large because it carries the upstream research lineage: GitHub reports 422 changed files and 1,707,820 additions. Removing inherited source or receipts would weaken provenance or replayability, so this closeout keeps them intact. The independently reviewable increment is exactly one commit over #259 SHA a53ec0ca4d59634f16630dd16adf035c9f12595c at head d2c63486d895860a65f25b80a51f4b2ec4f3c6ef: compare #259 to #261. That increment has 14 files:

1. README.md, PROOF.md — statement, arithmetic mechanism and limits.
2. fixed_prime_math.py, expected.json — exact recomputation and pinned result.
3. verify.py, SOURCE.json, MANIFEST.json — source pins, integrity checks and replay.
4. Prime.lean, lakefile.toml, lean-toolchain — kernel-checked prime/density facts only.
5. LICENSE, NOTICE.md, requirements.txt — provenance, license and dependencies.
6. .github/workflows/multicut-fixed-prime.yml — replay and Lean job.

Reproduction and verification

From the repository root (Python 3.11+, SymPy 1.14.0, GNU-compatible C++17 and Boost headers):

python3 -m pip install -r research/multicut-kernel-condensation/requirements.txt
python3 -B research/multicut-kernel-condensation-fixed-prime/verify.py --output /tmp/multicut-fixed-prime

The final head SHA d2c63486d895860a65f25b80a51f4b2ec4f3c6ef has successful Actions runs: full repository workflow (52/52: all 16 make verify-* groups across Python 3.11/3.13/3.14, ranked-pair finite replay and three formal jobs) and fixed-prime replay workflow (exact replay plus Lean prime/density kernel check). The Lean job does not prove the multiplication theorem. The gh pr checks view does not expose a PR check suite for these push runs; the linked Actions records identify the exact SHA directly.

Closeout status and scope

This is a conditional finite research result, not an unconditional integer-multiplication theorem or a practical runtime claim. It inherits #259/#249's all-size compiler, common weighted chart, restored-row, routing, prime-supply, precision/recovery, complex-correctness and analytic-reduction obligations; these interfaces are not proved by this certificate. The parent #259 remains open and unmerged, so this PR remains Draft. No further maintenance or record claim is planned.

The later comparable PR #263 reports 355516119100349/(5·10^17), exceeds this result by 2848213271121654119/10^27, and has 52/52 full repository jobs passing on head 87129308f762f86a9d8e9a60f5d6c8ef23ce1f6e, along with its terminal and recycled-bit workflows. Its author reports a physically checked retiming of 25 internal ADD gate frames on #259's construction. This PR therefore makes no current-frontier claim. PR #262's larger priced target is explicitly marked “not a witness” by its author and leaves the physical placement obligation B1 open; it is not treated as a completed construction. The 7.547361e-4 figure in #256 is a complex-supplier saving, not a final kappa value.

The other sennemmi Drafts are being archived with their histories retained: #140 (terminal-elision/shrunk-frame composition; targeted deferred-witness checks failed), #198 (endpoint-frame/leaf bootstrap; full CI failed), #213 (five-level cascade bootstrap; targeted checks passed, root workflow cancelled and inherited full replay remains limited), #230 (plateau retiming; full and targeted workflows passed), #247 (fixed-prime refinement on #244; workflows passed), #253 (fixed-prime refinement on #251; workflows passed), and #257 (eight-level refinement on #254; targeted workflows passed, root workflow cancelled). All depend on unmerged upstream branches; their independent ideas and negative verification results remain in their PR histories. A consolidated public research closeout is supplied with this task.