Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting

[chafreaky]

Result

Conditional κ = 0.001246073505487942 (623036752743971 / 500000000000000000), 0.6886% above PR442, with the bit supplier binding.

Compose PR443's R740 rank138 word and 32-replica layout with PR442's idle-data hosting interface. Hosting moves 159,168 helper residuals onto pending coordinates of untouched data pairs; all 53,952 remaining residuals fill complete banks. Live bit stock falls from 56,739 to 37,464, physical stock is 37,473, and the largest child falls from 42 to 8. Every local instruction, data owner, stage orientation, residual isometry and bridge remains identical to PR443.

The exact bit saving is 1247628141860215921989 / 10^24. The complex supplier is freshly reconstructed from PR442 at 1267453159029119 / 10^18; it is not merely cited.

Verification

research/r740-idle-hosted-replicas/verify.py checks 234 byte-identical pinned upstream files, runs both parent verifiers, then independently audits the hosted construction and finite composition. The audit checks all 288 calls, 213,120 helper occurrences, last-touch windows, replica ownership, residual widths, disjoint slots, remainder children and stage-specific full banks. It recomputes the 5,681,952-instruction digest after ownership changes and rejects 14 corruptions.

Two rational moment engines agree; both finite invoices pass. The bit prefix uses 123 bits charged at 128, the two-level coefficient uses 286 bits charged at 512, and all helper routes retain the residual-isometry charge plus the hosting route charge. Six bootstrap levels, 47 strict inequalities, next-grid and parameter controls pass.

Validation: full Linux reconstruction; macOS quick mode (all new checks and full PR442 replay); repository-native make verify-joint including 16 tests. CI runs the full wrapper on Python 3.11 and 3.14.

python -m pip install -r research/e8-r740-replica32/parent436/requirements.txt
python -B research/r740-idle-hosted-replicas/verify.py /tmp/r740-hosted-new

Scope and credit

This retains PR442's conditional idle-data hosting extension and all inherited all-size compiler, restored-work, common-grid, precision and analytic assumptions. The hosting operator lemma is the same mathematical review point as PR442. No unconditional multiplication theorem, κ = 1 result, or new Lean proof is claimed.

DaysSky's PR442 construction and assembly (Anthropic Claude assistance) and chafreaky's PR443 (OpenAI Codex assistance) are copied with complete notices/licenses and exact source pins. The new selection, integration, stronger independent audit and ownership digest were prepared by chafreaky with substantial OpenAI Codex assistance. Earlier contributions and disclosures remain in the unchanged upstream packages.

[Mister-Meeseeks]

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

Contribution: R740 whole-residual hosting on idle data coordinates.

Review and limits: Retains the explicit idle-data hosting extension, arbitrary dirty-state and complete-invocation obligations. It does not establish cross-family residual splitting.

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: 8c4242ea61c67827fe62b07b2edc093d7ddfa631. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.