Conditional κ = 0.001133928043872860: R740 shared-donor kernel with explicit residual bank routes

[chafreaky]

Result

Conditional κ = 56696402193643 / 50000000000000000 = 0.001133928043872860.

This applies our #417 shared-donor kernel to eumemic's #436 R740 bit word and composes it with #433's cited complex supplier. It is 0.4380% above #436 and 0.03703% above #438 at e3b56de7869f6af88d48ac0ab1d7959c4c9e6176. The bit side binds.

Quantity	Result
Kernel pivots / entrance rank / donor slots	115 / 136 / 103
Local helpers	740
Banks per stage	723 (parent: 740)
Live / physical stock	14,187 / 14,196
Exact bit lower endpoint	1135215296333206583339 / 10^24
Cited complex lower endpoint	1183986549206329 / 10^18
Rank deficit	2,496

Construction

Each selected pivot starts at its verified entrance. Its target response is the XOR of its donors' responses; an entrance correction and its full-frame inverse restore arbitrary dirt. The selection is regenerated against the exact #436 word, not transferred by helper index from another word.

The residual widths fill dimension-64 banks without padding. Following #435's mixed-bank composition, this package additionally supplies explicit orthogonal completions for all 15 entrance frames. For this selection the maps are coordinate permutations, with each elementary factor replayed. The independent audit checks the residual subspace, every bank interval and every stage owner, and rejects altered bases, wrong residual frames and omitted route factors. The fresh finite invoice conservatively charges the additional routing work.

The entire #436 package is vendored byte-for-byte, including its reconstruction, complete verifier, licenses, notices and file-level provenance. The kernel engine and transformation are pinned to our #417 blobs. The existing three-file #318 joint-dual pin repair is included so that the repository's joint check passes.

Validation

Verified source at commit fa3a588ec1e383083abb8de9c51904fb9d689ad4:

• Full package verifier on Linux/aarch64 (Spark), including all parent historical regressions: exit 0.
• Quick package verifier on macOS/arm64: exit 0. Quick mode omits only the parent's historical regressions; it runs the complete new-witness checks.
• make verify-joint: passed, including its 16 mutation tests.
• Frozen reconstruction, all 980 local columns, official five-stage and B4 column checkers, exact nondegeneracy, forward/reflected copy chronology, all 7,620 unbanked global columns and all 53,280 helper assignments passed.
• 3,279 exact nested/reflected projector pairs, 15 residual isometries, 11 independent template mutations, three construction mutations, retained record-level controls and all 18 omitted bridge-shear controls passed.
• Both rational moment engines, the fresh finite invoice, six bootstrap levels and all 47 strict outer inequalities passed. The next grid point and invalid outer parameters were rejected.
• Linux and macOS emitted the same complete program digest: 418958c0010bf48e5e40c8573f7927d04c0edf01a15db324996498c72a7b3b1d.
• All 112 package files match their staged manifest bytes. Unchanged vendor files retain their original trailing blank lines.

python3 -m pip install -r research/e8-r740-kernel/parent436/requirements.txt
bash research/e8-r740-kernel/verify.sh /tmp/e8-r740-kernel-full

The dedicated workflow covers Python 3.11 and 3.14. CI is pending; the unchanged whole-repository suite was not rerun locally.

Scope and weakest steps

This is a finite conditional witness for the retained O(n (log n)^(1-κ)) assembly. It does not prove κ = 1, provide a new Lean theorem, or establish unconditional multiplication complexity.

The complex result is cited from #433, including its #425 mutable-basis interface; its scalar construction and Gaussian-phase audits are not rebuilt here. Full cover enumeration and the all-size compiler, correlated dirty-bank, restored-work, precision, tape/common-grid and analytic interfaces remain inherited assumptions. The exact finite checks do not discharge them.

Credits

New selection, explicit residual routes and integration: chafreaky, with substantial OpenAI Codex assistance.

Retained contributions: eumemic (#436, with Anthropic Claude); chafreaky (#417, with Anthropic Claude); rohanarun (#435 mixed-bank composition); icekylinx (#433); jacklightChen (#386/#425); utcorvusvolat-dotcom / Dylan Crow (#370); applecrusher (#376); ai-nanachi (#389/#426); DaysSky (#363/#375/#381); DreamingOfClouds (#315); Jacob Sussman and the earlier credited contributors. Full attribution, licenses and source-specific AI disclosures are preserved in NOTICE.md, SOURCE.json and parent436/. Apache-2.0.

[Mister-Meeseeks]

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

Contribution: R740 shared-donor kernel with explicit residual routes.

Review and limits: Exact residual isometries, bank intervals, elementary factors and the route invoice matter, not merely matching widths. Whole residuals remain on physical families.

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: fa3a588ec1e383083abb8de9c51904fb9d689ad4. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.