Conditional κ = 0.001265931727546219: dual eleven-stage whole-residual hosting

[chafreaky]

Conditional κ = 1265931727546219 / 10¹⁸ = 0.001265931727546219, 0.00655% above #447 at d3cf3079091bb7e93a23e9cd98e82ccb89697328 and 0.2031% above #445, using the retained PR442 whole-residual hosting interface. The bit supplier binds: c = 0.001267536342017397153864, with freshly rebuilt complex b = 0.001268142296690675.

The R740 bit kernel now has 121 pivots, entrance rank149, 114 donor slots and 18 explicit residual isometries. The unchanged PR433 complex local word moves to eleven compressed stages in dimension89. Both suppliers use eight replicas and 9,600 live families, with every helper residual hosted whole: 65,120 bit occurrences and 49,280 complex occurrences. There are no helper banks or padding children.

The new complex row argument keeps the existing 20,161 allowance. The coarse three-level estimate is 21,948 and fails. Instead, 89^9 > 16*64^9 gives depth at most (9/4)log2(e)+9; the full row charge is (76005/4)log2(e)+35361, strictly below 20161 log2(e) for log2(e) >= 32. Smaller widths use retained finite leaves. No larger external row contract is assumed.

Passed before submission: full Linux/Python3.12.3 wrapper; macOS/Python3.14.8 quick wrapper; both with E8_REPLAY_PHASES=1, regenerating all 2,918 inherited local Clifford edges. Their exact results, deterministic template hashes and instruction digests agree. New checks include official C++ records, integer source spans, 3,366 nested/reflected bit projector pairs, 1,320 routes per supplier, 8,520 exact complex phase equalities, complete hosted ownership, 22 bit and 19 complex mutations, and all 24 omitted bridge shears on each side. Both rational moment engines, both finite invoices, six bootstrap levels and all 47 strict outer inequalities pass. make verify-joint passes all 16 controls. All 262 parent-file pins are unchanged. Receipts are in research/r740-hosted-rank149-validation/. CI remains pending separately.

Regarding #448: its higher claim introduces cross-family splitting. This PR includes a reproducible, scoped review in research/cross-family-residual-obstruction/. Same-family factorization does not justify assigning factors to separate families by a coordinate permutation: for q=3, m=2, a=b=1, D_(a+b) ⊕ I has trace90 while D_a ⊕ D_b has trace54. The exact complex analogue has traces 4+2i and 4+4i. General formulas and the proposed rank-eight/rank-nine cases are checked. This obstructs the stated single-residual routing derivation; it does not rule out a different, fully charged compiler or global regrouping theorem. The new certificate here retains whole-residual ownership.

Reproduce:

python -m pip install -r research/e8-r740-replica32/parent436/requirements.txt
E8_REPLAY_PHASES=1 python -B research/r740-hosted-rank149/verify.py /absolute/new/output
python -B research/cross-family-residual-obstruction/verify.py

The conditional all-size compiler, mutable-basis, correlated dirty-helper, compressed-window, restored-work, tape/common-grid, precision, row-divisor, PR442 hosting and analytic interfaces remain explicit. No unconditional multiplication theorem, κ=1 result, new Lean proof or earlier-assumption record is claimed.

Prepared by chafreaky with substantial OpenAI Codex assistance. Retained credits and exact source pins cover Zhihao Chen #377/#386, icekylinx #433, DaysSky #442, eumemic #436/#440/#441, Henry Grant #234, the prior response-kernel lineage, and all parent notices. #447 is a comparison point; #448 is the independently reviewed proposal, credited to Sreekanth Nair with Anthropic Claude assistance.

[higherbeing]

Thanks for the careful note (research/cross-family-residual-obstruction). I believe it compares two placements of one residual, all coordinates in family A versus split between A and B, rather than two complete programs.

In the note's own model the residual action is coordinate-wise (the exchange x_i ↔ y_i on each selected i), so the invocation's action on a residual placed in two families is D_a^A ⊗ D_b^B by definition of the placement. #448 does not claim this equals D_{a+b}^A ⊗ I^B, any more than #442's hosted factor on a data family equals the unhosted factor on a bank family (those traces differ in the same way). The claim in #448 §2, as in #442 §2, is equality of the complete program after absorption: on each family the hosted pieces times the remainder child equal the original child, so every family's per-stage rank-m budget is covered exactly, and the audit checks that cover on every (child, copy).

I have added §7 to research/cross-copy-hosting/PROOF.md (commit on the #448 branch) spelling this out. The review point that remains is the interface one your scope paragraph also names: whether a local register's route may land in two families. In the coordinate-wise model it does; a model in which a residual's action is not a tensor product over its coordinates would be a different interface, and the same question would then apply to banks that hold several helpers' residuals in one family.

[higherbeing]

Follow-up: research/cross-copy-hosting/toy_two_family_check.py (now on the #448 branch, run by its verify.py) does the comparison exactly in your model. At q = 3, m = 2 it reproduces your traces for the residual factor alone (90 fixed addresses whole-on-A, 54 split), and then composes each placement's factor(s) with its remainder children: both complete programs equal the original children Γ_A × Γ_B as permutations of the two-family address space. It also covers w > r, where only the split placement exists.

[Mister-Meeseeks]

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

Contribution: Dual eleven-stage hosting and a cross-family trace obstruction.

Review and limits: The trace counterexample rejects the stated direct-sum routing derivation, not every encoded interface. Whole-owner construction and sharper row accounting retain their explicit assumptions.

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: f3a38153678b55063fc005901cc968266166507d. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.