Prove the E8 d=0 kernel floor and scoped rigid C1 bounds

[conorholds]

Change and mathematical scope

Proves C7: cost(S)>=72 for the d=0, nonzero-kernel branch left open in PR3074, PR3154 and Discussion3156. The bound is sharp over every characteristic-zero field in the one-sided E8 lift model of PR2816. The reusable theorem bounds exact-swap-eligible root rank by correction-space basis cost, retaining actual scalar coefficients; the full-remainder inequality 24>21 then closes the kernel case.

The extension also establishes these scoped d=0 C1 results:

• CONJG2 Section 2(ii), kappa<=7: every weight-eight high normal is an E8 root direction, excluding nonroot D7/E6 types, independently of the actual low-basis total.
• Actual low-basis total Sigma=7: corrected invoice >=65 and pivot invoice >=64. Actual total eight is also excluded when minimum correction cost kappa<=7 or a minimum basis has profile (7,1); the latter uses the simultaneous-incidence contradiction 168>148.
• Residual F1: the implication holds through support size 24, sharply, with a literal failure at 25. Together with the (7,1) exclusion, this discharges F1 on all surviving d=0 rigid low-invoice candidates. The unrestricted cost-eight F1 extension is false; the boundary fixture is not a complete C1 decoder counterexample.

Every C1 decoder consequence assumes PR3154's rigid shape and high-member duality, corrected ledger cost(D') = sum_j w(A_j) - n_c, pivot rebate epsilon_k in {0,1}, and N8 (at most eight corrections with distinct frame indices). This package does not independently validate that broader reduction. Minimum cost kappa and actual total Sigma remain distinct. Other C1 cases, C2-C6, a complete L120 theorem, suppliers and the selected frontier remain outside this contribution.

Task and overlap

Base 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6. Scope changed from complex-rank-shape-decoder to the named decoder-floor handoff; the original card, reason, exact source heads and bounded coverage are retained in TASK.json and SOURCES.json. PR3154 is pinned at 708f0ad8928c383de8294aea3c126e9461d2066d.

The 2026-10-17 13:14 UTC overlap refresh covered 283 open PRs, 2,556 closed PRs and 1,932 Discussion records. No matching C7 or scoped C1 completion was found. Adjacent PR3158, PR3160, PR3161, PR3162 and PR3163 concern C5. The reduction's author is consolidating those results and actively working on C6, expressly leaving C1-C4 and C7 open. Public coordination remains pending donor approval. Only research/e8-d0-root-swaps/ is added.

Validation

Python 3.12, standard library only:

python3.12 -I -B scripts/research_package.py --package research/e8-d0-root-swaps

Registered replay passed in 8.40 s; a read-only Git-free optimized export passed in 7.97 s. Coverage includes 511 hyperplanes and the cap-64 check in both coordinate models, 256 parabolics, 120 root remainders, five normal families, actual coefficient fixtures, 56 connected-deletion configurations, 57,120 frame-plane cases per model and eighteen rejection controls. Two additional cap-65 source mutants fail their mathematical guards. The coordinate methods share expected profiles and authorship: they are an internal cross-check, not independent review.

Three unchanged Lean 4.21.0/mathlib files previously compiled without warnings, using only the standard reported axioms, and were separately rechecked by the auditor. They formalize partial coefficient and connected-cover cores. E8 instantiation and the complete C7/C1 arguments remain unformalized. A separate read-only AI audit independently reconstructed the finite inputs and found no demonstrated mathematical error. Hosted CI, predecessor long suites and independent human review have not been run.

Credit

Credit DreamingOfClouds' PR1590 (two-factor lemma), PR2816 (model and parabolic argument), PR2862 (carrier predecessor), PR2985 (cap context), PR3074 and PR3154 (open obligations and conditional decoder model), and CubeJerry's PR3095 (support chart and field scope). Adjacent C5 work retains its credit.

Open next questions

The package’s Open next questions and structured handoff retain the remaining C1 profiles/pivot cases, full formalization, and C6/reduction review. C6 is already actively claimed; coordinate before taking it. The smallest direct follow-up is a remaining cost-eight C1 profile with its actual shared coefficients and pivot conditions retained.