Close the marked d=0 C1 invoice and its singleton-pivot consumer

[conorholds]

Change and mathematical scope

Proves the complete marked d=0 invoice I - epsilon_k >= 64 for the tight t=2,3 packets in PR3154, CONJ-G Section 0 / CONJG2 Section 2, and Discussion3156. The theorem works over every characteristic-zero field, starting with an actual basis of S containing C, dim S=8+t, nonzero restricted duals, T=S_{<=7}, C intersect T=0, and the actual marked (k,g) witness. Here I=sum_F w-n_c is the actual corrected invoice.

• Derives the rigid basis shape and actual-total cap without N8 or K2. Every Sigma9/10 boundary and every actual Sigma8 profile is excluded, including (7,1) without a minimum-basis assumption. The remaining coefficient cases retain the actual map A(h)=-sum xi_a(h)b_a, repeated corrected LOW frame indices, weight-nine highs and the actual marked coefficient.
• Also proves the unmarked floor I>=64 in this d=0 class. The actual-decoder consumer consequently needs neither a free-pivot assumption nor an assumed losing root row. Actual rank pair (9,8+t) and the singleton pivot also derive the code-plus-unit row shape directly; no projected-normalization premise is imported. A paid singleton pivot pays its possible rebate; a free pivot supplies the actual marked witness when needed. Exact normalized factorizations attain cost64 for both t values and include paid-pivot equality.
• Proves the raw marked correction cap n_c<=m-2 for actual bases of size m<=11, for every d. This supplies the numerical N8 bound for marked t=2 without merging projected rows. Marked t=3 still permits n_c=9; no positive-d survivor is claimed closed.

The unmarked conclusion is I>=64. A separate exact I=64, epsilon=1 control has S_v=0, so the marked witness cannot be removed from I-epsilon>=64. The proof closes the operative invoice directly; it does not claim unrestricted Swap(+carrier) or unrestricted F1.

The actual consumer retains an exact N0 factorization, rank pair (9,8+t), a singleton pivot, nonzero pairwise nonproportional actual left factors, tight 9+t terms and d=0. Its row shape follows from these actual ranks. A masked-to-exact completion need not preserve the tight term count. Global branch coverage/normalization, d>=1 invoice closure, a complete P112/L120 floor, suppliers and the selected frontier remain outside this result.

Only research/e8-marked-d0-invoice/ is added. It contains the written proofs, actual consumer, exact pins, deterministic receipt, finite controls and four partial Lean cores.

Task and overlap

Scientific base 24db9bfffd4a7c812545ea6ae58853cb9ef094b2, priority revision 6. The scope change from complex-rank-shape-decoder to the named C1 handoff is retained with the original card in TASK.json. PR3154's source is pinned at 708f0ad8928c383de8294aea3c126e9461d2066d.

The auditor's 2026-10-18 00:30–00:31 UTC refresh covered 312 open PRs and 2,240 Discussion records, including all 68 comments and 18 replies in Discussion3156. Closed-PR coverage remains the earlier 2,761-record snapshot at 00:14:40 UTC. No matching whole marked-d0 completion or claim was found within the inspected coverage. Canonical main remained 6f100d4bb5083f8c35d96fd95a539811c62f245c: all nine pinned input/provenance files still match, and the frozen patch applies cleanly. The package retains its historically accurate 22:03–22:04 UTC snapshot. Collections are non-atomic; arbitrary diffs and unpublished work are not exhaustively covered. Public coordination remains pending donor approval.

PR3166 is the direct predecessor: it supplied root-swap/root-only/F1 geometry and selected invoice strata while retaining N8/rigid-shape assumptions and leaving the present coefficient and profile cases open. This package reuses its standalone geometric theorems, not those normalization premises. Adjacent PR3191 concerns C6 and is not a premise. PR3383's tariff-224 result fixes one fitting matrix's rowspace; PR3397 proves a four-piece support obstruction; PR3398 treats a bounded external-query replacement family for a fixed fit. Those contracts do not supply this actual-basis invoice theorem. Recent C5 shortening/capacity results likewise have different contracts; PR3300's rank-seven equality classification does not change the inherited capacity36 used here.

Validation

Python 3.12, standard library only:

python3.12 -I -B scripts/research_package.py --package research/e8-marked-d0-invoice

The registered replay authenticates the frozen inputs, recomputes the inherited geometric filters, and binds two different root/label algorithms through all 14,400 Gram pairs and an explicit rank-nine label isomorphism. Finite coverage includes 7,168 carrier triples, all 42,840 two-frame nonaxis carrying cases, restricted one-/two-/three-axis forms, private-quartet certificates and a marked repeated-low-frame control. Eleven actual factorizations check every N0 entry and cover paid/free and losing/no-loss branches. The paid equality control and the sharp marked t=2,3 families are separately reconstructed with D8+ Gram-row labels.

The author's registered CPython 3.12 replay passed in 9.21 s. All 28 committed package/input byte comparisons and official export integrity checks passed. The exact read-only, Git-free export passed under optimized Python 3.14 in 12.409 s. Optimized Python 3.12 runs reject changed live input, boolean-for-integer receipt corruption and a semantic mutation omitting the paid pivot toll; both export and control-copy bytes are unchanged afterward. The frozen candidate has one local submission commit, cf6487b0915d8a7ba81245c565d503edb6f15b84. Attribution and final patch checks pass for the exact artifacts under review.

Four Lean 4.21.0/mathlib308445d7 files compile without warnings. They derive private-incidence bounds/quartet structure, private-label independence/frame exclusion, carrier quadratic contradiction, and the actual coefficient-map identity. Their printed axiom sets contain only the reported standard axioms. E8 instantiation and the complete invoice/consumer assembly remain unformalized.

A separate read-only AI-assisted audit recommended submission and found no blocking mathematical error. It reran metadata/pin validation, the registered replay, all 28 committed-byte comparisons, export integrity, the optimized Git-free replay (12.364 s) and all four Lean files with warnings treated as errors. Three delegated reviews additionally checked 120 boundary ledger classes, 39 exact three-axis zero-pattern flats and 6,384 root-coordinate deletions; all passed. This is not independent human certification. Hosted CI, repository-wide builds and human review remain unrun; the auditor did not separately repeat the author's corruption mutations.

Credit

Credit DreamingOfClouds' PR2816 (E8 model, parabolic framework and earlier root-swap family), PR2832 (actual decoder/tariff model), PR2862 (carrier/parity predecessor), PR2985 (cap context), PR1590 (the reused two-factor lemma), and PR3154 (the open obligation). CubeJerry's PR3095 supplies field/normalization context. The frozen PR3166 inputs retain their full predecessor credits. Earlier cost64 witnesses, including PR2854's (1,1) examples, retain priority in their scopes; the present controls establish sharpness for the displayed class.

Open next questions

Audit the original global branch-coverage and actual normalization transitions before importing this result into a complete P112 floor. Independently validate the positive-d filters and address the marked t=3 n_c=9/slack9 boundary. Formalization can bind the E8 labels, incidence and quotient splitting to the four generic Lean cores, then assemble the finite geometry and case proof. TASK.json records first steps and blockers.