Prove actual P112 invoices at d=1 and d=2 for t=2,3

[conorholds]

Change and mathematical scope

Proves the actual d=1 and d=2 C1 invoices for t=2 and t=3 over every characteristic-zero field: the root-marked invoice is I>=65, hence I-epsilon_k>=64, and the unmarked floor is I>=64. For d=1 or d=2, the same bounds prove the cost-64 floor for the exact singleton-pivot consumer with actual ranks (9,8+t) and nonparallel left factors. All actual basis cases are included: nonadapted low families, repeated corrected-low indices, corrected weight-nine highs and the raw t3/slack-nine boundary. No N8, minimum actual basis or historical-list completeness is assumed.

D1.md and D2.md give the actual hypotheses and complete proofs. The new geometric inputs are:

• A sharp five-site remainder outside one rank-six and one rank-seven binary flat, allowing arbitrary overlap and cancellation. Every nonframe root attains five with a coefficient partition of sizes 5,16,32, ranks 4,6,7, and exactly three optimal flat pairs. Frame roots have sharp remainder eight and 945 optimal pairs each.
• Intrinsic code-intersection-at-most-one theorems for independent pieces of ranks 7,7,5,5, 7,7,6,4 and 7,6,6,5,1. The last two close the remaining d2 profile obligations. The 7764 proof forces actual root representations to be the sharp partitions; shared coefficient ratios then rule out a second root. The 76651 proof uses saturated support capacities and translation fibers of a six-dimensional plus quadric.

The proof derives the d2 low-space dimension directly and exhausts all minimum profiles through costs 24/25, then treats every actual-basis surplus. It does not exclude the genuine d1 T-space of profile 7,6,3,1 and cost 17: its actual correction budget is handled explicitly. Nine direct and 85 credited root-pencil implications for literal PR3154 lists are also recorded, without claiming 85 open or first closures.

RAW-SHAPE.md supplies a further reduction for every positive d: under the raw weight budget and a putative I<=64, the full threshold space has dimension d+t. The raw budget alone suffices through d=6 for both t2/t3. Two actual boundary-profile lemmas close the remaining d=7/8 dimension defects. A further actual-coefficient transfer closes all d3 basis defects r>=2 and every nonadapted eta=r branch. Only adapted r0 and r1 with eta2/3 remain at d3; higher-d invoices are still open. This does not force the actual basis to adapt to the threshold space.

Only research/p112-six-seven-supplement/ is added. Complete actual d>=3 invoices, complete C1, global rank/pivot normalization, supplier admission and the selected frontier remain outside the result. The complete E8 instantiation and invoice assembly remain outside Lean.

Task and overlap

Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. TASK.json retains the original complex-rank-shape-decoder card and explains the user-directed scope change to Discussion3156.

The entry snapshot covered 326 open PRs, 2,959 closed PRs and 2,444 Discussion records through 2026-10-18 09:09 UTC. Read-only refreshes through 13:50 UTC inspected the latest 100 updated all-state PRs (50 updates from 10:20 to 13:30, then ten updated since 13:30), latest 100 Discussion records, all 3156/1163 comments and replies, and the latest 100 of 573 handoff comments with replies. The changed mechanism-thread comment was also read. Canonical main, PR3154's exact head and the C1 thread were unchanged. New adjacent work includes PR3632's physical scatter/restoration, PR3625's lost-free plane-port bound, PR3604's tetrad fusion and PR3639/3643's named-class C5 charging. PR3658's five-cell block-basis obstruction is a different problem; the newer PR3674 independently certifies PR3625's lost-free plane bound, without this actual-basis invoice. No matching actual d=2 completion appeared in the inspected delta. A final read-only delta at 2026-10-18 14:00 UTC found canonical main and the C1 thread unchanged, with no matching completion in the inspected updates. These are bounded non-atomic searches, not a novelty guarantee. No public claim or coordination post has been made under the local-only restriction.

PR3455 supplies the marked d=0 invoice, raw correction cap and row-edit inputs; this package supplies actual d=1/d=2 extensions. PR3166's matched two-axis frame cuts are retained and recomputed. PR2816's one-extra-site exclusion remains valid: its printed nine-site minimum pertains to a pruned domain that omits the present 32+16 cover. PR3269/PR3300 supply analytic global capacities, PR3171 the credited root-pencil mechanism, and PR3418 the different sparse-four plus 5,6,6 shape. Exact heads and source bytes are pinned in SOURCES.json.

Validation

Python 3.12, standard library only:

python3.12 -I -B scripts/research_package.py --package research/p112-six-seven-supplement

Two separately implemented exact engines agree on 43,180 affine six-spaces, 7,896 optimal root covers, 42,840 matched frame-plane cuts, 4,900 root planes, 12,000 complementary 28-root sections and all 2,604 affine four-sections of the six-dimensional plus quadric. They check all 70 four-axis form inventories, 589 bounded d2 minimum profiles and the retained coefficient-sign halves and caller implications.

Four actual factorization controls cover t2/t3 at d=1 and d=2 and check 50,176 exact Gram entries, actual ranks, nonparallel factors, restricted duals and term costs. The d2 controls have live-root marks and three corrected weight-nine highs each. Complete low-space containment is certified by rank-six quotient minors modulo3 outside each of 43,180 flats; their nonzero integer minors apply in characteristic zero. The low-space certificates are independently replayed using ordinary-dot RREF coordinates and full evaluation matrices modulo five. The exact factorization and term-cost construction itself uses one implementation with separate read-only review and attains actual costs 71,72,77,77; no sharpness at 64 or exhaustive actual-factorization search is claimed.

Sixteen CLI control groups exercise nine scientific mutants, including wrong pivot compensation and a wrong d=2 marked shear, together with source tampering, engine disagreement and strict receipt controls. Default verification preserves input bytes.

Nineteen Lean declarations are checked: seventeen unchanged Mathlib cores cover the quadratic quarter and dimension deletion; two new Init-only cores prove the shared-piece determinant identity and self-site obstruction, with explicit algebra/embedding premises. The new cores compile with one thread in 1.173 s, peak RSS 242.48 MiB; their axioms are none and propext, respectively. The old seventeen retain their earlier standard-axiom checks. The full E8, field-transfer and invoice/consumer assembly are written proofs, not full kernel certification.

Frozen local candidate 3193d1ea9a4fc5d178570a61e6019454b53c1db9 is one commit above canonical main. Committed-byte comparison and source-history checks pass for all 31 package/input files; the official export integrity check passes. Registered CPython3.12 replay passed in 117.158 s, peak aggregate RSS 133.41 MiB across receipt generation and replay. All three optimized commands also pass on CPython3.14 in a sanitized, Git-free export in 119.636 s, peak 83.53 MiB, with all 32 exported files, modes and inventory unchanged. The regenerated receipt differs from the audited version only in the two clarified proof-file hashes; every scientific value and all Python/Lean source bytes are unchanged. The exact package, draft, commit history and exported patch pass attribution preflight. Scientific replays ran sequentially under an aggregate 256 MiB RSS watchdog; no Docker or repository-wide build was used.

• Full-index patch SHA256: 533f798a680b33ec44454e9a360350ef5cd8af0938c1c5ff84a1f27ddcc1e1a6.
• Portable export SHA256: 21ed40da325299d298e2f270cfc180207c501dcbb60830f042b2e82e86aa3980.

Internal skeptical AI reviews found no missing implication in the actual d=2 case analysis or intrinsic geometric arguments; the raw-dimension proof received two separate read-only reviews and a further 13-case boundary-arithmetic probe. A subsequent read-only AI audit of 015ab5a3c7b8bf3f2c954af9c13a4876f5ec9d47 reconstructed the finite inputs using its own E8 construction and flat enumeration and found no demonstrated mathematical error. This revision adds its three requested clarifications: the minimum-basis cost sandwich, the nine-bit upper weight bound, and the boundary argument's allowance for repeated corrected indices.

That audit reviewed the hardest D2 branches in detail and others more briefly; further case-analysis review should prioritize 7772, 77711, 77721 and 77722. Complete human review remains open. It recompiled the Init-only core, while the seventeen unchanged Mathlib declarations retain their earlier checks and were not recompiled by that auditor under the 256 MiB cap. Hosted CI and independent human review remain unrun.

Credit

Credit DreamingOfClouds for PR3154's actual contract and literal lists, PR2816's P112 model and circuit methods; Nick Petalas for PR3171's coordinate-kernel/root-pencil method, PR3269/3300's capacities, the retained PR3161/3162 geometry and adjacent PR3418; David Arias for the earlier masked-root-star-assembly quadratic model; and the retained E8/P112 lineage including joaqcoo. Predecessor files are unchanged. The PR3154 excerpts retain their exact bytes and notice. Assistance and licensing are disclosed in TASK.json and NOTICE.md.

Open next questions

The raw threshold-dimension defects are closed for every positive d in the putative bad-invoice regime. Resolve the remaining d3 cases (adapted r0, or r1 with eta2/3), then the higher-d invoices, using the new intrinsic obstructions while preserving the actual coefficient/correction ledger. T-level deletion alone does not transfer an invoice. Formalize the remaining E8/completion bridge and invoice assembly. Global physical rank/pivot normalization remains a separate obligation; TASK.json gives first steps and blockers.