[petalas]

Summary

Prove the sharp 68-point union bound for two affine codimension-two live sets in the full 120-line E8 quadratic shell, and classify every equality case. Claim type: scoped written proof with finite exact corroboration. The new classification reduces the counting part of Matarener's PR6218 hand-proof question to one puncture-specific obligation.

two constraint planes in the full E8 quadratic model
  -> every union <= 68
  -> equality iff disjoint 36/32 sets share a constraint line with inconsistent values
  -> every other union <= 66
  -> P104 union <= 66 reduces to two removed lines meeting each exceptional union

The last puncture assertion remains open here, as does compatible-plane disjointness. This ships the first verified partial step from the advance CLAIM; the narrower delivered scope was recorded in the progress update.

Only research/p104-quadratic-union-reduction/ is added. This contribution was produced by Nick Petalas. Credit Matarener's PR6218 at 159352d273bd7b9f92c55cee3a9ac9b955de9b86 for the question, B-form/compatibility reduction and E8/2E8 model; DreamingOfClouds and the retained decoder/P104 lineage; Ali Romman and the retained native/caller/compiler contributors. Classical quadratic Gauss sums retain their credit. The older PR3253 already reports 68 for a punctured P112 two-flat census, and PR3269 uses signed Gauss sums for individual affine supports. The new contribution is the full-E8 proof, complete equality classification and P104 reduction; those predecessor results are not audited or claimed again.

Evidence

Before: PR6218 uses a finite P104 66-union bound in its open geometric proof route. After: the written proof isolates exactly the exceptional full-E8 configuration that needs puncture information; a literal 68-point fixture shows why full-E8 counting alone cannot prove 66.

Local CPython 3.12.15 checks pass:

• All 288,072 consistent and 1,368 inconsistent relative quadratic configurations on spans of dimensions two through four. The normalization covers every relative configuration; these are representative forms, not a census of all literal eight-space pairs.
• Every one of 43,180 direct affine plane systems agrees with the Fourier formula. Their sizes 24/28/32/36 occur 5,355/16,320/16,065/5,440 times.
• Explicit 36+32 disjoint attainment of 68, and two 36-point sets with intersection six and union 66. An opposite-sign form attains 72 and exposes the necessary plus-type hypothesis. Seven mathematical/domain rejection controls pass.
• Both warning-strict normal and optimized registered commands pass locally, timed individually at 0.306 s and 0.317 s, with identical generated source-hash receipts. Each has a 60-second timeout.
• Committed-byte/mode and full source-history checks pass for eight package/input files at 5f5e7fc87844f265ec5b206b2ed2e690f202f879. Canonical main was refreshed before committing and remained a5bff3bf133e482541c09eba4e773884bdb2b1b7.
• Official export integrity and both declared-input checks pass. Both replay modes pass in a read-only export with empty PATH and no Git; all nine exported files retain their bytes and modes. Disposable Gauss-sign, weakened-bound and source-byte mutations reject in both modes. Source compilation and all function-length checks pass; a separate static linter is unavailable.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -Werror -B research/p104-quadratic-union-reduction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -Werror -B research/p104-quadratic-union-reduction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-quadratic-union-reduction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-quadratic-union-reduction --check-source-history

Elsewhere substitute a CPython 3.12 executable. Expected markers: PASS_P104_QUADRATIC_UNION_REDUCTION, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=partial, and PASS_RESEARCH_COMMIT_BYTES.

Limitations

This is a proof reduction, not the source's puncture inequality, compatibility lemma or decoder floor. It supplies no physical word, restoration/ownership contract, native implementation, complete signed invoice, supplier admission, new kappa or complete multiplication theorem. The selected result and five framework assumptions remain unchanged. Independent human and formal proof-kernel review are unclaimed. No predecessor scientific replay, Lean, Docker or full selected cold replay was run or needed for the stated theorem.

Open next questions

• Show every exceptional 68-point union meets at least two of the sixteen removed lines. Start from the shared-line symplectic/isotropic normal form and the two removed orthogonal frames. Their relative position is the remaining input.
• Then prove the source's compatible-plane disjointness statement with its hyperplane and at-most-four pairwise non-orthogonal exceptions. The present union theorem does not track orthogonality of the uncovered ports.

Related open work

• research: P104 B-form floor follow-up: RootFact/SpanFact proved (hand + Lean), Cases II/III without the census (no κ claim) #6218 poses the exact hand-proof question and reports the finite punctured-P104 union bound 66. This result proves a full-E8 theorem and isolates the missing puncture step; it does not recheck that finite bound or close Lemma P.
• research: P104 wire routing obstructions (scoped negative, no κ) #6217 studies P104 wire routing through A2 triples and intersected partner frames. This proof concerns unions of affine label-plane live sets and provides no wire/helper construction.
• Prove the P104 disjoint A2 packing maximum is 33 (no κ) #6219 proves the disjoint A2 packing optimum 33 for P104. Its packing and parity quantity differs from this two-plane union classification; no numerical comparison is made.
• Prove E8 span inputs over every odd-characteristic field (no new κ) #6198 proves odd-characteristic E8 root-span facts. This result concerns the binary quadratic shell and affine equations, without extending or auditing those span facts.
• Exhibit complex rank-seven P104 approximations below the real 1/4 floor (no κ) #6211 studies complex rank-seven P104 Gram approximation. This result supplies no approximation error or correction program.
• Prove sharp 1/4 error for every real rank-seven P104 approximation (no κ) #6191 studies the sharp real rank-seven P104 approximation floor. Its approximation quantity differs from the present live-set union size.
• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies non-dense fitting pairings and square root-code decoder floors. This result does not change its pairing model or establish a decoder-cost floor.
• Rank-8 rigidity of the E8 pattern with arbitrary extra zeros after deleting any at most 8 lines (hypothesis (1) of Lemma RIGD; no kappa claim) #3040 studies rank-eight E8 pattern rigidity under deletion. The present result classifies a different two-plane cardinality problem and supplies no rigidity theorem.

Live open PRs, the latest handoffs and active Discussions, recent account work, local reservations and main/merged searches were checked before claiming and before committing. The overlap helper had complete pagination for 337 open PRs and 5,947 Discussion records; its sparse local-note coverage was incomplete, so targeted main-tree searches and relevant full notes were read separately. This bounded coverage is not proof of global novelty. No extra computations checked another contribution's numbers.

CI notes

Hosted CI is pending at 5f5e7fc87844f265ec5b206b2ed2e690f202f879. No hosted pass or failure is claimed. Both registered commands and every required local package/publication check passed. No required local proof obligation was left unrun; the mathematical limitations above are explicit open questions.

Merge Danger

Door: two-way.

Blast Radius: isolated.

One new research package. Shared code, frozen evidence and selected artifacts are unchanged.