[petalas]

Summary

Prove the binary case left open by #5248: a deficiency-two symplectic partial spread in every F2 dimension 2n completes exactly when its integer signless Laplacian is singular over the rationals. This is a scoped written mathematical proof with exact finite corroboration.

Q = 2^(n-1) I + adjacency(nonorthogonality graph of holes)
if Q is singular over the rationals:
    recover the unique missing pair from the signs of a kernel vector
else:
    no completion exists

The new geometric lemma makes any bipartite component the whole hole graph. Its color class spans a Lagrangian; every outside component would lie in that isotropic span and have degree zero. This replaces the earlier graph-order argument, which does not cover binary n >= 3.

Credit David Ganz's atlas B098 and deficiency-two proposal in #2849, and Nick Petalas's odd-field criterion and explicit binary handoff in #5248. Their mathematical verifiers are not imported, audited or replayed. The five added files in research/symplectic-binary-completion/ contain the proof, standalone verifier, metadata and exact provenance. No supplier, kappa or selected-frontier change is claimed. Global literature priority is unclaimed.

Evidence

Before this contribution, #5248 explicitly left the binary all-dimension criterion open. The new proof settles that recognition and unique-recovery implication.

The standalone check constructs punctured field spreads in binary dimensions 2, 4, 6 and 8. It checks every old subspace, hole degree, exact rational kernel, integer determinant and recovered Lagrangian. A separate exhaustive search through closed isotropic spans among the holes finds exactly one completion pair in each fixture. The hole counts are 2, 6, 14 and 30; the Q ranks are 1, 5, 13 and 29.

Three invalid-spread controls reject wrong deficiency, duplicate members and a nonisotropic member. An abstract 14-vertex degree-four graph has a singular kernel with six zero entries, demonstrating why matching binary counts alone cannot replace the new geometric argument. A second, nonbipartite abstract graph has trivial rational kernel and integer determinant 49,760,288. Neither abstract graph is claimed to be a valid hole graph.

Local CPython 3.12.15 reproduction, from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/symplectic-binary-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/symplectic-binary-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/symplectic-binary-completion
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/symplectic-binary-completion --check-source-history --check-modes

Outputs: PASS_SYMPLECTIC_BINARY_COMPLETION normally and under optimization, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Both exact registered commands were timed locally at 0.061 and 0.064 seconds, below their 60-second limits. Syntax and receipt-path rejection checks also pass. An optional generated receipt binds results to all five package file hashes; no receipt is handwritten or committed.

Base and provenance: canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. Source-history preflight confirms that base is an ancestor of publication 83f1fd986966558491ca938f7816cda1f46eacc6. The selected-baseline metadata control passes after sparse inputs are materialized. The advance CLAIM, live open/merged searches, main README searches and local question reservations record the bounded overlap scope. Sparse notes and unpublished work remain coverage limits.

Limitations

The universal theorem is an informal written proof. Finite corroboration uses only four specified punctured field spreads, three invalid inputs and two abstract graphs. No proof-assistant build or independent human proof review was run. No valid noncompletable binary deficiency-two fixture is supplied, and unconditional completion is not claimed. Larger deficiencies, nonisotropic spreads and physical supplier obligations are outside scope. The rational spectral algorithm is polynomial in the explicit hole inventory, not in n. No multiplication theorem or inherited framework assumption is discharged.

Open next questions

• Determine whether every binary deficiency-two symplectic partial spread completes or exhibit a source-bound nonsingular-Q example. Start with one specified dimension or a literature construction; a failed finite search would not prove unconditional completion.
• Find a geometric invariant for deficiency at least three, beginning with its hole degree and one literal fixture. The two-color kernel argument does not supply higher-color recovery.

Related open work

No open PR found by the completion refresh studies binary deficiency-two symplectic spread completion or the same hole-graph quantity. Merged #5248 is the direct predecessor; merged #2849 supplies the atlas question. Open #5803's mathematical atlas cites #5248 in its source inventory but does not provide this binary extension. No computations were run to validate its results.

The refresh inspected this account's other open PRs created since the advance CLAIM: #5854, #5856, #5857, #5858, #5859, #5860, #5861 and #5862. Their subjects are radical reachability, asymmetric UT3 pricing, clean square-zero compilation, Hermitian commuting repair, directed inverse envelopes, header dependence, local-product right-vector costs and unitary-cycle conditioning. None studies this completion criterion or computes the same hole-graph quantity. No numerical agreement or disagreement is asserted across these different objects.

CI notes

At head 83f1fd986966558491ca938f7816cda1f46eacc6, Registered research packages run 40479163791 finished with plan failed and replay skipped.

The sole failed step is Onboarding helper tests, running python -B -m unittest discover -s tests -p 'test_research_*.py' -v. Its log reports Ran 511 tests and FAILED (failures=8, errors=2, skipped=22). All ten reported failures/errors are in tests/test_research_current_assignment.py. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight assertion failures concern frozen-card routing, expected missing-reference rejection and sparse-helper selection.

The failed step, all ten failure/error headers and the final counts match #5854's run 40478863245 and #5856's run 40479202777, which share main base de5ec2e7a6074068452f45b1a191b5c72832df4a. This is the shared onboarding-test failure, outside the added package. No shared tests or implementation files were changed for it.

The hosted replay job was skipped. Hosted replay has not certified this package. Only the local package checks recorded above have run. The normal/optimized local verifier, registered runner and committed-byte/source-history preflight all passed. The written all-dimension theorem remains without independent human or formal proof-kernel review.

Merge danger

Door: two-way. Blast radius: isolated. Only one five-file research package is added. Shared code, frozen sources and the selected result are unchanged.