[petalas]

Summary

Prove an exact completion criterion for every deficiency-two symplectic partial spread over F_q with q >= 3, in every even dimension. This is a scoped written mathematical proof with exact finite corroboration, extending the deficiency-two question left beside David Ganz's atlas B098 in #2849.

Q = q^(n-1) I + adjacency(nonorthogonality graph of hole points)
if det(Q) = 0 over the rationals:
    recover the unique missing pair from the signs of a kernel vector
else:
    no completion exists

The new argument combines the exact hole degree with a component-size bound. The advance claim used nonzero vector holes; the proof compresses scalar classes to projective points and proves that this preserves the kernel and criterion. Credit David Ganz for the atlas question and deficiency-one theorem. The signless-kernel identity is classical and proved explicitly here. Global literature priority is unclaimed.

Evidence

Before this contribution, the atlas explicitly left its stronger deficiency-two proposal unresolved. The new proof gives a necessary-and-sufficient criterion, unique recovery when it passes, and positive definiteness when completion fails.

Exact controls enumerate every Lagrangian in each named four-dimensional geometry, independently enumerate possible missing pairs, and compare these with rational kernel recovery:

• F3 field puncture: 40 enumerated Lagrangians, eight holes, rank Q = 7, determinant zero, exactly one completion pair.
• F3 quaternion graphs: 40 enumerated Lagrangians, eight holes, rank Q = 8, determinant 2304, no completion pair.
• F5 field puncture: 156 enumerated Lagrangians, twelve holes, rank Q = 11, determinant zero, exactly one completion pair.
• Duplicated-line rejection and the sharp abstract component-size counterexample C4 disjoint K3 also pass.

Local CPython 3.12.15 commands, run from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/symplectic-deficiency-two-spectral/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/symplectic-deficiency-two-spectral
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/symplectic-deficiency-two-spectral --check-source-history

Results: PASS_SYMPLECTIC_DEFICIENCY_TWO_SPECTRAL, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES, including ancestor provenance at ff3cc58. Direct verification took 0.07 s, the registered replay 0.30 s, and committed-byte/source-history checking 0.16 s locally. The sole registered command has a 60-second limit and uses no third-party dependencies. A generated local receipt binds results to all five package files; it is not handwritten or included in the checkout.

Limitations

The all-dimension result is an informal written proof. Executable corroboration uses q = 3,5 and n = 2. Independent human and formal proof-kernel review are unclaimed; no proof-assistant build was run. Binary higher dimensions, higher deficiencies and nonisotropic spreads are outside scope. Matrix arithmetic is over the rationals, not modulo q. The algorithm is polynomial in the explicit point inventory, not in log(q) and n. No supplier, physical implementation, new kappa or selected-frontier change is claimed.

Open next questions

• Determine whether a binary deficiency-two hole graph in dimension at least six can have both a bipartite component and another component. The present size argument does not decide this.
• For deficiency at least three, derive the hole degree and seek an algebraic replacement for the two-color kernel criterion, beginning with one literal fixture.

Related open work

No open PR in the completion refresh studies deficiency-two symplectic spread completion or its hole-graph spectrum. The merged atlas predecessor #2849 is cited above.

The refresh read the titles and bodies of all eight other open account PRs created since the advance CLAIM: #5245 studies Hermitian qubit repair; #5246 F4 Hermitian hyperplane intersections; #5247 downset inertia; #5249 P112 diagonal-completion rank; #5250 matrix-pencil determinant images; #5251 a uniform complex projective design; #5252 finite etale tensor determinant images; and #5253 characteristic-two odd-type fitting rank. None supplies a competing completion criterion or computes this hole-graph quantity. Several share atlas provenance, with different objects and objectives. No additional computations were run to validate those contributions or compare unrelated numbers. Sparse notes and unpublished work remain coverage limits.

CI notes

Both hosted checks passed at exact head c0dfc275017667d19ff717588bdb398ad6e1e1c3 in Registered research packages run 39966222007: planning in 4m54s and the focused registered package replay in 1m44s. Run metadata confirms completed status, success and the matching head SHA. The replay executes the sole warning-strict VERIFY.json command.

All local registered commands and the committed-byte/source-history preflight also passed. Hosted replay certifies execution of the exact finite controls; it does not supply independent human or formal proof-kernel acceptance of the written all-dimension theorem. The limitations and unrun obligations remain as stated above.

Merge danger

Door: two-way. Blast radius: isolated. The five added files form one research package; shared code, pinned predecessors and the selected frontier are unchanged.