Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Symplectic double-orthogonal and dimension identities

Statement

If W is a subspace of the finite-dimensional symplectic vector space (V,ω), then

dim⁡W+dim⁡Wω=dim⁡V,(Wω)ω=W.

Facts & Assumptions

Given: A finite-dimensional symplectic vector space (V,ω) and a subspace W≤V.

[F1]

The symplectic orthogonal is (ω♭)−1(ann⁡W), where ω♭ is an isomorphism. Symplectic orthogonal complement.

Proof

technique · direct
1.1

By [F1], ω♭ restricts to an isomorphism Wω→ann⁡W. Finite-dimensional annihilator algebra gives dim⁡ann⁡W=dim⁡V−dim⁡W, proving the dimension identity.

F1algebra
2.1

Alternation shows W⊆(Wω)ω: if w∈W and v∈Wω, then ω(w,v)=−ω(v,w)=0. Applying step 1.1 to Wω gives dim⁡(Wω)ω=dim⁡W, so the inclusion is equality. For W=0 or W=V the same calculation gives the stated endpoint identities.

F1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources