LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

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.

Ground eigenvalue as the form variational infimum

Statement

Assume Countable Choice and use $Q$ , $q_{h}$ , and the operator $h$ from Self-adjoint unbounded polynomial interaction at fixed regulator. Let $λ_{1}$ be the least eigenvalue of $h$ . Then $\inf {q_{h} (u) : u \in Q, ∥ u ∥_{2} = 1} = λ_{1} .$ The infimum is attained by every normalized vector in the lowest eigenspace; no simplicity or nondegeneracy is assumed.

Facts & Assumptions

Given: Countable Choice, the form domain $Q$ , the form $q_{h}$ , and the complete eigenfamily of $h$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable eigenbasis choices used in the form realization are permitted.

[F1]

Self-adjoint unbounded polynomial interaction at fixed regulator: the eigenvalues $λ_{n}$ are real, ordered increasingly with multiplicity, have a least value $λ_{1}$ , and for $u \in Q$ , $q_{h} (u) = \sum_{n \geq 1} λ_{n} ∣ ⟨ u, ψ_{n} ⟩ ∣^{2} .$

[F2]

Fourier expansion in a Hilbert space: $\sum_{n} ∣ ⟨ u, ψ_{n} ⟩ ∣^{2} = ∥ u ∥_{2}^{2}$ .

Proof

technique · direct

Given: $u \in Q$ and the ordered real eigenvalue list.

1.1A1F1F2algebra

Since $λ_{n} \geq λ_{1}$ for every $n$ , [A1], [F1] and [F2] imply $q_{h} (u) - λ_{1} ∥ u ∥_{2}^{2} = \sum_{n} (λ_{n} - λ_{1}) ∣ ⟨ u, ψ_{n} ⟩ ∣^{2} \geq 0.$ Thus every unit vector in $Q$ has form value at least $λ_{1}$ .

2.1F1F2step 1.1algebra∎

A normalized eigenvector $ψ_{1}$ belongs to $D (h) \subseteq Q$ by Self-adjoint unbounded polynomial interaction at fixed regulator, has norm one, and its sole nonzero coefficient is at an eigenvalue $λ_{1}$ . Therefore $q_{h} (ψ_{1}) = λ_{1}$ , so the lower bound is attained and equals the infimum. If the lowest eigenspace has dimension greater than one, the same coefficient calculation applies to every normalized vector in it.

Ground eigenvalue as the form variational infimum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources