CounterexampleConstruction: AI-adaptedVerification: 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.

Bounded free-photon energy need not yield a Fock-vector limit

Statement refuted

Every normalized coherent family in the transverse free-photon Fock space with uniformly bounded expectation of $H_{p h} = ℏ c d Γ (∣ k ∣)$ converges strongly in that Fock space as its infrared cutoff $ϵ ↓ 0$ .

Construction

Take a unit vector $e \in R^{3}$ , set $b (n) = P_{T} (n) e$ for $n \in S^{2}$ , fix $κ > 0$ , and use $f_{ϵ} (k) = 1_{{ϵ < ∣ k ∣ < κ}} ∣ k ∣^{- 3 / 2} b (\hat{k})$ . This gives the explicit transverse coherent family from Logarithmic coherent infrared tail: finite energy, no Fock limit.

Facts & Assumptions

Given: The planned transverse free-photon space and the exact coherent-family result of Logarithmic coherent infrared tail: finite energy, no Fock limit.

[F1]

Logarithmic coherent infrared tail: finite energy, no Fock limit: for any transverse $b$ with $C_{b} = \int_{S^{2}} ∣ b ∣^{2} d Ω > 0$ , the vectors $C (f_{ϵ})$ have norm one, energy expectation $ℏ c C_{b} (κ - ϵ)$ , weak limit zero, and no strong Fock-space limit.

[A1]

$e \in R^{3}$ is a unit vector, $κ > 0$ , and $ℏ, c > 0$ ; cutoffs satisfy $0 < ϵ < κ$ . These are the chosen model parameters in the Counterexample.

[F2]

Agreement with the existing polar sphere measure: the Euclidean chart measure on $S^{2}$ equals polar surface measure and is invariant under orthogonal transformations.

[F3]

Surface integration on compact C1 hypersurfaces: surface integration is the Euclidean chart integral with density $\sqrt{\det (D X^{T} D X)}$ .

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume the same Countable Choice carried by the coherent-tail and surface-measure suppliers; no stronger choice principle is used.

Counterexample

Given: [A1]–[A2], [F1]–[F3], and the profile in the Counterexample.

1.1A1A2F1F2F3algebra

For $b (n) = P_{T} (n) e$ , $∣ b (n) ∣^{2} = 1 - (n \cdot e)^{2}$ . By [F2], rotate $e$ to the third coordinate axis. The regular sphere chart $X (y, z) = (\sqrt{1 - y^{2} - z^{2}}, y, z)$ on $∣ y ∣, ∣ z ∣ < 1 / 4$ lies in the belt $∣ n \cdot e ∣ < 1 / 2$ and has area density $(1 - y^{2} - z^{2})^{- 1 / 2} \geq 1$ by [F3]. Its parameter rectangle has area $1 / 4$ , so this belt has positive surface measure. On the belt $∣ b (n) ∣^{2} \geq 3 / 4$ , hence $C_{b} > 0$ ; also $b$ is continuous, square-integrable on the finite sphere measure, and transverse.

2.1A1A2F1step 1.1algebra∎

Apply [F1] to this $b$ . It gives unit vectors with $0 \leq ⟨ C (f_{ϵ}), H_{p h} C (f_{ϵ}) ⟩ \leq ℏ c C_{b} κ < \infty$ , yet no strong Fock-space limit. This contradicts the asserted implication and is the required counterexample.

Bounded free-photon energy need not yield a Fock-vector limit

Statement refuted

Construction

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources