DefinitionDefinition: Literature-sourcedProof: 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.

Positive transverse photon one-particle space

Definition

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $H = L^{2} (R^{3}, d^{3} k; C^{3}), ⟨ u, v ⟩_{2} = \int_{R^{3}} \overline{u (k)} \cdot v (k) d^{3} k,$ with the second-variable-linear convention of Schwartz/distribution spaces and Hilbert Fock conventions; the real momentum dot product and norm are those of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . Explicitly extend the real dot product complex-bilinearly to $a \cdot b = \sum_{j = 1}^{3} a_{j} b_{j}$ for $a, b \in C^{3}$ . Each complex fiber instead has the positive Hermitian pairing $⟨ a, b ⟩_{2} = \overline{a} \cdot b = \sum_{j} {\overline{a}}_{j} b_{j}$ and norm $∥ a ∥^{2} = \sum_{j} ∣ a_{j} ∣^{2}$ ; the displayed integral uses this pairing. Hilbert completeness follows from $L^{2}$ with the integral pairing is a Hilbert space. For $k \neq 0$ , write $n (k) = k / ∣ k ∣$ and $P_{T} (k) = I_{3} - n (k) n (k)^{T} .$ Assign any value at $k = 0$ , which is a Lebesgue-null set. Let $M_{P_{T}}$ be multiplication by this measurable matrix field and define the positive transverse photon space $H_{T} = Ran M_{P_{T}} = {u \in H : k \cdot u (k) = 0 for almost every k \neq 0} .$ The inner product on $H_{T}$ is the restriction of that on $H$ . Each nonzero momentum fiber is the two-complex-dimensional plane ${a \in C^{3} : k \cdot a = 0}$ perpendicular to the real vector $k$ in that Hermitian pairing; no global continuous polarization frame is part of the definition.

Facts & Assumptions

Given: Countable Choice and the $L^{2}$ Hilbert-space and Fock-pairing conventions stated above.

[A1]

Under Countable Choice, complex $L^{2} (μ)$ with the integral pairing is a Hilbert space. $L^{2}$ with the integral pairing is a Hilbert space

[A2]

Assume Countable Choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The Fock convention uses a second-variable-linear Hilbert pairing. Schwartz/distribution spaces and Hilbert Fock conventions

[F2]

The Euclidean dot product is $u \cdot v = \sum_{i = 1}^{3} u_{i} v_{i}$ and $∣ u ∣ = \sqrt{u \cdot u}$ . The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$

Proof

Given: [A1], [A2], [F1], [F2], and the measurable matrix field in the definition.

1.1F1F2givenalgebra

For each $k \neq 0$ , $n^{T} n = n \cdot n = 1$ by [F2], so $P_{T}^{*} = P_{T}$ and $P_{T}^{2} = (I - n n^{T})^{2} = I - 2 n n^{T} + n (n^{T} n) n^{T} = I - n n^{T} = P_{T} .$ Its complex kernel is ${span}_{C} {n}$ and its range is the Hermitian perpendicular plane $n^{⊥}$ , of complex dimension two. For $z \in C^{3}$ , put $a = n^{T} z$ . The specified Hermitian pairing and realness of $n$ give $∥ P_{T} z ∥^{2} = \sum_{j} ∣ z_{j} - n_{j} a ∣^{2} = ∥ z ∥^{2} - ∣ a ∣^{2} \leq ∥ z ∥^{2}$ . Thus it is contractive, with norm one because it fixes every unit vector in that nonzero perpendicular plane. The arbitrary value at the single point $k = 0$ does not affect its multiplication operator on $L^{2}$ .

2.1A1A2F1F2step 1.1given∎

Under [A2], [A1] makes each complex scalar $L^{2}$ component complete for its first-variable-linear pairing $(u, v)_{1} = \int u \overline{v}$ . The three-component direct sum is complete componentwise. Its second-variable-linear pairing is $⟨ u, v ⟩_{2} : = (v, u)_{1} = \int \overline{u} \cdot v$ , with the same norm; [F1] fixes this convention. Thus $H$ is complete. The multiplier $M_{P_{T}}$ is bounded and idempotent by step 1.1. If $u_{r} = M_{P_{T}} v_{r} \to u$ in $H$ , then $M_{P_{T}} u = \lim_{r} M_{P_{T}} u_{r} = \lim_{r} u_{r} = u$ by boundedness, so $u$ belongs to its range. Thus $H_{T}$ is closed and inherits a positive Hilbert inner product. Since $P_{T} u = u$ is equivalent to $n \cdot u = 0$ for unit $n$ by [F2], the pointwise range formula gives the displayed transverse condition almost everywhere.

Positive transverse photon one-particle space

Definition

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources