LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Faithful irreducible finite-excitation representation

Statement

Assume Countable Choice, $Γ = Z^{d}$ with $d \geq 1$ , and finite on-site dimension $q \geq 1$ . Let $H_{v a c}$ and $π : A \to B (H_{v a c})$ be the finite-excitation representation of Finite-excitation vacuum spin representation and normality. Then $π$ is faithful and irreducible: $π (A)^{'} = C I, π (A)^{''} = B (H_{v a c}) .$

Here irreducible means precisely that the commutant consists only of scalar multiples of the identity. No classification of other representations is claimed.

Facts & Assumptions

Given: Countable Choice; the local spin algebras of Local spin matrix algebras and their factor inclusions; the configuration basis and isometric representation of Finite-excitation vacuum spin representation and normality.

[A1]

Assume ${A C}_{ω}$ , the exact countable-choice principle used to form the norm completion in the preceding definition; no stronger choice principle is used (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Finite-excitation vacuum spin representation and normality: $H_{v a c} = ℓ^{2} (S_{f i n})$ has basis $(e_{s})$ , and every finite local matrix acts on exactly its finite tensor factors in $π$ .

[F2]

Local spin matrix algebras and their factor inclusions: for each site $x$ and labels $a, b \in {0, \dots, q - 1}$ after the fixed basis relabeling, the local matrix unit $E_{x}^{a b}$ maps a configuration with label $b$ at $x$ to the configuration with label $a$ there and leaves every other site unchanged.

Proof

technique · show that an operator commuting with the local algebra is diagonal in the configuration basis, then use local matrix units to equate its diagonal entries

Given: $T \in π (A)^{'}$ , the basis $(e_{s})_{s \in S_{f i n}}$ , and [F1]–[F2].

1.1givenA1F1F2construct

Let $s \neq t$ in $S_{f i n}$ . They differ at some site $x$ ; set $a = s_{x}$ and $Q : = π (E_{x}^{a a})$ . Then $Q e_{s} = e_{s}$ and $Q e_{t} = 0$ . Since $T Q = Q T$ , self-adjointness of $Q$ gives $⟨ T e_{s}, e_{t} ⟩ = ⟨ Q T e_{s}, e_{t} ⟩ = ⟨ T e_{s}, Q e_{t} ⟩ = 0.$ Thus every off-diagonal coefficient of $T e_{s}$ vanishes, so completeness of the specified orthonormal basis implies $T e_{s} = c_{s} e_{s}$ for some scalar $c_{s}$ . The representation and completion are under the exact Countable Choice hypothesis A1.

2.1F1F2step 1.1construct

If two configurations $s, t$ differ at only one site $x$ , choose the matrix unit $E_{x}^{t_{x} s_{x}}$ . By [F2], $π (E_{x}^{t_{x} s_{x}}) e_{s} = e_{t}$ . Commuting this operator through $T$ gives $c_{t} e_{t} = T e_{t} = T π (E_{x}^{t_{x} s_{x}}) e_{s} = π (E_{x}^{t_{x} s_{x}}) T e_{s} = c_{s} e_{t}$ , hence $c_{t} = c_{s}$ . Any $s, t \in S_{f i n}$ differ on a finite set, and changing those finitely many labels one site at a time connects them; therefore all $c_{s}$ equal a single scalar $c$ .

3.1step 2.1F1algebra∎

Step 2.1 proves every $T$ in the commutant is $c I$ , while scalars commute with all operators, so $π (A)^{'} = C I$ . The definition of bicommutant then gives $π (A)^{''} = (C I)^{'} = B (H_{v a c})$ . Faithfulness follows from the isometry established in Finite-excitation vacuum spin representation and normality.

Faithful irreducible finite-excitation representation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources