Pipeline-generated

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.

Lieb–Robinson locality, clustering and gapped spectral transport

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-10-22 (gpt-6.1-sol)Open item page →

Weighted interactions and gapped finite-volume paths

Definition

Fix $d \geq 1$ , the site set $Γ = Z^{d}$ , its max-coordinate metric $d_{\infty}$ , and the local spin algebras $A_{Λ}$ of the uniformly finite-dimensional spin system. A finite interaction is a family $Φ (Z) = Φ (Z)^{*} \in A_{Z}$ indexed by finite nonempty $Z \subset Γ$ . For a finite $Λ \subset Γ$ , define

$H_{Λ}^{Φ} : = \sum_{\emptyset \neq Z \subseteq Λ} Φ (Z), τ_{t}^{Λ, Φ} (A) : = e^{i t H_{Λ}^{Φ} / ℏ} A e^{- i t H_{Λ}^{Φ} / ℏ},$

where $ℏ > 0$ , $A$ is embedded in a common finite local algebra, and the finite sum is a self-adjoint matrix. When a path $Φ_{s}$ , $s \in [0, 1]$ , is used, each $Φ_{s} (Z)$ is $C^{1}$ in matrix norm and the derivative is taken term by term.

For $r \geq 0$ put $F (r) = (1 + r)^{- d - 1}$ . The admissible weights used here are

$F_{a} (r) : = e^{- a r} F (r) (a > 0), F_{ρ} (r) : = (1 + r)^{- ρ} (ρ > d) .$

For an admissible weight $G$ , define the weighted interaction norm

$∥ Φ ∥_{G} : = \sup_{x, y \in Γ} \frac{1}{G (d_{\infty} (x, y))} \sum_{\begin{array}{c} Z \subset Γ f i n i t e, n o n e m p t y \\ x, y \in Z \end{array}} ∥ Φ (Z) ∥ .$

The derivative norm of a path is

$∥ \partial Φ ∥_{G} : = \sup_{s \in [0, 1]} \sup_{x, y \in Γ} \frac{1}{G (d_{\infty} (x, y))} \sum_{Z ∋ x, y} ∣ Z ∣ ∥ \partial_{s} Φ_{s} (Z) ∥ .$

A finite-volume spectral band is a nonempty selected subset $Σ_{Λ} (s) \subseteq σ (H_{Λ}^{Φ_{s}})$ together with the orthogonal projection onto the sum of its eigenspaces. It may be specified by an interval $I_{Λ} (s)$ with smoothly varying endpoints, in which case $Σ_{Λ} (s) = σ (H_{Λ}^{Φ_{s}}) \cap I_{Λ} (s)$ . The band is uniformly isolated with gap $γ > 0$ along a specified finite-volume exhaustion when

$dist ⁣ (Σ_{Λ} (s), σ (H_{Λ}^{Φ_{s}}) ∖ Σ_{Λ} (s)) \geq γ$

for every volume in that exhaustion and every $s \in [0, 1]$ . For spectral-flow arguments, the selected eigenvalues must also form a locally fixed spectral cluster: for each $s_{0}$ , a contour in the resolvent set encloses exactly the selected eigenvalues for all $s$ in a neighborhood of $s_{0}$ . In finite dimension this implies that its projection is $C^{1}$ when the Hamiltonian path is $C^{1}$ . This cluster condition holds, for example, for the first $r_{Λ}$ eigenvalues counted with multiplicity when the integer $r_{Λ}$ is fixed along the path and the gap after them stays open. Uniform distance between the selected and unselected spectra alone does not prevent an interval from changing membership discontinuously. A ground band is such a band containing the lowest eigenvalue or a finite-volume cluster of lowest eigenvalues; no uniqueness is implicit.

For physical units, take interaction terms and Hamiltonians in joules, time in seconds, and $ℏ$ in joule-seconds. The path parameter $s$ , lattice distance, and admissible weights are dimensionless; the interaction norm therefore has energy units and the derivative norm has energy units per dimensionless path parameter.

Statement references

Local spin matrix algebras and their factor inclusions

Uniformly bounded finite-range quantum spin interaction

LemmaStatement: AI-adaptedProof: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Summability and convolution control for lattice weights

Statement

Let $d \geq 1$ , $Γ = Z^{d}$ , and $d_{\infty}$ be the max-coordinate metric. For $F (r) = (1 + r)^{- d - 1}$ , $F_{a} (r) = e^{- a r} F (r)$ with $a > 0$ , and $F_{ρ} (r) = (1 + r)^{- ρ}$ with $ρ > d$ , each weight $G \in {F, F_{a}, F_{ρ}}$ is summable on $Γ$ . Its convolution constant

$C_{G} : = \sup_{x, y \in Γ} \frac{\sum_{z \in Γ} G (d_{\infty} (x, z)) G (d_{\infty} (z, y))}{G (d_{\infty} (x, y))},$

where $C_{F}$ uses this same formula with the base weight $F = F_{d + 1}$ . Each such constant is finite and positive. In particular, for all $x, y \in Γ$ ,

$\sum_{z \in Γ} G (d_{\infty} (x, z)) G (d_{\infty} (z, y)) \leq C_{G} G (d_{\infty} (x, y)) .$

The constant $C_{F_{a}}$ may be taken to be the convolution constant $C_{F}$ of $F$ , independently of $a$ .

Statement references

Weighted interactions and gapped finite-volume paths

Local spin matrix algebras and their factor inclusions

Facts & Assumptions

Given: $d \geq 1$ , $Γ = Z^{d}$ , the metric $d_{\infty}$ , and the weights $F, F_{a}, F_{ρ}$ as stated.

[F1]

Local spin matrix algebras and their factor inclusions: $d_{\infty} (x, y) = \max_{1 \leq j \leq d} ∣ x_{j} - y_{j} ∣$ , and the site set is $Z^{d}$ .

[F2]

Weighted interactions and gapped finite-volume paths: $F (r) = (1 + r)^{- d - 1}$ , $F_{a} (r) = e^{- a r} F (r)$ , and $F_{ρ} (r) = (1 + r)^{- ρ}$ .

Proof

Given: [F1], [F2], $a > 0$ , and $ρ > d$ .

1.1givenF1F2algebra

For $k \geq 1$ , the $d_{\infty}$ -shell of radius $k$ has $(2 k + 1)^{d} - (2 k - 1)^{d} \leq c_{d} (k + 1)^{d - 1}$ sites, by factoring the difference of powers; the radius-zero shell has one site. Therefore $\sum_{z} F_{ρ} (d_{\infty} (x, z)) \leq c_{d} \sum_{k \geq 0} (k + 1)^{d - 1 - ρ} < \infty$ , and $\sum_{z} F_{a} (d_{\infty} (x, z)) \leq \sum_{z} F (d_{\infty} (x, z)) < \infty$ . The bounds are independent of $x$ by translation invariance.

2.1givenstep 1.1algebra∎

Write $r = d_{\infty} (x, y)$ . For $G = F_{ρ}$ , the triangle inequality places every $z$ in at least one of the sets $d_{\infty} (x, z) \geq r / 2$ or $d_{\infty} (z, y) \geq r / 2$ . On the first set, $G (d_{\infty} (x, z)) \leq 2^{ρ} G (r)$ ; summing the other factor over all $z$ is bounded by the finite shell sum in step 1.1. The second set gives the same bound with $x, y$ exchanged. Hence $C_{F_{ρ}} \leq 2^{ρ + 1} \sum_{z} F_{ρ} (d_{\infty} (0, z)) < \infty$ . For $F_{a}$ , the triangle inequality gives $e^{- a (d (x, z) + d (z, y))} \leq e^{- a r}$ ; the same splitting argument applied to $F = F_{d + 1}$ , for which $d + 1 > d$ , yields $C_{F} < \infty$ , and then $\sum_{z} F_{a} (d (x, z)) F_{a} (d (z, y)) \leq C_{F} F_{a} (r)$ . Positivity of each weight and the $z = x$ summand show $C_{G} > 0$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-10-22 (gpt-6.1-sol)Open item page →

Normalized-trace expectation onto a finite spin region

Definition

Let $X, Y \subset Z^{d}$ be finite regions and $A \in A_{Y}$ . Embed $A$ in $A_{X \cup Y}$ , use the canonical site-order permutation to group factors in $X \cap Y$ and $Y ∖ X$ , and apply the normalized trace on the omitted factors:

$E_{X} (A) : = ({id}_{A_{X \cap Y}} \otimes τ_{Y ∖ X}) (A) \otimes I_{X ∖ Y} \in A_{X}, τ_{D} : = q^{- ∣ D ∣} {Tr}_{D} .$

The empty tensor factor has normalized trace $1$ . Since $τ_{D} (I_{D}) = 1$ , adding an identity factor does not change the result, so this formula consistently defines $E_{X}$ on the directed union $A_{l o c}$ . This definition alone asserts no norm extension to its completion.

Statement references

Local spin matrix algebras and their factor inclusions

Norm-completed quasi-local quantum spin algebra

LemmaStatement: AI-adaptedProof: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Contractive local expectations and their commutator bound

Statement

For every finite $X \subset Z^{d}$ , the map $E_{X}$ of the normalized-trace-expectation definition is a unital positive contraction on $A_{l o c}$ and has a unique contractive extension $E_{X} : A \to A_{X}$ . For $B \in A_{Y}$ , $Y$ finite,

$∥ B - E_{X} (B) ∥ \leq \sum_{z \in Y ∖ X} \sup_{\begin{array}{c} V \in A_{{z}} \\ V^{*} V = V V^{*} = I \end{array}} ∥ [B, V] ∥ .$

Statement references

Normalized-trace expectation onto a finite spin region

Local spin matrix algebras and their factor inclusions

Facts & Assumptions

Given: $d \geq 1$ , a fixed finite spin dimension $q \geq 1$ , finite regions $X, Y \subset Z^{d}$ , the local algebras $A_{Λ}$ , and $B \in A_{Y}$ .

[F1]

Normalized-trace expectation onto a finite spin region: $E_{X}$ is normalized partial trace over the factors in $Y ∖ X$ , followed by identity inclusion into $X$ .

[F2]

Local spin matrix algebras and their factor inclusions: $A_{Y} ≅ ⨂_{z \in Y} M_{q} (C)$ , with the ordinary operator norm, normalized trace, and consistent factor inclusions.

Proof

technique · direct

Given: [F1], [F2], and $B \in A_{Y}$ .

1.1givenF1F2algebra

On $C^{q}$ in the standard basis $e_{j}$ , $0 \leq j < q$ , define $S e_{j} = e_{j + 1 m o d q}$ and $D e_{j} = e^{2 π i j / q} e_{j}$ . The $q^{2}$ unitaries $S^{a} D^{b}$ , $0 \leq a, b < q$ , satisfy $q^{- 2} \sum_{a, b = 0}^{q - 1} (S^{a} D^{b})^{*} R (S^{a} D^{b}) = q^{- 1} Tr (R) I_{q} (R \in M_{q} (C)) .$ Indeed, averaging over $b$ cancels each off-diagonal matrix unit because its phase is a nontrivial $q$ -th root sum, and averaging the surviving diagonal units over $a$ gives their arithmetic mean times $I_{q}$ ; for $q = 1$ the formula is the identity. Applying this one-site twirl on each factor in $Y ∖ X$ yields the normalized partial trace of [F1]. Each twirl is an average of unitary conjugations, so it is unital, positive, and contractive. The formula is unchanged when an extra identity factor is added, since its normalized trace is $1$ ; hence $E_{X}$ is consistent on $A_{l o c}$ . Contractivity extends it uniquely to the completion, and its range remains in the closed finite-dimensional subalgebra $A_{X}$ .

2.1step 1.1F1F2algebra∎

Enumerate $Y ∖ X = {z_{1}, \dots, z_{k}}$ and write $E_{z_{j}}$ for the one-site twirl. The commuting twirls satisfy $B - E_{X} (B) = \sum_{j = 1}^{k} E_{z_{1}} \dots E_{z_{j - 1}} (B - E_{z_{j}} (B)) .$ For any $C \in A_{Y}$ , the average formula gives $∥ C - E_{z_{j}} (C) ∥ \leq \sup_{V_{z}} ∥ C - V_{z}^{*} C V_{z} ∥ = \sup_{V_{z}} ∥ [C, V_{z}] ∥$ , where $V_{z}$ ranges over the one-site unitaries. Each preceding twirl is contractive and commutes with conjugation by $V_{z}$ , so its term in the displayed sum is at most $\sup_{V_{z}} ∥ [B, V_{z}] ∥$ . The triangle inequality and the finite sum over sites give the stated bound.

TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Weighted interaction commutator bound

Statement

Let $G$ be either $F_{a}$ , $a > 0$ , or $F_{ρ}$ , $ρ > d$ , as in the weighted-interaction definition. Let $Φ_{u}$ be a norm-continuous family of finite interactions on $Z^{d}$ , indexed by $u \in [0, T]$ , with $M_{G} : = \sup_{u \in [0, T]} ∥ Φ_{u} ∥_{G} < \infty$ . For finite $Λ \subset Z^{d}$ , define $H_{Λ} (u) = \sum_{\emptyset \neq Z \subseteq Λ} Φ_{u} (Z)$ and let $U_{Λ} (0) = I$ solve

$\frac{d}{d u} U_{Λ} (u) = - i H_{Λ} (u) U_{Λ} (u) / ℏ .$

Set $α_{u}^{Λ} (A) = U_{Λ} (u)^{*} A U_{Λ} (u)$ , extended by the identity outside $Λ$ . If $A \in A_{X}$ , $B \in A_{Y}$ for disjoint finite nonempty $X, Y \subset Λ$ , then

$∥ [α_{u}^{Λ} (A), B] ∥ \leq \frac{2 ∥ A ∥ ∥ B ∥}{C_{G}} (e^{2 C_{G} M_{G} u / ℏ} - 1) \sum_{x \in X, y \in Y} G (d_{\infty} (x, y)) .$

The same estimate holds for the inverse path. In particular, for a constant interaction $Φ$ evolved in physical time $t$ by $τ_{t}^{Λ, Φ}$ , take $u = ∣ t ∣$ and the constant path $Φ_{u} = Φ$ to obtain

$∥ [τ_{t}^{Λ, Φ} (A), B] ∥ \leq \frac{2 ∥ A ∥ ∥ B ∥}{C_{G}} (e^{2 C_{G} ∥ Φ ∥_{G} ∣ t ∣ / ℏ} - 1) \sum_{x \in X, y \in Y} G (d_{\infty} (x, y)) .$

In physical units, $u, T, t$ are times, $M_{G}$ and $∥ Φ ∥_{G}$ have energy units, and $ℏ$ has energy-times-time units; the weights and convolution constants are dimensionless, so the exponential arguments are dimensionless.

Statement references

Weighted interactions and gapped finite-volume paths

Summability and convolution control for lattice weights

Local spin matrix algebras and their factor inclusions

Facts & Assumptions

Given: $d \geq 1$ , a finite volume $Λ$ , finite nonempty disjoint supports $X, Y \subseteq Λ$ , a norm-continuous finite-interaction path $Φ_{u}$ , and the bounds in the Statement.

[F1]

Weighted interactions and gapped finite-volume paths: $H_{Λ} (u)$ is a finite self-adjoint matrix, $∥ Φ_{u} ∥_{G}$ is the supremum of the pair-weighted interaction sums, and the finite-volume dynamics are matrix conjugations.

[F2]

Summability and convolution control for lattice weights: $G$ is summable and $\sum_{z} G (d_{\infty} (x, z)) G (d_{\infty} (z, y)) \leq C_{G} G (d_{\infty} (x, y))$ , with $C_{G} > 0$ .

[F3]

Local spin matrix algebras and their factor inclusions: finite local algebras are matrix algebras; operators in disjoint tensor factors commute, and their factor inclusions preserve the operator norm.

Proof

technique · direct

Given: [F1]–[F3], $M_{G} < \infty$ , and $ℏ > 0$ in the constant-interaction statement.

1.1givenF1F3algebra

For a finite nonempty $S \subseteq Λ$ , define $C_{B}^{Λ} (S, u) = \sup_{D \in A_{S}, D \neq 0} ∥ [α_{u}^{Λ} (D), B] ∥ / ∥ D ∥$ . Let $U_{S} (u)$ solve $U_{S}^{'} (u) = - i H_{S} (u) U_{S} (u) / ℏ$ , $U_{S} (0) = I$ , where $H_{S} (u) = \sum_{\emptyset \neq Z \subseteq S} Φ_{u} (Z)$ . Put ${\tilde{α}}_{u}^{S} (D) = U_{S} (u) D U_{S} (u)^{*}$ and $f (u) = [α_{u}^{Λ} ({\tilde{α}}_{u}^{S} (D)), B]$ . Product differentiation and the Jacobi identity give $f^{'} (u) = \frac{i}{ℏ} [K_{S} (u), f (u)] + \frac{i}{ℏ} \sum_{\begin{array}{c} Z \subseteq Λ \\ Z \cap S \neq \emptyset, Z \subseteq̸ S \end{array}} [α_{u}^{Λ} ({\tilde{α}}_{u}^{S} (D)), [B, α_{u}^{Λ} (Φ_{u} (Z))]],$ where $K_{S} (u) = \sum_{Z \subseteq Λ : Z \cap S \neq \emptyset, Z \subseteq̸ S} α_{u}^{Λ} (Φ_{u} (Z))$ is self-adjoint. Indeed, the local propagator cancels the terms contained in $S$ , and terms disjoint from $S$ commute with $D$ by [F3]. The homogeneous equation $g^{'} = (i / ℏ) [K_{S} (u), g]$ preserves norm because its propagator is unitary; existence follows from the uniformly convergent finite-dimensional Picard series. Variation of constants, $∥ [R, Q] ∥ \leq 2 ∥ R ∥ ∥ Q ∥$ , and enlargement of the crossing-support sum to all terms meeting $S$ therefore give $C_{B}^{Λ} (S, u) \leq C_{B}^{Λ} (S, 0) + \frac{2}{ℏ} \sum_{\begin{array}{c} Z \subseteq Λ \\ Z \cap S \neq \emptyset \end{array}} \int_{0}^{u} ∥ Φ_{v} (Z) ∥ C_{B}^{Λ} (Z, v) d v .$ The same inequality applies to the inverse path, which is generated over a reversed parameter interval by the interaction $- Φ_{u - v}$ and has the same norm bound.

2.1givenF1F3step 1.1algebra

At $u = 0$ , disjointness gives $C_{B}^{Λ} (S, 0) = 0$ when $S \cap Y = \emptyset$ , while $C_{B}^{Λ} (S, u) \leq 2 ∥ B ∥$ for all supports and times by the commutator inequality and isometry of the dynamics. Iteration of step 1.1 produces time-ordered integrals over $0 \leq v_{n} \leq \dots \leq v_{1} \leq u$ and chains $Z_{1}, \dots, Z_{n}$ with $Z_{1} \cap S \neq \emptyset$ , $Z_{j + 1} \cap Z_{j} \neq \emptyset$ and $Z_{n} \cap Y \neq \emptyset$ , with product $\prod_{j = 1}^{n} ∥ Φ_{v_{j}} (Z_{j}) ∥$ . All support sums are finite. For the iteration remainder, put $Q_{Λ} = \sum_{\emptyset \neq Z \subseteq Λ} \sup_{v \in [0, T]} ∥ Φ_{v} (Z) ∥ < \infty$ , which is finite by norm continuity and compactness. Its absolute size after $n$ iterations is at most $2 ∥ B ∥ (2 Q_{Λ} u / ℏ)^{n} / n!$ , so it tends to zero. This volume-dependent number is used only to discard the remainder. For the chain terms, define the time-dependent kernel $K_{v} (x, y) = \sum_{Z \subseteq Λ : x, y \in Z} ∥ Φ_{v} (Z) ∥$ . At each fixed time tuple, summing over all endpoint and overlap sites overcounts each nonnegative interaction-chain term at least once. Thus the order- $n$ contribution for $S = X$ is at most $2 ∥ B ∥ (2 / ℏ)^{n} \int_{0 \leq v_{n} \leq \dots \leq v_{1} \leq u} \sum_{x \in X, y \in Y} (K_{v_{1}} \dots K_{v_{n}}) (x, y) d v_{n} \dots d v_{1} .$ Kernel products here sum only over sites of the finite volume. No supremum over time has been moved inside an interaction sum for this bound.

3.1givenF1F2step 2.1algebra∎

By [F1], for each $v \in [0, T]$ separately, $K_{v} (x, y) \leq M_{G} G (d_{\infty} (x, y))$ . Induction using [F2] gives $(K_{v_{1}} \dots K_{v_{n}}) (x, y) \leq M_{G}^{n} C_{G}^{n - 1} G (d_{\infty} (x, y)) (n \geq 1) :$ the intermediate-site convolution over the finite volume is bounded by the full nonnegative lattice convolution. This bound is uniform in every time tuple. Its integral over the time-ordered simplex is $u^{n} / n!$ times that bound. Substituting into step 2.1 and summing the convergent exponential series gives $C_{B}^{Λ} (X, u) \leq \frac{2 ∥ B ∥}{C_{G}} (e^{2 C_{G} M_{G} u / ℏ} - 1) \sum_{x \in X, y \in Y} G (d_{\infty} (x, y))$ . The finite-volume remainder justifies the iteration limit; alternatively the displayed factorial majorant dominates every partial chain sum. Multiplication by $∥ A ∥$ proves the path estimate, including $A = 0$ . Applying the same time-ordered argument to the reversed interaction path, whose norm at every time is bounded by $M_{G}$ , proves the inverse estimate. For a constant interaction take $u = ∣ t ∣$ ; reversal changes its sign but not its norm. All constants in the resulting estimates depend only on $C_{G}, M_{G}$ , not on the volume-dependent remainder bound.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Scalar spectral measures of a separable positive generator under countable choice

Statement

Assume Countable Choice. Let $H \neq {0}$ be a separable complex Hilbert space with inner product linear in its first argument. Let $K : D (K) \to H$ be densely defined, self-adjoint and positive, meaning $⟨ K x, x ⟩ \geq 0$ for $x \in D (K)$ . Let $(U_{t})_{t \in R}$ be a strongly continuous unitary group with generator $i K$ : its strong derivative at zero exists exactly on $D (K)$ and equals $i K x$ . The spectrum $σ (K)$ is the complement of the set of $z$ for which $K - z : D (K) \to H$ has a bounded everywhere-defined inverse.

For every $x, y \in H$ there is a finite complex Borel measure $μ_{x, y}$ on $[0, \infty)$ such that

$⟨ U_{t} x, y ⟩ = \int_{[0, \infty)} e^{i t E} d μ_{x, y} (E), ∣ μ_{x, y} ∣ ([0, \infty)) \leq ∥ x ∥ ∥ y ∥ .$

The measures are sesquilinear in $(x, y)$ , $μ_{x} : = μ_{x, x}$ is positive with total mass $∥ x ∥^{2}$ , and every $μ_{x, y}$ is carried by $σ (K)$ . If $P_{0}$ is the orthogonal projection onto $\ker K$ , then

$μ_{x, y} ({0}) = ⟨ P_{0} x, y ⟩ .$

In particular, if $σ (K) \subseteq {0} \cup [g, \infty)$ with $g > 0$ , the measures $μ_{(I - P_{0}) x, (I - P_{0}) y}$ are carried by $[g, \infty)$ . Spectral measures and a projection-valued spectral resolution are not assumptions of this statement.

Statement references

Symmetric, self-adjoint and essentially self-adjoint operators

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

Facts & Assumptions

Given: The Hilbert space, positive self-adjoint generator and unitary group in the Statement.

[A1]

Countable Choice is assumed throughout. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Under Countable Choice a self-adjoint operator is closed, its range orthogonal complement is the kernel of its adjoint, and every closed Hilbert subspace has an orthogonal complement decomposition. Symmetric, self-adjoint and essentially self-adjoint operators The adjoint is well defined, closed, and reverses inclusions Orthogonal decomposition by a closed subspace

[F2]

Finite Hermitian matrices have orthonormal eigenbases with real eigenvalues; the polynomial norm estimate needed here is derived in step 2.1. Hilbert adjoints satisfy the adjoint and norm identities under Countable Choice. Finite matrix spectral and state calculus Hilbert-adjoint identities

[F3]

Real polynomials are uniformly dense in continuous functions on a compact interval, and operator-norm Cauchy sequences of bounded Hilbert operators converge. Complex approximation follows by approximating real and imaginary parts. Polynomials are uniformly dense in $C ([a, b], R)$ for every closed interval If (Y) is Banach then (\mathcal B(X,Y)) is Banach

[F4]

A positive normalized functional on real continuous functions on a compact metric space has a Borel probability representation under Countable Choice, with no Dependent Choice assumption. A compact-metric probability representation using countable choice

[F5]

Dominated convergence and scalar Fubini apply with their stated measure and integrability hypotheses. Dominated convergence Fubini's theorem for L^1 functions on a sigma-finite product

[F6]

A lambda-system containing a pi-system contains the sigma-algebra it generates. Dynkin's pi-lambda theorem

[F7]

The real exponential is smooth with derivative itself, and its inverse logarithm has derivative $1 / s$ . The $C^{1}$ change-of-variables theorem applies to nonnegative Lebesgue integrands under Countable Choice; applying it to the four signed real/imaginary parts gives the absolutely integrable complex version. The continuous-curve fundamental theorem applies on each compact interval, also for Hilbert-valued curves. A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions The exponential function is smooth and $(\exp)^{'} = \exp$ The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t Fundamental theorem of calculus for Banach-valued continuous curves

Proof

Given: [A1], [F1]–[F7] and the Statement data.

1.1A1F1givenalgebra

Positivity gives $∥ (I + K) x ∥ \geq ∥ x ∥$ on $D (K)$ , by pairing $(I + K) x$ with $x$ . Its range is closed: if $(I + K) x_{n}$ converges, the bound makes $x_{n}$ Cauchy, and $K x_{n} = (I + K) x_{n} - x_{n}$ converges, so closedness in [F1] identifies the limit in $D (K)$ . The range has zero orthogonal complement because $\ker (I + K)^{*} = \ker (I + K) = 0$ ; [F1] makes the closed range all of $H$ . Thus $B = (I + K)^{- 1}$ exists, $∥ B ∥ \leq 1$ , $\ker B = 0$ and $ran B = D (K)$ is dense. For $x = (I + K) u$ and $y = (I + K) v$ , symmetry gives $⟨ B x, y ⟩ = ⟨ x, B y ⟩$ , while $⟨ B x, x ⟩ = ∥ u ∥^{2} + ⟨ K u, u ⟩ \geq 0$ . Hence $B$ is a positive self-adjoint contraction and $K B = I - B$ .

1.2F5F6givenconstruct

We shall use uniqueness of finite complex measures on $[0, 1]$ from their continuous integrals. If their difference $ν$ integrates every continuous function to zero, then for any relatively open $O \subseteq [0, 1]$ the continuous functions $f_{n} (b) = \min (1, n dist (b, [0, 1] ∖ O))$ increase to $1_{O}$ ; take $f_{n} = 1$ when $O = [0, 1]$ . Dominated convergence against its finite total variation gives $ν (O) = 0$ . Since also $ν ([0, 1]) = 0$ , the sets where $ν$ vanishes form a lambda-system. The relatively open sets form a generating pi-system, so [F6] proves $ν = 0$ on all Borel sets. This also applies to any finite linear combination of positive measures.

1.3F3F5F7givenconstruct

Scalar Laplace uniqueness will identify the evolution. If a bounded continuous $h : [0, \infty) \to C$ satisfies $\int_{0}^{\infty} e^{- z t} h (t) d t = 0$ for every real $z > 0$ , apply [F7] to the $C^{1}$ diffeomorphism $t \mapsto s = e^{- t}$ from $(0, \infty)$ to $(0, 1)$ , with inverse $- \log s$ and absolute Jacobian $e^{- t} = s$ , and take $z = n + 1$ . The endpoints have Lebesgue measure zero; boundedness of $h$ makes the complex integrands absolutely integrable. Then $\int_{0}^{1} s^{n} h (- \log s) d s = 0$ for all integers $n \geq 0$ , with any value assigned at $s = 0$ . Polynomial approximation [F3] implies the same equality against every continuous test on $[0, 1]$ . If $h (t_{0}) \neq 0$ , multiplication by a constant complex phase and continuity give a time interval on which its real part is strictly positive. A nonnegative continuous piecewise-linear hat supported inside the corresponding interval of $s$ has strictly positive integral against that real part, contradicting the preceding test identity. For $t_{0} = 0$ use a one-sided time neighborhood and a smaller interior interval. Therefore $h = 0$ .

2.1F2F3step 1.1constructalgebra

Fix a dense sequence in $H$ and let $P_{n}$ be the projection onto the span of its first $n$ terms. Finite Gram–Schmidt, skipping zero remainders and using positive real normalizations, constructs these projections deterministically, and density gives $P_{n} x \to x$ for every $x$ . The compression $B_{n} = P_{n} B P_{n}$ is a positive contraction on that finite-dimensional span and zero on its complement. For every complex polynomial $p$ , diagonalize the compression using [F2], with eigenvalues $b_{j} \in [0, 1]$ . For $x = \sum_{j} c_{j} e_{j} + x_{⊥}$ its squared image norm is $\sum_{j} ∣ p (b_{j}) ∣^{2} ∣ c_{j} ∣^{2} + ∣ p (0) ∣^{2} ∥ x_{⊥} ∥^{2} \leq (\sup_{0 \leq b \leq 1} ∣ p (b) ∣)^{2} ∥ x ∥^{2}$ . Thus $∥ p (B_{n}) ∥ \leq \sup_{0 \leq b \leq 1} ∣ p (b) ∣$ , including the zero complement. Boundedness and strong convergence $B_{n} \to B$ imply $p (B_{n}) x \to p (B) x$ by induction on powers. Hence the same norm bound holds for $p (B)$ . By [F3], polynomial approximations define $f (B)$ for every $f \in C ([0, 1]; C)$ , independently of the approximation, with $∥ f (B) ∥ \leq ∥ f ∥_{\infty}$ . Norm limits preserve sums, products, the identity and conjugate adjoints. If $f \geq 0$ , approximating its continuous square root gives $f (B) = \sqrt{f} (B)^{*} \sqrt{f} (B) \geq 0$ . This constructs the required continuous calculus without a spectral theorem.

2.2F1F7step 1.1givenconstructalgebra

For $z > 0$ the strong improper integral $R_{z} x = \int_{0}^{\infty} e^{- z t} U_{t} x d t$ exists. The identity $∥ U_{t} x - U_{s} x ∥ = ∥ (U_{t - s} - I) x ∥$ and strong continuity at zero give uniform norm-continuity of the orbit, and the scalar exponential factor is uniformly continuous on each finite interval by its bounded derivative. Hence on a finite interval the integrand is uniformly norm-continuous, so uniform-mesh Riemann sums are Cauchy in $H$ ; their tails have norm at most $e^{- z T} ∥ x ∥ / z$ . Thus $R_{z}$ is bounded by $1 / z$ . The group law makes $U_{t} D (K) = D (K)$ and $K U_{t} x = U_{t} K x$ by difference quotients. For $x \in D (K)$ the same Riemann sums converge also after applying $K$ , and closedness gives differentiation/integration in its graph. The continuous-curve fundamental theorem [F7], followed by the exponentially small endpoint limit, applied to $(e^{- z t} U_{t} x)^{'} = e^{- z t} (i K - z) U_{t} x$ yields $(z - i K) R_{z} x = x$ . The estimate $∥ (z - i K) x ∥^{2} = z^{2} ∥ x ∥^{2} + ∥ K x ∥^{2}$ , closedness and the zero kernel of the adjoint $z + i K$ show that $z - i K$ is onto with inverse norm at most $1 / z$ , just as in step 1.1. Density extends $R_{z} = (z - i K)^{- 1}$ to every $x$ .

3.1F4step 1.2step 2.1constructalgebra

For $x \neq 0$ , the real functional $f \mapsto ⟨ f (B) x, x ⟩ / ∥ x ∥^{2}$ is positive and normalized. Apply [F4] on $[0, 1]$ and multiply its probability measure by $∥ x ∥^{2}$ , obtaining a positive measure $ν_{x}$ ; set $ν_{0} = 0$ . Its continuous integrals are $⟨ f (B) x, x ⟩$ , first for real $f$ , then for complex $f$ by linearity. Define $ν_{x, y} = \frac{1}{4} \sum_{j = 0}^{3} i^{j} ν_{x + i^{j} y} .$ Polarization gives $\int f d ν_{x, y} = ⟨ f (B) x, y ⟩$ . Uniqueness from step 1.2 makes these measures linear in $x$ , conjugate-linear in $y$ , with $ν_{y, x} = \overline{ν_{x, y}}$ and diagonal $ν_{x, x} = ν_{x}$ . Their total masses are $⟨ x, y ⟩$ . Because the representing measure is unique, this defines the entire indexed family without an uncountable choice of representatives.

4.1step 3.1algebra

For every Borel $D$ , the form $(x, y) \mapsto ν_{x, y} (D)$ is positive sesquilinear. Expanding its nonnegative value on $x + c y$ , and minimizing over $c \in C$ (with the zero-diagonal case treated by arbitrary large multiples), gives $∣ ν_{x, y} (D) ∣^{2} \leq ν_{x} (D) ν_{y} (D)$ . On any finite Borel partition $(D_{j})$ , finite scalar Cauchy–Schwarz now gives $\sum_{j} ∣ ν_{x, y} (D_{j}) ∣ \leq \sqrt{\sum_{j} ν_{x} (D_{j})} \sqrt{\sum_{j} ν_{y} (D_{j})} \leq ∥ x ∥ ∥ y ∥$ . Taking the partition supremum yields $∣ ν_{x, y} ∣ ([0, 1]) \leq ∥ x ∥ ∥ y ∥$ . In particular all bounded measurable scalar functions are integrable against these measures.

4.2step 1.2step 2.1step 3.1algebra

For continuous $f, h$ , multiplicativity of the calculus gives $\int h d ν_{f (B) x, y} = ⟨ h (B) f (B) x, y ⟩ = \int h f d ν_{x, y} .$ Step 1.2 implies $d ν_{f (B) x, y} = f d ν_{x, y}$ as measures. Conjugate symmetry yields $d ν_{f (B) x} = ∣ f ∣^{2} d ν_{x}$ ; in particular $∥ f (B) x ∥^{2} = \int ∣ f ∣^{2} d ν_{x} .$ These identities use only continuous factors and scalar measures, not a postulated Borel operator calculus.

5.1step 1.1step 4.1step 4.2algebra

Since $ν_{B z} ({0}) = 0$ by step 4.2, step 4.1 makes its mixed measure with any vector zero on ${0}$ . Hence $ν_{x} ({0}) = ν_{x - B z} ({0}) \leq ∥ x - B z ∥^{2}$ . The dense range from step 1.1 allows this upper bound to be arbitrarily small, proving $ν_{x} ({0}) = 0$ and then $ν_{x, y} ({0}) = 0$ for all $x, y$ . Thus no zero-endpoint mass is hidden by the reciprocal change of variable below.

5.2step 4.1step 4.2algebra

If a real $b_{0}$ is in the resolvent of $B$ , let $L = (B - b_{0})^{- 1}$ . For every positive integer $n$ , step 4.2 with $x = L^{n} z$ gives $d ν_{z} = ∣ b - b_{0} ∣^{2 n} d ν_{L^{n} z}$ . Thus on $∣ b - b_{0} ∣ < ε$ , $ν_{z} ({∣ b - b_{0} ∣ < ε}) \leq (ε ∥ L ∥)^{2 n} ∥ z ∥^{2} .$ Choose $ε ∥ L ∥ < 1$ and let $n \to \infty$ . The neighborhood is null for every diagonal measure and therefore for every mixed measure by step 4.1. The real resolvent set is open by the inverse Neumann series. It is covered by the rational intervals contained in these null neighborhoods; this is a subfamily of a fixed countable base. Countable subadditivity proves that all $ν_{x, y}$ are carried by $σ (B) \cap [0, 1]$ .

5.3F1F5step 1.1step 4.1step 4.2constructalgebra

The identity in step 4.2 makes $(B^{n} x)$ Cauchy: for $n, m \geq N$ its squared difference is at most $4 \int_{[0, 1)} b^{2 N} d ν_{x}$ , which tends to zero by [F5]. Let its limit be $P x$ . Again $∥ (B - I) B^{n} x ∥^{2} = \int (1 - b)^{2} b^{2 n} d ν_{x} \to 0$ , so $P x \in \ker (I - B)$ . For $y$ in this kernel, self-adjointness gives $⟨ B^{n} x, y ⟩ = ⟨ x, y ⟩$ , and therefore $x - P x$ is orthogonal to the kernel. Thus $P$ is its orthogonal projection by [F1]. This kernel equals $\ker K$ , since $K B = I - B$ . Dominated convergence in $⟨ B^{n} x, y ⟩ = \int b^{n} d ν_{x, y}$ gives $ν_{x, y} ({1}) = ⟨ P_{0} x, y ⟩$ .

6.1F5step 1.3step 2.1step 2.2step 3.1step 4.1step 5.1algebra

On $(0, 1]$ put $λ (b) = b^{- 1} - 1$ and define $g_{x, y} (t) = \int e^{i t λ (b)} d ν_{x, y} (b)$ , omitting the null endpoint by step 5.1. It is bounded and continuous by dominated convergence and step 4.1. For $z > 0$ , scalar Fubini, applied to the finite positive measures in the polarization formula, gives its Laplace transform as $\int (z - i λ (b))^{- 1} d ν_{x, y} (b)$ . The reciprocal is the continuous function $r_{z} (b) = \frac{b}{z b - i (1 - b)}, r_{z} (0) = 0.$ By step 2.1, $r_{z} (B) = B (z B - i (I - B))^{- 1}$ , and $K B = I - B$ shows $(z - i K) r_{z} (B) = I$ on $H$ . Step 2.2 identifies this with $R_{z}$ . Thus $g_{x, y}$ and $⟨ U_{t} x, y ⟩$ have equal Laplace transforms. Step 1.3 gives equality for $t \geq 0$ ; conjugate symmetry and $U_{- t} = U_{t}^{*}$ give it for negative $t$ as well.

6.2step 1.1step 4.1step 5.1step 5.2step 5.3constructalgebra

Push $ν_{x, y} ∣_{(0, 1]}$ forward by $λ$ to obtain $μ_{x, y}$ . Positivity, total mass, sesquilinearity and the variation bound pass to the pushforward. If $E \geq 0$ is in the resolvent of $K$ and $b_{0} = (1 + E)^{- 1}$ , the bounded inverse identity $(B - b_{0})^{- 1} = - (1 + E) I - (1 + E)^{2} (K - E)^{- 1}$ follows by multiplying on $D (K) = ran B$ . Therefore $b_{0}$ is in the resolvent of $B$ . Step 5.2 and the homeomorphism $(0, 1] \to [0, \infty)$ show that the pushed measures are carried by $σ (K)$ . Their atom at $E = 0$ is the atom at $b = 1$ , computed in step 5.3. All these statements include vectors in the ground subspace and the possibility that the excited subspace is zero.

7.1A1step 6.1step 6.2algebra∎

Step 6.1 proves the Fourier pairing formula, and step 6.2 proves the remaining measure properties. For $Q = I - P_{0}$ , its measures have zero atom at zero by that formula for the atom and are carried by $σ (K)$ ; the stated gap therefore places them in $[g, \infty)$ . Every construction used only [A1] and the displayed suppliers; no general spectral theorem or existence of a PVM was invoked.

Remarks

The source's unbounded spectral theorem supplies context for the standard spectral interface. The proof above reconstructs only the scalar/cross measures needed here, on a separable Hilbert space under Countable Choice, rather than importing the source theorem or the canonical full-Choice spectral construction.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Gaussian frequency separation of connected ground correlations

Statement

Assume Countable Choice. Let $A$ be a unital complex $C^{*}$ -algebra with a strongly continuous automorphism group $α_{t}$ . Let $π : A \to B (H)$ be a contractive unital $*$ -representation on a separable Hilbert space, with linear-first inner product, and let $Ω$ be a unit vector. Suppose a strongly continuous unitary group $U_{t}$ implements $π (α_{t} (A)) = U_{t} π (A) U_{t}^{*}$ , fixes $Ω$ , and has positive self-adjoint generator $i K$ satisfying

$\ker K = C Ω, σ (K) \subseteq {0} \cup [g, \infty), g > 0.$

Write $ω (A) = ⟨ π (A) Ω, Ω ⟩$ . Let $A, B \in A$ commute, put $f (t) = ω ([A, α_{t} (B)])$ , and suppose $∣ f (t) ∣ \leq L ∣ t ∣$ for $∣ t ∣ \leq η$ , for some finite $L$ and $η > 0$ . For every $q > 0$ , the scalar integral

$I_{q} (A, B) : = \int_{R} \frac{e^{- q t^{2}}}{2 π i t} f (t) d t$

is absolutely convergent, with any value assigned to its integrand at $t = 0$ , and satisfies

$∣ ω (A B) - ω (A) ω (B) - I_{q} (A, B) ∣ \leq ∥ A ∥ ∥ B ∥ e^{- g^{2} / (4 q)} .$

The gap is that of this infinite- or finite-dimensional representation; no finite-volume approximating gaps are assumed.

Statement references

Scalar spectral measures of a separable positive generator under countable choice

Facts & Assumptions

Given: The covariant positive-energy representation, its simple ground vector and frequency gap, the commuting observables and the stated $O (t)$ bound.

[A1]

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

[F1]

The preceding scalar spectral-measure lemma gives the Fourier pairings, support above $g$ for vectors orthogonal to the ground vector, and mixed-measure total variation bounded by the product of the vector norms. Scalar spectral measures of a separable positive generator under countable choice

[F2]

The Gaussian Fourier transform, after the change from the $2 π$ -normalized convention to angular frequency, is $\int_{R} e^{- q t^{2}} e^{i t E} d t = \sqrt{π / q} e^{- E^{2} / (4 q)}$ . Fourier transform of a Gaussian

[F3]

Scalar Fubini and dominated convergence apply to absolutely integrable product integrands and to dominated sequences, respectively. Fubini's theorem for L^1 functions on a sigma-finite product Dominated convergence

[F4]

Scalar changes of variables below are $C^{1}$ affine diffeomorphisms (the nonnegative formula extends to absolutely integrable complex functions by four signed parts). The real and complex exponential derivatives and the continuous-curve fundamental theorem evaluate the scalar exponential primitive on finite intervals, followed by its decaying endpoint limit. A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions The exponential function is smooth and $(\exp)^{'} = \exp$ The complex exponential is entire and its complex derivative is itself Fundamental theorem of calculus for Banach-valued continuous curves

Proof

Given: [A1], [F1]–[F4] and the Statement hypotheses; put $Q = I - ∣ Ω ⟩ ⟨ Ω ∣$ and abbreviate $π (A), π (B)$ by $a, b$ .

1.1A1F1givenalgebra

The positive and negative frequency measures $μ_{+} = μ_{Q b Ω, Q a^{*} Ω}, μ_{-} = μ_{Q a Ω, Q b^{*} Ω}$ are carried by $[g, \infty)$ and each has total variation at most $∥ A ∥ ∥ B ∥$ , since $π$ and $Q$ are contractions. Using $U_{t} Ω = Ω$ and the linear-first pairing gives $ω (A B) - ω (A) ω (B) = μ_{+} ([g, \infty)), f (t) = \int e^{i t E} d μ_{+} (E) - \int e^{- i t E} d μ_{-} (E) .$ Indeed $ω (A α_{t} (B)) = ⟨ U_{t} b Ω, a^{*} Ω ⟩$ and $ω (α_{t} (B) A) = ⟨ U_{- t} a Ω, b^{*} Ω ⟩$ . Their ground-vector terms are both $ω (A) ω (B)$ and cancel. Commutativity also gives $f (0) = 0$ .

1.2F2F3F4givenconstructalgebra

For $ε > 0$ , the exponential primitive and its limit at infinity in [F4] give $1 / (ε + i t) = \int_{0}^{\infty} e^{- ε w} e^{- i t w} d w$ . The Gaussian factor makes the resulting double integral absolutely integrable, with majorant $e^{- q t^{2}} e^{- ε w}$ of finite integral. Hence [F2] and [F3] give $h_{q, ε} (E) : = \int_{R} \frac{e^{- q t^{2}} e^{i t E}}{2 π (ε + i t)} d t = \frac{1}{2 \sqrt{π q}} \int_{0}^{\infty} e^{- ε w} e^{- (E - w)^{2} / (4 q)} d w .$ It lies between zero and one and converges as $ε ↓ 0$ to $h_{q} (E)$ given by the same right side without $e^{- ε w}$ . For $E \geq g$ , substituting $u = (E - w) / (2 \sqrt{q})$ in its complementary Gaussian tail gives $0 \leq 1 - h_{q} (E) = h_{q} (- E) \leq \frac{1}{2} e^{- E^{2} / (4 q)} \leq \frac{1}{2} e^{- g^{2} / (4 q)} .$ For completeness, the tail estimate is $\int_{z}^{\infty} e^{- u^{2}} d u \leq e^{- z^{2}} \int_{0}^{\infty} e^{- v^{2}} d v$ for $z \geq 0$ , obtained by $u = z + v$ and $(z + v)^{2} \geq z^{2} + v^{2}$ ; the Gaussian integral in [F2] normalizes the final half.

1.3F3givenalgebra

The unregularized time integrand is integrable near zero because $∣ f (t) ∣ / ∣ t ∣ \leq L$ there. Away from zero, $∣ f (t) ∣ \leq 2 ∥ A ∥ ∥ B ∥$ , so the Gaussian divided by $∣ t ∣$ is an integrable majorant on each tail. For $t \neq 0$ one has $∣ ε + i t ∣ \geq ∣ t ∣$ . Dominated convergence in time thus proves $I_{q, ε} \to I_{q} (A, B)$ , and also establishes the asserted absolute convergence without a principal-value convention.

2.1F1F3step 1.1step 1.2algebra

Insert the two measures of step 1.1 into the regularized integral $I_{q, ε} = \int e^{- q t^{2}} f (t) / (2 π (ε + i t)) d t$ . Scalar Fubini is legitimate at fixed $ε > 0$ : their finite variations and the integrable Gaussian divided by $ε$ give an absolute majorant. Equivalently apply it to the four positive measures in each polarization formula of [F1]. Therefore $I_{q, ε} = \int h_{q, ε} (E) d μ_{+} (E) - \int h_{q, ε} (- E) d μ_{-} (E) .$ The bound $0 \leq h_{q, ε} \leq 1$ and finite variations permit dominated convergence on these measures. Its limit is $\int h_{q} (E) d μ_{+} - \int h_{q} (- E) d μ_{-}$ .

3.1step 1.1step 1.2step 2.1step 1.3algebra∎

By steps 1.1, 2.1 and 1.3, the difference from the connected correlation is the sum of an integral of $1 - h_{q} (E)$ against $μ_{+}$ and an integral of $h_{q} (- E)$ against $μ_{-}$ . Both measures are carried by $E \geq g$ . Step 1.2 bounds each multiplier by $\frac{1}{2} e^{- g^{2} / (4 q)}$ , and step 1.1 bounds each variation by $∥ A ∥ ∥ B ∥$ . The two halves give exactly the stated estimate, with no factor from replacing $A, B$ by centered operators.

Remarks

The source's Gaussian separation argument motivates this local real-time estimate. Here the spectral measures are constructed by the preceding Countable-Choice lemma, and the regularized integral and both limiting operations are proved explicitly. No external clustering theorem is used as a premise.

TheoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Exponential connected clustering from a unique infinite-volume ground-state gap

Statement

Assume Countable Choice and the finite-matrix $C^{*}$ -identity used by the quasi-local completion. Let $A$ be the quasi-local algebra of uniformly finite-dimensional spins on $Z^{d}$ , $d \geq 1$ , with max-coordinate distance $d_{\infty}$ . Let $α_{t}$ be a strongly continuous automorphism group satisfying the exponential Lieb–Robinson estimate

$∥ [A, α_{t} (B)] ∥ \leq \frac{2 ∥ A ∥ ∥ B ∥}{C} (e^{v ∣ t ∣} - 1) \sum_{x \in X, y \in Y} F_{a} (d_{\infty} (x, y)), F_{a} (r) = e^{- a r} (1 + r)^{- d - 1},$

for disjoint finite nonempty $X, Y$ , $A \in A_{X}$ , $B \in A_{Y}$ , all real $t$ , and fixed $a, C > 0$ , $v \geq 0$ . In particular this holds for the thermodynamic dynamics of a bounded finite-range interaction $Φ$ , with $C = C_{F_{a}}$ and $v = 2 C ∥ Φ ∥_{F_{a}} / ℏ$ from the weighted commutator theorem and any $ℏ > 0$ .

Here a positive-energy ground state means an algebraic state $η$ admitting a contractive unital $*$ -representation $π$ on a complex Hilbert space, a unit vector $Ω$ , and a strongly continuous unitary group $U_{t}$ fixing $Ω$ , such that $η (A) = ⟨ π (A) Ω, Ω ⟩$ , $π (α_{t} (A)) = U_{t} π (A) U_{t}^{*}$ , and the generator is $i K$ for a positive self-adjoint densely defined $K$ . Inner products are linear in their first argument. Such a representation can be restricted to the cyclic subspace generated by $Ω$ ; no spectral resolution is included in the definition.

Suppose $ω$ is the unique positive-energy ground state and its cyclic ground representation has an infinite-volume energy gap $γ > 0$ : writing $H = ℏ K$ , require

$σ (K) \subseteq {0} \cup [g, \infty), g = γ / ℏ > 0.$

Then for every pair of disjoint finite nonempty supports, with $r = {dist}_{\infty} (X, Y) \geq 1$ , one has

$∣ ω (A B) - ω (A) ω (B) ∣ \leq ∥ A ∥ ∥ B ∥ (\frac{2 ∣ X ∣ ∣ Y ∣}{π C} + 1 + \frac{2 v + g}{π a g r}) e^{- μ r}, μ = \frac{a g}{2 v + g} > 0.$

This is exponential connected clustering for the infinite-volume state. The hypotheses do not require uniformly gapped finite-volume approximations, translation invariance, or a particular choice of finite-volume boundary ground states. The same estimate holds for any positive-energy state representation with $\ker K = C Ω$ and the stated gap, even without uniqueness among all ground states. Uniqueness is a sufficient hypothesis for that kernel condition, proved below. For other weighted interactions the assertion applies whenever their strongly continuous dynamics and the displayed Lieb–Robinson estimate are supplied; no missing thermodynamic-existence theorem for a general interaction is assumed to have been proved here.

For physical bookkeeping $t$ is in seconds, $H, γ$ are in joules, $ℏ$ is in joule-seconds and $K, g, v$ have inverse-second units. Lattice distances, $a, C, μ$ and the exponential argument are dimensionless. The theorem does not prove a gap or uniqueness for an unspecified physical model.

Statement references

Weighted interactions and gapped finite-volume paths

Weighted interaction commutator bound

Strongly continuous thermodynamic spin automorphism group

Gaussian frequency separation of connected ground correlations

Facts & Assumptions

Given: The quasi-local spin model, positive-energy ground-state convention, dynamics, exponential commutator estimate and infinite-volume gap in the Statement.

[A1]

Countable Choice and the exact finite-matrix $C^{*}$ -identity are the completion/dynamics hypotheses, with the identity supplied locally by the finite-matrix theorem. The Axiom of Countable Choice ( ${A C}_{ω}$ ) Finite matrix C-star algebra and density-state duality

[F1]

Local spin algebras are finite matrix factors, disjoint factors commute, and their completion is a unital $C^{*}$ -algebra. An algebraic state has consistent finite densities. Local spin matrix algebras and their factor inclusions Norm-completed quasi-local quantum spin algebra Quasi-local algebra completion and locality Infinite algebraic state and finite-region restrictions

[F2]

Bounded finite-range interactions have thermodynamic norm-limit dynamics under [A1]. The finite-volume weighted Lieb–Robinson estimate has constants uniform in volume. Uniformly bounded finite-range quantum spin interaction Strongly continuous thermodynamic spin automorphism group Weighted interactions and gapped finite-volume paths Summability and convolution control for lattice weights Weighted interaction commutator bound

[F3]

Under Countable Choice, a state with consistent local densities has a contractive cyclic Hilbert representation with the linear-first pairing. Cyclic representation of a quasi-local state

[F4]

The Gaussian ground-correlation filter gives an error at most $∥ A ∥ ∥ B ∥ e^{- g^{2} / (4 q)}$ when the ground vector is simple and the local time commutator is $O (t)$ . Its spectral-measure supplier constructs the measures under Countable Choice rather than assuming their existence. Scalar spectral measures of a separable positive generator under countable choice Gaussian frequency separation of connected ground correlations

[F5]

Closed Hilbert subspaces have orthogonal projections under Countable Choice, and the self-adjoint adjoint/range identities hold with their actual operator domains. Orthogonal decomposition by a closed subspace The adjoint is well defined, closed, and reverses inclusions

[F6]

The $C^{1}$ change-of-variables theorem applies to the positive-tail substitution $u = q t^{2}$ on $(τ, \infty)$ . The real exponential has derivative itself, the mean value theorem gives $e^{v t} - 1 \leq v t e^{v t}$ for $v, t \geq 0$ , and the continuous-curve fundamental theorem evaluates its scalar primitive on compact intervals and then by a decaying endpoint limit. A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions The exponential function is smooth and $(\exp)^{'} = \exp$ Fundamental theorem of calculus for Banach-valued continuous curves The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$

Proof

Given: [A1], [F1]–[F6] and the Statement hypotheses.

1.1A1F1F3givenconstructalgebra

A cyclic representation of the spin algebra is separable. Indeed, finite tuples of the countable lattice enumerate its finite subsets, and rational real/imaginary entries form a countable dense family in each finite local matrix algebra. Countable Choice permits their countable union. Density follows, for a $D \times D$ matrix, from $∥ T ∥ \leq D \max_{i, j} ∣ T_{i j} ∣$ , obtained by finite Cauchy–Schwarz. The resulting countable family is dense in the completion, and its vectors $π (A) Ω$ are dense in the cyclic Hilbert space by contractivity. The cyclic representation supplied by [F3] has the same state pairings as any given cyclic representation: the map $[A] \mapsto π (A) Ω$ preserves $⟨ [A], [B] ⟩ = ω (B^{*} A)$ and extends by density to a unitary. Thus the chosen cyclic ground representation is a separable realization of the actual GNS state, with no extra representation hypothesis.

1.2A1F1F2givenalgebra

We verify the advertised finite-range instance of the commutator premise. If $Φ$ has range $R$ and per-site norm bound $κ$ , a pair separated by more than $R$ contributes no interaction term, while for a pair of distance at most $R$ the numerator of the weighted norm is at most $κ$ . Hence $∥ Φ ∥_{F_{a}} \leq κ / F_{a} (R) < \infty$ . The uniform finite-volume bound [F2] passes to the thermodynamic limit for each local $A, B$ and fixed $t$ , by norm continuity of commutators. The group is isometric, so $∥ [A, α_{t} (B)] ∥ = ∥ [α_{- t} (A), B] ∥$ , matching exactly the supplier's time/support convention. This establishes the stated constants without replacing a general infinite-volume gap by a finite-volume gap.

1.3F4F5givenconstructalgebra

The cyclic restriction in the ground-state definition preserves positive energy. Let $L = \overline{π (A) Ω}$ in a ground representation. Covariance and $U_{t} Ω = Ω$ imply that $U_{t}$ and $U_{- t}$ preserve $L$ ; its orthogonal projection commutes with $U_{t}$ . For $x \in D (K)$ , difference quotients show that this projection preserves $D (K)$ and commutes with $K$ . The generator domain inside $L$ is dense: the vectors $h^{- 1} \int_{0}^{h} U_{t} x d t$ lie in it and tend to $x$ for $x \in L$ , as their difference quotients give $i K$ equal to $(U_{h} x - x) / h$ . The restriction is self-adjoint because $K \pm i$ are onto, their bounded inverses preserve $L$ by the commuting projection, and the restricted symmetric operator has both ranges onto; directly, if $y$ is in its adjoint domain, solve $(K + i) x = (K_{L}^{*} + i) y$ in $L$ , and the zero kernel of $K_{L}^{*} + i$ forces $x = y$ . It is positive by restriction. More generally, for every $z$ in the resolvent of $K$ , commutation of the projection with $K$ on its domain implies commutation with $(K - z)^{- 1}$ : apply $(K - z)$ and its injectivity to the two candidate images. This inverse therefore restricts to the inverse on $L$ , so $σ (K_{L}) \subseteq σ (K)$ and the spectral gap is inherited, as is a simple ground kernel. The resolvent surjectivity follows from the self-adjoint closed-range estimate exactly as in the spectral-measure helper. Thus the definition using a possibly noncyclic representation agrees with its cyclic restriction, without a Stone or spectral-existence theorem.

2.1F3F4step 1.1step 1.3givenalgebra

Let $P_{0}$ be the ground-kernel projection in the separable representation. A unit vector $ψ \in \ker K$ satisfies $U_{t} ψ = ψ$ , since its orbit derivative is $i U_{t} K ψ = 0$ . Its vector state $η_{ψ} (A) = ⟨ π (A) ψ, ψ ⟩$ is a positive-energy ground state using the same representation; step 1.3 also supplies its cyclic version if desired. Uniqueness gives $η_{ψ} = ω$ . Scaling and polarization inside $\ker K$ therefore yield $P_{0} π (A) P_{0} = ω (A) P_{0}$ for every $A$ . If $ψ \in \ker K$ is orthogonal to $Ω$ , this identity implies $⟨ π (A) Ω, ψ ⟩ = 0$ for every $A$ . Cyclicity forces $ψ = 0$ . Since the fixed vector $Ω$ has zero orbit derivative, $Ω \in D (K)$ and $K Ω = 0$ , so we have $\ker K = C Ω$ . The alternative hypothesis stated in the theorem supplies this directly and needs no uniqueness argument.

2.2F1F2F4step 1.2givenalgebra

Put $N = ∥ A ∥ ∥ B ∥$ and $f (t) = ω ([A, α_{t} (B)])$ . The displayed Lieb–Robinson premise, $F_{a} (s) \leq e^{- a s}$ and $d_{\infty} (x, y) \geq r$ give $∣ f (t) ∣ \leq \frac{2 N}{C} ∣ X ∣ ∣ Y ∣ e^{- a r} (e^{v ∣ t ∣} - 1), ∣ f (t) ∣ \leq 2 N .$ The first bound is $O (t)$ at zero, including $v = 0$ . The local observables commute by [F1], so all time-integral hypotheses in [F4] hold. The same estimates apply to the finite-range instance justified in step 1.2.

3.1F4F6step 2.1step 2.2algebra

Apply [F4] using step 2.1. For any $q, τ > 0$ , split its absolutely convergent integral at $∣ t ∣ = τ$ . Step 2.2 gives the short-time bound $\frac{2 N ∣ X ∣ ∣ Y ∣}{π C} e^{- a r} \int_{0}^{τ} \frac{e^{v t} - 1}{t} d t \leq \frac{2 N ∣ X ∣ ∣ Y ∣}{π C} e^{- a r + v τ},$ by [F6], because $(e^{v t} - 1) / t \leq v e^{v t}$ and $e^{v τ} - 1 \leq e^{v τ}$ , with zero left side when $v = 0$ . On the long-time part the trivial bound gives $\frac{2 N}{π} \int_{τ}^{\infty} \frac{e^{- q t^{2}}}{t} d t = \frac{N}{π} \int_{q τ^{2}}^{\infty} \frac{e^{- u}}{u} d u \leq \frac{N}{π q τ^{2}} e^{- q τ^{2}} .$ Adding the filter error $N e^{- g^{2} / (4 q)}$ bounds the connected correlation. These estimates use only the supplied dynamics and its infinite-volume ground representation.

4.1step 3.1algebra∎

Choose $τ = a r / (v + g / 2)$ and $q = g / (2 τ)$ . They are positive for $r \geq 1$ and $g, a > 0$ , even when $v = 0$ . Then $a r - v τ = q τ^{2} = \frac{g^{2}}{4 q} = \frac{a g}{2 v + g} r = μ r, \frac{1}{q τ^{2}} = \frac{2 v + g}{a g r} .$ Substitution into step 3.1 proves the displayed bound. Zero observables give $N = 0$ and require no division by $N$ . The independent finite-volume band multiplicity and transport statements on this page are unchanged; this proof adds no claim that a degenerate gapped state has ordinary connected clustering. The clock conversion $g = γ / ℏ$ gives exactly the units in the Statement.

Remarks

The source distinguishes the direct infinite-volume spectral-gap route from limits of finite-volume uniform gaps. This theorem proves the former under its precise ground-state and locality hypotheses. Its estimate uses the actual weighted commutator constant of this page, not the source's different interaction norm or a silently copied sharp prefactor. The full-Choice canonical unbounded spectral theorem is not a premise.

LemmaStatement: Literature-sourcedProof: AI-adaptedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Filtered derivative of an isolated finite-volume band

Statement

Assume Countable Choice. Let $H_{Λ} (s)$ be the finite-volume self-adjoint matrix of a $C^{1}$ interaction path on the dimensionless parameter $s \in [0, 1]$ , and let the selected spectral band be a locally fixed spectral cluster, separated from its complement by $γ > 0$ uniformly in $s$ , in the sense of the interaction definition. Choose a positive reference energy $E_{0}$ and measure physical energies in units of $E_{0}$ ; $H_{Λ}$ , $H_{Λ}^{'}$ , and $γ$ in this statement are dimensionless, and the Fourier evolution uses $ℏ = 1$ . Let $P_{Λ} (s)$ be the orthogonal spectral projection onto that band. Then $P_{Λ}$ is $C^{1}$ . There is a real, odd, integrable function $W_{γ} \in L^{1} (R)$ , with Fourier convention

${\hat{W}}_{γ} (ω) : = \int_{R} W_{γ} (t) e^{i t ω} d t,$

such that ${\hat{W}}_{γ} (ω) = i / ω$ whenever $∣ ω ∣ \geq γ$ , and for every integer $N \geq 0$ there is $c_{N} < \infty$ with

$\int_{∣ t ∣ > T} ∣ W_{γ} (t) ∣ d t \leq c_{N} (1 + T)^{- N} (T \geq 0) .$

The finite matrix

$D_{Λ} (s) : = \int_{R} W_{γ} (t) e^{i t H_{Λ} (s)} H_{Λ}^{'} (s) e^{- i t H_{Λ} (s)} d t$

is self-adjoint and satisfies

$P_{Λ}^{'} (s) = i [D_{Λ} (s), P_{Λ} (s)] .$

Statement references

Weighted interactions and gapped finite-volume paths

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

Finite matrix spectral and state calculus

Fourier inversion on Schwartz space

Fourier transform acts continuously on Schwartz space

The residue theorem for a null-homologous cycle

Facts & Assumptions

Given: Countable Choice, a finite region $Λ$ , a $C^{1}$ path $H_{Λ} (s) = H_{Λ} (s)^{*}$ from the interaction definition, a selected nonempty spectral band that is a locally fixed spectral cluster in the sense of the interaction definition, and its uniform gap $γ > 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice, as required by the supplied Fourier inversion theorem.

[F1]

Weighted interactions and gapped finite-volume paths: $H_{Λ} (s)$ is a finite self-adjoint matrix and $H_{Λ}^{'} (s)$ exists in matrix norm.

[F2]

Finite matrix spectral and state calculus: each finite Hermitian matrix has an orthonormal eigenbasis with real eigenvalues and its spectral projections are basis independent.

[F3]

Fourier inversion on Schwartz space and Fourier transform acts continuously on Schwartz space: the negative-sign $2 π$ -normalized Fourier transform maps Schwartz space to itself and Fourier inversion holds there; after $ω = 2 π ξ$ and reflection, the inverse for $\int f (t) e^{i t ω} d t$ is $(2 π)^{- 1} \int \hat{f} (ω) e^{- i t ω} d ω$ .

[F4]

The residue theorem for a null-homologous cycle: entrywise contour integration of the finite-matrix resolvent encircling selected real eigenvalues returns the sum of their orthogonal eigenspace projections.

[F5]

The standard flat function is smooth and flat at zero: the function $u : R \to R$ given by $u (t) = \exp (- 1 / t)$ for $t > 0$ and $u (t) = 0$ for $t \leq 0$ is smooth on $R$ , with $u^{(m)} (0) = 0$ for every $m \geq 0$ . The one-variable chain rule and sum, scalar-multiple and product derivative rules apply on $R$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F6]

Dominated convergence: on a measure space, measurable complex functions converging pointwise almost everywhere and dominated in modulus by one nonnegative integrable function have convergent integrals. Apply this scalar result to each entry in a fixed finite orthonormal matrix basis; the existing Countable Choice assumption [A1] supplies the adopted Lebesgue-measure interfaces.

Proof

technique · direct

Given: [A1], [F1]–[F6], a locally fixed selected spectral cluster separated by $γ > 0$ , and dimensionless energy units.

1.1givenA1F3F5constructalgebra

Let $u$ be the standard flat function of [F5] and put $q (ω) : = 1 - (ω / γ)^{2}$ and $χ (ω) : = e u (q (ω))$ . Since $γ > 0$ , $q$ is a smooth real polynomial. The chain and product rules in [F5], iterated at each derivative order, make $χ$ smooth on all of $R$ : every positive-order derivative is a finite sum of products of a derivative $u^{(m)} \circ q$ , with $m \geq 1$ , and derivatives of $q$ . At $ω = \pm γ$ , one has $q (ω) = 0$ and every such $u^{(m)} (0) = 0$ , so all these derivatives vanish and match the exterior zero branch. The value itself also vanishes there because $u (0) = 0$ . Since $q > 0$ exactly when $∣ ω ∣ < γ$ , this formula is exactly $e \exp (- 1 / (1 - (ω / γ)^{2}))$ inside that interval and zero outside. Thus $χ$ is real, even, smooth, compactly supported in $[- γ, γ]$ , and $χ (0) = e \exp (- 1) = 1$ . Define $w (t) = (2 π)^{- 1} \int χ (ω) e^{- i t ω} d ω$ . By [A1], [F3], and the sign/scale conversion stated in [F3], $w$ is Schwartz, real and even, its Fourier transform $\hat{w} (ω) = \int w (t) e^{i t ω} d t$ equals $χ (ω)$ , and $\int w = 1$ . For $t > 0$ put $W_{γ} (t) = \int_{t}^{\infty} w (v) d v$ ; for $t < 0$ put $W_{γ} (t) = - \int_{- \infty}^{t} w (v) d v$ ; set $W_{γ} (0) = 0$ . Evenness and unit integral make $W_{γ}$ odd with one-sided limits $W_{γ} (0^{\pm}) = \pm \frac{1}{2}$ . For every $N$ , a Schwartz bound $∣ w (v) ∣ \leq c (1 + ∣ v ∣)^{- N - 2}$ on either tail gives $∣ W_{γ} (t) ∣ \leq c^{'} (1 + ∣ t ∣)^{- N - 1}$ ; hence $W_{γ} \in L^{1}$ and has the stated tail estimate. Integration by parts on the two half-lines, accounting for the jump of size $1$ at zero and using $W_{γ}^{'} = - w$ away from zero, gives $- \hat{w} (ω) = - 1 - i ω {\hat{W}}_{γ} (ω)$ . Thus ${\hat{W}}_{γ} (ω) = i (1 - χ (ω)) / ω$ for $ω \neq 0$ , so it equals $i / ω$ for $∣ ω ∣ \geq γ$ .

1.2F1F2F4givenalgebra

Fix $s_{0}$ . By the locally fixed spectral-cluster hypothesis and [F2], choose a finite union of positively oriented contours $Γ$ surrounding exactly the selected eigenvalues and staying a positive distance from the remaining spectrum, with the same selected eigenvalues enclosed for nearby $s$ . For $s$ near $s_{0}$ , $∥ H_{Λ} (s) - H_{Λ} (s_{0}) ∥$ is small; factoring $z I - H_{Λ} (s)$ through $z I - H_{Λ} (s_{0})$ and using the norm-convergent Neumann series keeps every $z \in Γ$ in the resolvent set. The contour projection $P_{Λ} (s) = \frac{1}{2 π i} \int_{Γ} (z I - H_{Λ} (s))^{- 1} d z$ is therefore $C^{1}$ , since the inverse matrix is $C^{1}$ in $s$ .

2.1givenA1F1F2F6step 1.1step 1.2algebra∎

In an orthonormal eigenbasis of $H_{Λ} (s)$ , write its eigenvalues as $E_{m}$ and the band indicator as $p_{m} \in {0, 1}$ . Differentiating $P^{2} = P$ gives $P_{m n}^{'} = 0$ when $p_{m} = p_{n}$ . Differentiating $H P = P H$ gives $P_{m n}^{'} = (p_{m} - p_{n}) H_{m n}^{'} / (E_{m} - E_{n})$ when $p_{m} \neq p_{n}$ ; the denominator has magnitude at least $γ$ . The integral defining $D_{Λ}$ converges in norm because $W_{γ} \in L^{1}$ ; it is self-adjoint because $W_{γ}$ is real and each conjugated $H_{Λ}^{'}$ is self-adjoint. For each fixed $t$ , the matrix exponential series for $e^{\pm i t H_{Λ} (s)}$ converges uniformly in $s \in [0, 1]$ : if $M = \sup_{s} ∥ H_{Λ} (s) ∥ < \infty$ , its order- $r$ term has norm at most $(∣ t ∣ M)^{r} / r!$ . The terms are continuous in $s$ , so both exponentials, and hence their product with the continuous $H_{Λ}^{'} (s)$ , are norm-continuous in $s$ . For any sequence $s_{m} \to s$ in $[0, 1]$ , each matrix entry of the difference of the filtered integrands tends pointwise to zero and has modulus at most $2 \sup_{v} ∥ H_{Λ}^{'} (v) ∥ ∣ W_{γ} (t) ∣$ , an integrable majorant by step 1.1 and compact-interval boundedness of $H_{Λ}^{'}$ . The entries are measurable in $t$ , since the exponentials are continuous in $t$ and $W_{γ}$ is measurable. Applying [F6] entrywise gives convergence of every entry of $D_{Λ} (s_{m}) - D_{Λ} (s)$ . If the matrix dimension is $m_{Λ}$ , then $∥ R ∥ \leq m_{Λ} \max_{i, j} ∣ R_{i j} ∣$ : for a unit vector $x$ , $∣ (R x)_{i} ∣ \leq \max_{i, j} ∣ R_{i j} ∣ \sum_{j} ∣ x_{j} ∣ \leq \sqrt{m_{Λ}} \max_{i, j} ∣ R_{i j} ∣$ , and summing squares over $i$ proves the bound. Thus entrywise convergence gives norm convergence, proving norm-continuity of $D_{Λ}$ without an undeclared norm-equivalence theorem. Its matrix entries are $(D_{Λ})_{m n} = H_{m n}^{'} {\hat{W}}_{γ} (E_{m} - E_{n}) = i H_{m n}^{'} / (E_{m} - E_{n})$ across the band gap. Hence $i [D_{Λ}, P_{Λ}]_{m n} = (p_{n} - p_{m}) i (D_{Λ})_{m n} = (p_{m} - p_{n}) H_{m n}^{'} / (E_{m} - E_{n}) = P_{m n}^{'}$ across the gap, and both sides vanish within either band block. This proves the identity in every matrix entry.

LemmaStatement: AI-adaptedProof: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Filtered spectral generators have summable local shells

Statement

$D_{Λ_{n}} (s) = \int_{R} W_{γ} (t) τ_{t}^{Λ_{n}, Φ_{s}} ⁣ (H_{Λ_{n}}^{'} (s)) d t,$

where energies are dimensionless ( $ℏ = 1$ ) and $W_{γ}$ and the filtered derivative are as in the finite-volume spectral-generator lemma. For every $ρ > d$ , there are self-adjoint local interactions $Ψ_{Λ_{n}} (Z, s)$ , supported in $Z \subseteq Λ_{n}$ , such that

$D_{Λ_{n}} (s) = \sum_{\emptyset \neq Z \subseteq Λ_{n}} Ψ_{Λ_{n}} (Z, s), \sup_{n} \sup_{s \in [0, 1]} ∥ Ψ_{Λ_{n}} (s) ∥_{F_{ρ}} < \infty .$

For every fixed finite nonempty $Z \subset Z^{d}$ , the coefficient $Ψ_{Λ_{n}} (Z, s)$ , set to zero when $Z \subseteq̸ Λ_{n}$ , converges in norm uniformly in $s$ to a self-adjoint $Ψ_{\infty} (Z, s)$ . The limiting interaction has finite $F_{ρ}$ -norm for every $ρ > d$ .

For every finite $Λ_{n}$ and nonempty support $Z \subseteq Λ_{n}$ , the coefficient $s \mapsto Ψ_{Λ_{n}} (Z, s)$ is norm-continuous.

Statement references

Weighted interactions and gapped finite-volume paths

Weighted interaction commutator bound

Filtered derivative of an isolated finite-volume band

Normalized-trace expectation onto a finite spin region

Contractive local expectations and their commutator bound

Local spin matrix algebras and their factor inclusions

Norm-completed quasi-local quantum spin algebra

Quasi-local algebra completion and locality

Finite matrix C-star algebra and density-state duality

Strongly continuous thermodynamic spin automorphism group

Facts & Assumptions

Given: Countable Choice, the finite-matrix $C^{*}$ -identity, a $C^{1}$ finite-range interaction path with the uniform bounds above, an increasing finite exhaustion $Λ_{n} ↑ Z^{d}$ , locally fixed spectral clusters with uniform gap $γ > 0$ , and the dimensionless $ℏ = 1$ convention for the filtered generator.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed by the norm-completion and finite-range dynamics suppliers.

[F6]

Finite matrix C-star algebra and density-state duality: each finite local matrix algebra with its ordinary operator norm satisfies the finite-matrix $C^{*}$ -identity used by the quasi-local completion and finite-range dynamics suppliers.

[F1]

Weighted interactions and gapped finite-volume paths: $H_{Λ} (s) = \sum_{Y \subseteq Λ} Φ_{s} (Y)$ , its derivative is the corresponding finite sum, and $∥ \partial Φ ∥_{F_{a}} < \infty$ controls $\sup_{s} \sum_{Y ∋ x} ∣ Y ∣ ∥ Φ_{s}^{'} (Y) ∥$ .

[F2]

Filtered derivative of an isolated finite-volume band: $D_{Λ} (s)$ is a norm-continuous, norm-convergent filtered integral whose kernel has tails bounded by every inverse power.

[F3]

Weighted interaction commutator bound: the finite-volume Heisenberg evolution of the finite-range path obeys the $F_{a}$ -weighted commutator bound with constants uniform in volume and path parameter.

[F4]

Normalized-trace expectation onto a finite spin region and Contractive local expectations and their commutator bound: $E_{X}$ is a consistent contraction onto $A_{X}$ , and its error is bounded by the sum of commutators with one-site unitaries outside $X$ .

[F5]

Strongly continuous thermodynamic spin automorphism group: under Countable Choice and the finite-matrix $C^{*}$ -identity, finite-volume dynamics of each bounded finite-range interaction converge in norm on local observables, uniformly on compact time intervals, to a dynamics on the norm-completed quasi-local algebra. The proof's boundary estimate is uniform over the stated path because its finite-range and interaction bounds are uniform in $s$ .

Proof

technique · direct

Given: [A1], [F1]–[F5], the finite-volume gaps, and the exhaustion.

1.1givenA1F2F4F5construct

For each finite nonempty $Y \subseteq Λ_{n}$ , set $B_{Y, n} (s) = \int_{R} W_{γ} (t) τ_{t}^{Λ_{n}, Φ_{s}} (Φ_{s}^{'} (Y)) d t .$ For fixed $n, Y, t$ , the finite matrix integrand is continuous in $s$ , and it is bounded by $\sup_{s} ∥ Φ_{s}^{'} (Y) ∥ ∣ W_{γ} (t) ∣$ , an integrable majorant. Dominated convergence makes $B_{Y, n} (s)$ norm-continuous. Linearity of the finite sum and [F2] give $D_{Λ_{n}} (s) = \sum_{\emptyset \neq Y \subseteq Λ_{n}} B_{Y, n} (s)$ . Define $Y_{k} = {x : d_{\infty} (x, Y) \leq k}$ , $E_{Y_{k}}^{(n)} : = E_{Y_{k} \cap Λ_{n}}$ , and shells $Δ_{Y, n}^{0} = E_{Y_{0}}^{(n)} (B_{Y, n}), Δ_{Y, n}^{k} = E_{Y_{k}}^{(n)} (B_{Y, n}) - E_{Y_{k - 1}}^{(n)} (B_{Y, n}) (k \geq 1) .$ For all sufficiently large $k$ , $Y_{k} \cap Λ_{n} = Λ_{n}$ , so the sum of these shells is $B_{Y, n}$ . Each shell is self-adjoint and supported in $Y_{k} \cap Λ_{n}$ . Grouping the finitely many shells by this support defines $Ψ_{Λ_{n}} (Z, s)$ and proves the displayed decomposition. Each expectation is a fixed linear map, so the shell and grouped coefficients are norm-continuous in $s$ .

2.1givenF2F3F4step 1.1algebra

Let $a > 0$ be the weight parameter and $v = 2 C_{F_{a}} \sup_{s} ∥ Φ_{s} ∥_{F_{a}}$ , with $V = 1 + v$ . For $z \in Λ_{n} ∖ Y_{k}$ , the supports $Y$ and ${z}$ are disjoint; [F3] and the local-expectation estimate [F4] give $∥ τ_{t}^{Λ_{n}, Φ_{s}} (Φ_{s}^{'} (Y)) - E_{Y_{k}}^{(n)} τ_{t}^{Λ_{n}, Φ_{s}} (Φ_{s}^{'} (Y)) ∥ \leq C ∣ Y ∣ ∥ Φ_{s}^{'} (Y) ∥ e^{v ∣ t ∣} e^{- a k / 2},$ where the lattice shell count and the exponential factor in $F_{a}$ give one constant $C$ independent of $Y, n, s, t$ . The same difference is at most $2 ∥ Φ_{s}^{'} (Y) ∥$ by contractivity. Split the filtered integral at $T_{k} = a k / (4 V)$ . On $∣ t ∣ \leq T_{k}$ the first bound is at most $C ∣ Y ∣ ∥ Φ_{s}^{'} (Y) ∥ e^{- a k / 4}$ ; on $∣ t ∣ > T_{k}$ , the Schwartz-tail construction in [F2] gives $\int_{∣ t ∣ > T_{k}} ∣ W_{γ} (t) ∣ d t \leq C_{N} (1 + k)^{- N}$ for each $N$ . Since $Δ_{Y, n}^{k}$ is a difference of the (k)- and $(k - 1)$ -expectations, its norm is bounded by the sum of these two expectation errors. Thus, after increasing $C_{N}$ , for every $N$ there is a constant $C_{N}$ independent of $Y, n, s, k$ such that $∥ Δ_{Y, n}^{k} (s) ∥ \leq C_{N} ∣ Y ∣ ∥ Φ_{s}^{'} (Y) ∥ (1 + k)^{- N} .$ For $k = 0$ , the same estimate with an enlarged $C_{N}$ follows from $∥ E_{Y} (B_{Y, n}) ∥ \leq ∥ W_{γ} ∥_{1} ∥ Φ_{s}^{'} (Y) ∥$ .

3.1givenF1step 2.1algebra

The finite-range path has a uniform range $R$ , so $x, y \in Y_{k}$ implies $d_{\infty} (x, y) \leq R + 2 k$ . Also [F1] at $x = y$ gives $\sup_{s} \sum_{Y ∋ x} ∣ Y ∣ ∥ Φ_{s}^{'} (Y) ∥ < \infty$ . Since $x \in Y_{k}$ forces $Y \cap B_{k} (x) \neq \emptyset$ , the sum of $∣ Y ∣ ∥ Φ_{s}^{'} (Y) ∥$ over such $Y$ is at most $C (k + 1)^{d}$ . For each fixed $s$ , summing the shell estimate from step 2.1 over all $Y, k$ with $x, y \in Y_{k}$ gives $\sum_{\begin{array}{c} Z \subseteq Λ_{n} \\ x, y \in Z \end{array}} ∥ Ψ_{Λ_{n}} (Z, s) ∥ \leq C_{N} \sum_{k \geq (d_{\infty} (x, y) - R) / 2} (1 + k)^{d - N} .$ For any prescribed $ρ > d$ , choose $N > ρ + d + 1$ ; the last tail is bounded by $C_{ρ} (1 + d_{\infty} (x, y))^{- ρ}$ . The bound is independent of $s$ and $n$ , so taking their suprema outside the support sum proves the stated uniform $F_{ρ}$ -norm bound. No sum of separate time suprema is asserted; all finite-volume sums are finite and their nonnegative shell bounds are dominated by the displayed convergent scalar series.

4.1givenA1F2F4F5F6step 3.1algebra∎

Fix a finite support $Z$ . Once $Λ_{n}$ contains the $(2 diam Z + 1)$ -neighborhood of $Z$ , any shell with support $Y_{k} \cap Λ_{n} = Z$ has $Y \subseteq Z$ and $k \leq diam Z$ ; otherwise $Y_{k} \cap Λ_{n}$ would contain a point in that neighborhood outside $Z$ . Thus only finitely many pairs $(Y, k)$ contribute to $Ψ_{Λ_{n}} (Z, s)$ , and their shell projections are the fixed $E_{Y_{k}}$ once $n$ is large. By [A1], [F6], and [F5], for each fixed $s$ , $τ_{t}^{Λ_{n}, Φ_{s}} (A)$ converges locally uniformly on compact $t$ -intervals. The same boundary estimate in the proof of [F5] is uniform in $s$ , because the path has uniform finite range and interaction bounds. For $A = Φ_{s}^{'} (Y)$ , uniformity over $s$ follows by covering the compact finite-dimensional set ${Φ_{s}^{'} (Y) : s \in [0, 1]}$ by a finite norm net and using contractivity of the dynamics. The tails $∣ t ∣ > T$ are bounded uniformly by $2 \sup_{s} ∥ Φ_{s}^{'} (Y) ∥ \int_{∣ t ∣ > T} ∣ W_{γ} (t) ∣ d t$ , which tends to zero by [F2]. Hence every contributing filtered term converges uniformly in $s$ , defining $Ψ_{\infty} (Z, s)$ . The $F_{ρ}$ -norm bound passes to the limit by taking the limit on each finite set of support terms and then the supremum.

TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Thermodynamic spectral transport along a uniformly gapped path

Statement

Assume Countable Choice and the finite-matrix $C^{*}$ -identity for the local spin algebras. Let $Φ_{s}$ , $s \in [0, 1]$ , be a $C^{1}$ finite-range interaction path on $Z^{d}$ , with uniform range, support-cardinality, $F_{a}$ -interaction norm, and $F_{a}$ -derivative norm for some $a > 0$ . Let $Λ_{n}$ be any increasing finite exhaustion of $Z^{d}$ . Choose a positive reference energy $E_{0}$ and measure the interaction, Hamiltonians, and gap below in units of $E_{0}$ , so their values are dimensionless and the filtered evolution uses $ℏ = 1$ ; the path parameter, lattice distance, and weights are dimensionless. For each $n$ , fix an integer $r_{n}$ with $1 \leq r_{n} < \dim H_{Λ_{n}}$ , independent of $s$ , and let the selected band consist of the lowest $r_{n}$ eigenvalues of $H_{Λ_{n}} (s)$ , counted with multiplicity. Suppose the gap between the $r_{n}$ -th and $(r_{n} + 1)$ -st eigenvalues is at least one $γ > 0$ , uniformly in $n, s$ . Thus each selected band is a locally fixed spectral cluster in the interaction-definition sense. This includes a finite-volume ground eigenspace when $r_{n}$ is its constant multiplicity along the path.

Write $P_{n} (s)$ for the spectral projection onto that band. A finite-volume band state is a positive density matrix $ρ$ on $A_{Λ_{n}}$ with ${Tr}_{Λ_{n}} ρ = 1$ and $P_{n} (s) ρ P_{n} (s) = ρ$ . Extend it to a state on the quasi-local algebra by $ω_{ρ}^{Λ} (A) = {Tr}_{Λ_{n}} (ρ E_{Λ_{n}} (A))$ , where $E_{Λ_{n}}$ is the normalized-trace expectation. Let $S (s)$ be the set of pointwise limits on the quasi-local algebra of subsequences of such band states with $n \to \infty$ . If the band is exactly the finite-volume ground eigenspace, $S (s)$ is the corresponding limiting ground-state set.

Let $D_{n} (s)$ be the filtered derivative of $P_{n} (s)$ , in dimensionless energy units, and set $K_{ρ} : = \sup_{n, s} ∥ Ψ_{Λ_{n}} (s) ∥_{F_{ρ}} < \infty$ , using the local interaction decomposition of the filtered-generator locality lemma. Let $U_{n} (0) = I$ solve $U_{n}^{'} (s) = i D_{n} (s) U_{n} (s)$ , and set $α_{n, s} (A) = U_{n} (s)^{*} A U_{n} (s)$ . Then $α_{n, s} (A)$ converges in norm for every local $A$ , uniformly in $s \in [0, 1]$ , to a norm-continuous automorphism $α_{s}$ of the quasi-local algebra. For disjoint finite local supports $X, Y$ ,

$∥ [α_{s} (A), B] ∥ \leq \frac{2 ∥ A ∥ ∥ B ∥}{C_{F_{ρ}}} (e^{2 C_{F_{ρ}} K_{ρ}} - 1) \sum_{x \in X, y \in Y} F_{ρ} (d_{\infty} (x, y)) .$

The limiting band-state sets are transported by this automorphism:

$S (s) = {ω \circ α_{s} : ω \in S (0)} .$

Statement references

Weighted interactions and gapped finite-volume paths

Weighted interaction commutator bound

Filtered derivative of an isolated finite-volume band

Filtered spectral generators have summable local shells

Normalized-trace expectation onto a finite spin region

Contractive local expectations and their commutator bound

Local spin matrix algebras and their factor inclusions

Local spin factor inclusions preserve the operator norm

Norm-completed quasi-local quantum spin algebra

Quasi-local algebra completion and locality

Finite matrix C-star algebra and density-state duality

Finite matrix spectral and state calculus

Infinite algebraic state and finite-region restrictions

Sequential compactness of the quasi-local spin state space

Facts & Assumptions

Given: Countable Choice, the finite-matrix $C^{*}$ -identity, a $C^{1}$ bounded finite-range interaction path, an increasing exhaustion $Λ_{n} ↑ Z^{d}$ , fixed ranks $r_{n} \geq 1$ for the selected lowest bands, and a uniform gap after each such band.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the completion hypothesis used by the quasi-local algebra and its finite-range dynamics.

[F1]

Weighted interactions and gapped finite-volume paths: $H_{Λ_{n}} (s)$ and its derivative are finite matrix sums; a locally fixed spectral cluster with a uniform separating gap has a $C^{1}$ spectral projection.

[F2]

Filtered derivative of an isolated finite-volume band: $P_{n}^{'} (s) = i [D_{n} (s), P_{n} (s)]$ , with $D_{n} (s) = D_{n} (s)^{*}$ norm-continuous in $s$ .

[F3]

Filtered spectral generators have summable local shells: $D_{n} (s) = \sum_{\emptyset \neq Z \subseteq Λ_{n}} Ψ_{Λ_{n}} (Z, s)$ , each finite-volume coefficient is norm-continuous, the $F_{ρ}$ -norms are uniformly bounded, and every fixed finite nonempty support coefficient converges uniformly in $s$ .

[F4]

Weighted interaction commutator bound: the time-dependent finite-volume flows generated by $- Ψ_{Λ_{n}}$ , and their reversed flows, satisfy the $F_{ρ}$ -weighted commutator bound with constants uniform in $n, s$ .

[F5]

Normalized-trace expectation onto a finite spin region and Contractive local expectations and their commutator bound: $E_{Λ_{n}}$ is a consistent unital positive contraction extending to the quasi-local algebra.

[F6]

Norm-completed quasi-local quantum spin algebra, Local spin factor inclusions preserve the operator norm, and Quasi-local algebra completion and locality: isometric local inclusions make the local observables norm dense in a unital quasi-local $C^{*}$ -algebra.

[F7]

Finite matrix C-star algebra and density-state duality: each finite local algebra has the stated $C^{*}$ -identity and every positive normalized functional is a trace pairing with a density matrix.

[F8]

Sequential compactness of the quasi-local spin state space: every sequence of states on the quasi-local spin algebra has a subsequence converging pointwise on every observable to a state.

[F9]

Finite matrix spectral and state calculus: the finite trace is cyclic, so $Tr (U ρ U^{*} C) = Tr (ρ U^{*} C U)$ .

[F10]

Infinite algebraic state and finite-region restrictions: a state on the quasi-local algebra is a bounded positive normalized functional; its finite-region restrictions are trace pairings with consistent density matrices.

Proof

technique · direct

Given: [A1], the finite-matrix $C^{*}$ -identity, [F1]–[F10], and the path and gap hypotheses.

1.1givenF1F2F5F6F7F9F10algebra

First justify positivity in [F5] on the completion. For $C \in A$ , take local $C_{m} \to C$ . Continuity of multiplication and contractivity give $E_{Λ_{n}} (C^{*} C) = \lim_{m} E_{Λ_{n}} (C_{m}^{*} C_{m})$ . Each matrix on the right is positive by the local twirl; its norm limit is positive because its quadratic form on every vector is the limit of nonnegative quadratic forms. Thus $A \mapsto Tr (ρ E_{Λ_{n}} (A))$ is positive for every finite density $ρ$ , and is bounded and normalized by contractivity and unitality. For each $n$ , [F2] gives a norm-continuous self-adjoint $D_{n} (s)$ . The finite-dimensional matrix ODE $U_{n}^{'} = i D_{n} U_{n}$ , $U_{n} (0) = I$ , has a solution by its uniformly convergent Picard series; differentiating $U_{n}^{*} U_{n}$ gives $U_{n}^{*} U_{n} = I$ . Differentiate $Q_{n} (s) : = U_{n} (s)^{*} P_{n} (s) U_{n} (s)$ . Since $U_{n}^{'} = i D_{n} U_{n}$ , $U_{n}^{*^{'}} = - i U_{n}^{*} D_{n}$ , and $P_{n}^{'} = i [D_{n}, P_{n}]$ , the three terms in $Q_{n}^{'}$ cancel. Hence $Q_{n} (s) = P_{n} (0)$ , or $P_{n} (s) = U_{n} (s) P_{n} (0) U_{n} (s)^{*}$ . If $ρ_{0}$ is a finite-volume band density at $s = 0$ , then $U_{n} ρ_{0} U_{n}^{*}$ is a band density at $s$ . The normalized-trace expectation satisfies $E_{Λ_{n}} (U_{n}^{*} A U_{n}) = U_{n}^{*} E_{Λ_{n}} (A) U_{n}$ , first on finite tensor products and then by continuity. Cyclicity [F9] therefore gives $ω_{ρ_{0}}^{Λ} \circ α_{n, s} = ω_{U_{n} ρ_{0} U_{n}^{*}}^{Λ}$ . The density/state correspondence [F7, F10] makes both extensions states on $A$ . Reversing $U_{n}$ proves equality of the finite-volume state sets in both directions.

1.2F3F4algebra

By [F3], the generator interactions $Ψ_{Λ_{n}}$ have a common $F_{ρ}$ -norm bound $K_{ρ}$ for every $ρ > d$ . The path $U_{n}^{'} = i D_{n} U_{n}$ is the Heisenberg flow of the interaction $- Ψ_{Λ_{n}}$ with $ℏ = 1$ ; its inverse is the reversed path with the same norm bound.

2.1givenF2F3F4step 1.2algebra

Fix a local $A \in A_{X}$ . For a finite ball $B_{R} \supset X$ , let $α_{n, s}^{(R)}$ be the flow generated by the restriction of $- Ψ_{Λ_{n}}$ to supports $Z \subseteq B_{R}$ . For two finite-volume propagators $U_{J}, U_{K}$ , set $F (u) = α_{u, 0}^{J} (α_{s, u}^{K} (A))$ . The composition rule gives $F (0) = α_{s, 0}^{K} (A)$ and $F (s) = α_{s, 0}^{J} (A)$ . The endpoint evolution equations are $\partial_{u} α_{u, 0}^{J} (B) = i α_{u, 0}^{J} ([J (u), B])$ and $\partial_{u} α_{s, u}^{K} (A) = - i [K (u), α_{s, u}^{K} (A)]$ . Product differentiation and integration therefore give $α_{s, 0}^{J} (A) - α_{s, 0}^{K} (A) = i \int_{0}^{s} α_{u, 0}^{J} ⁣ ([J (u) - K (u), α_{s, u}^{K} (A)]) d u .$ Apply this with $J = - Ψ_{Λ_{n}}$ and $K = J ∣_{Z \subseteq B_{R}}$ . In the Duhamel integrand, the removed terms are exactly those $Ψ_{Λ_{n}} (Z, u)$ with $Z \subseteq̸ B_{R}$ . For terms with $Z \cap X = \emptyset$ , the initial supports are disjoint, so [F4] gives the commutator estimate. Their total contribution is at most $L_{ρ} ∥ A ∥ K_{ρ} \sum_{x \in X} \sum_{y \in Z^{d}} F_{ρ} (d_{\infty} (x, y)) \sum_{z \notin B_{R}} F_{ρ} (d_{\infty} (y, z)),$ where $L_{ρ} = 2 C_{F_{ρ}}^{- 1} (e^{2 C_{F_{ρ}} K_{ρ}} - 1)$ is the uniform commutator coefficient from [F4]. For each removed support choose an outside site $z \in Z ∖ B_{R}$ , sum over $y \in Z$ , and use $\sum_{Z ∋ y, z} ∥ Ψ_{Λ_{n}} (Z, u) ∥ \leq K_{ρ} F_{ρ} (d_{\infty} (y, z))$ . The displayed tail tends to zero by dominated convergence: for fixed $y$ , the inner tail tends to zero and is bounded by the full finite lattice sum of $F_{ρ}$ , while the outer sum is finite by [F3]. If $Z \cap X \neq \emptyset$ , the disjoint-support hypothesis of [F4] is unavailable, so use $∥ [α_{s, u}^{K} (A), Ψ_{Λ_{n}} (Z, u)] ∥ \leq 2 ∥ A ∥ ∥ Ψ_{Λ_{n}} (Z, u) ∥$ . Summing over $x \in Z \cap X$ and $z \in Z ∖ B_{R}$ bounds this contribution by $2 ∥ A ∥ K_{ρ} \sum_{x \in X} \sum_{z \notin B_{R}} F_{ρ} (d_{\infty} (x, z)),$ which also tends to zero by summability. Thus the full-flow versus restricted-flow error tends to zero uniformly in $n, s$ . For fixed $R$ , only finitely many $Z \subseteq B_{R}$ occur. By [F3], their generator coefficients converge uniformly in $s$ ; the finite-dimensional Duhamel bound makes $α_{n, s}^{(R)} (A)$ uniformly Cauchy. Combining this local convergence with the uniform tail shows $α_{n, s} (A)$ is uniformly Cauchy as $n \to \infty$ . Applying the same argument to the inverse paths gives a uniform pointwise limit for $α_{n, s}^{- 1}$ .

3.1givenA1F3F4F6step 2.1algebra

Each $α_{n, s}$ and inverse preserves the norm and the $C^{*}$ -operations. Their uniform limits on the dense local algebra extend by isometry to $A$ ; continuity of multiplication and adjoint passes the $C^{*}$ -operations to the limits. Taking limits in the two composition identities on local observables shows that the two limits are inverses. Thus $α_{s}$ is an automorphism, norm-continuous in $s$ on each local observable. Taking limits in [F4] gives the stated $F_{ρ}$ commutator bound, with $K_{ρ}$ from [F3].

4.1givenA1F5F7F8F10step 1.1step 2.1algebra∎

For every $n$ , the normalized projection $P_{n} (s) / Tr P_{n} (s)$ is a band density, since the band is nonempty. [F5, F7, F10] extend its trace pairing to a state on $A$ , and [F8] gives at least one subsequential limit; hence each $S (s)$ is nonempty. Let $ω_{0} \in S (0)$ , witnessed by band states $ω_{k}^{0}$ along a subsequence $n_{k}$ . Step 1.1 gives $ω_{k}^{s} = ω_{k}^{0} \circ α_{n_{k}, s} \in S_{Λ_{n_{k}}} (s)$ . For local $A$ , $∣ ω_{k}^{s} (A) - (ω_{0} \circ α_{s}) (A) ∣ \leq ∥ α_{n_{k}, s} (A) - α_{s} (A) ∥ + ∣ ω_{k}^{0} (α_{s} (A)) - ω_{0} (α_{s} (A)) ∣,$ which tends to zero by pointwise convergence of the states and uniform norm convergence of the automorphisms. Density and the state norm-one bounds extend convergence to all $A \in A$ . Thus $ω_{0} \circ α_{s} \in S (s)$ . Conversely, for any $ω_{s} \in S (s)$ , use the inverse finite-volume transport in step 1.1 and the convergent inverse flows from step 2.1 to obtain $ω_{s} \circ α_{s}^{- 1} \in S (0)$ . Both inclusions prove the asserted set equality.

Remarks

For each finite volume, the unitary $U_{n} (s)$ restricts to a unitary from the initial band subspace onto the final band subspace. The arrows $ι_{0}$ and $ι_{s}$ are the inclusion maps of those subspaces into $H_{Λ_{n}}$ . The square records these inclusions and the restricted map; it commutes by the projection identity proved in step 1.1. It summarizes only finite-volume transport, while the thermodynamic limit is proved in steps 2.1–4.1.

diagram failed to render:
\begin{tikzcd}
\operatorname{Ran}P_n(0) \arrow[r,"U_n(s)|_{\operatorname{Ran}P_n(0)}"] \arrow[d,hook,"\iota_0"'] & \operatorname{Ran}P_n(s) \arrow[d,hook,"\iota_s"] \\
\mathcal H_{\Lambda_n} \arrow[r,"U_n(s)"'] & \mathcal H_{\Lambda_n}
\end{tikzcd}

5 · Examples, experiments and counterexamples

None yet.

Lieb–Robinson locality, clustering and gapped spectral transport

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Weighted interactions and gapped finite-volume paths

Definition

Statement references

Summability and convolution control for lattice weights

Statement

Statement references

Facts & Assumptions

Proof

Normalized-trace expectation onto a finite spin region

Definition

Statement references

Contractive local expectations and their commutator bound

Statement

Statement references

Facts & Assumptions

Proof

Weighted interaction commutator bound

Statement

Statement references

Facts & Assumptions

Proof

Scalar spectral measures of a separable positive generator under countable choice

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Gaussian frequency separation of connected ground correlations

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Exponential connected clustering from a unique infinite-volume ground-state gap

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Filtered derivative of an isolated finite-volume band

Statement

Statement references

Facts & Assumptions

Proof

Filtered spectral generators have summable local shells

Statement

Statement references

Facts & Assumptions

Proof

Thermodynamic spectral transport along a uniformly gapped path

Statement

Statement references

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources