DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

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

Depends on

Dependency tree · 2 levels

Sources

S. Bachmann, S. Michalakis, B. Nachtergaele, and R. Sims, Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems, §4.3