ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

Pauli commutator and binary symplectic sign

Example

For one qubit, the labels of $X$ and $Z$ have symplectic pairing $1$ , so the binary form records their anticommutation. The two operators $Z$ and $- Z$ have the same binary label but different mod- $4$ phases.

Facts & Assumptions

Given: one qubit and $P (k, x, z) = i^{k} X^{x} Z^{z}$ .

[F1]

Pauli commutator is symplectic pairing: Pauli commutation is determined by $(- 1)^{ω}$ .

[F2]

The Pauli matrices and Pauli matrix algebra and trace identities: $X$ and $Z$ are the displayed one-qubit matrices and $X Z = - Z X$ .

Verification

1.1givenF1F2algebra

The labels are $λ (X) = (1, 0)$ and $λ (Z) = (0, 1)$ . Their pairing is $1 \cdot 1 + 0 \cdot 0 = 1$ in $F_{2}$ , so [F1] gives $X Z = - Z X$ , which also follows by multiplying the matrices in [F2].

2.1givenF2algebra∎

The operators $Z = P (0, 0, 1)$ and $- Z = P (2, 0, 1)$ have equal binary label $(0, 1)$ and distinct scalar phases $1$ and $- 1$ . Thus the label quotient records commutation but alone does not recover the exact signed Pauli.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Preskill, Quantum Error Correction, Physics 229 Lecture Notes, Chapter 7 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)