LemmaStatement: AI-generatedProof: AI-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.

The selected spin-four and spin-five codes have nontrivial cubic moments

Statement

In $V_{4}$ let $u_{0} = \sqrt{1 / 3} e_{4, - 4} + \sqrt{2 / 3} e_{4, 2}$ and $u_{1} = \sqrt{2 / 3} e_{4, - 2} - \sqrt{1 / 3} e_{4, 4}$ . In $V_{5}$ let $v_{0} = \sqrt{5 / 7} e_{5, - 2} + \sqrt{2 / 7} e_{5, 5}$ and $v_{1} = \sqrt{2 / 7} e_{5, - 5} + \sqrt{5 / 7} e_{5, 2}$ . Each ordered pair is orthonormal. For the corresponding two-column isometries $U$ and $V$ , $U^{*} J_{z} U = V^{*} J_{z} V = 0$ , whereas $U^{*} J_{z}^{3} U = diag (- 16, 16)$ and $V^{*} J_{z}^{3} V = diag (30, - 30)$ . Their second-moment compressions are $8 I_{2}$ and $10 I_{2}$ , respectively.

Facts & Assumptions

Given: The finite objects, measure conventions and explicit hypotheses in the Statement.

[F1]

The magnetic vectors form an orthonormal basis and $J_{z}$ acts by its magnetic label. Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Proof

Goal: Establish the stated formula, convergence and qualifications.

1.1F1algebra

Each column has squared norm one and the two columns in either block have disjoint magnetic supports, hence are orthogonal. All cross matrix elements of powers of $J_{z}$ vanish for the same reason. For the spin-four columns, the first moments are $(- 4 + 4) / 3 = 0$ and $(- 4 + 4) / 3 = 0$ ; the second moments are $(16 + 8) / 3 = (8 + 16) / 3 = 8$ ; the third moments are $(- 64 + 16) / 3 = - 16$ and $(- 16 + 64) / 3 = 16$ .

2.1F1step 1.1algebra∎

For the spin-five columns the first moments are $(- 10 + 10) / 7 = (- 10 + 10) / 7 = 0$ . Their second moments are $(20 + 50) / 7 = (50 + 20) / 7 = 10$ . Their third moments are $(- 40 + 250) / 7 = 30$ and $(- 250 + 40) / 7 = - 30$ . Together with the zero cross matrix elements and orthonormality established in step 1.1, these are exactly the asserted compression matrices.

Depends on

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Used by · 3

Dependency tree · 2 levels

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