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 codes have magnetic-support gaps two and three

Statement

For the two ordered code pairs in the cubic-moment lemma, define support to be the set of magnetic labels with nonzero coefficient. The spin-four union of supports is ${- 4, - 2, 2, 4}$ , with consecutive gaps $(2, 4, 2)$ ; the spin-five union is ${- 5, - 2, 2, 5}$ , with consecutive gaps $(3, 4, 3)$ . Their minimum distinct-label separations are therefore two and three. Their hosting spin blocks have dimensions nine and eleven, respectively.

Facts & Assumptions

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

[F1]

The displayed coefficients and magnetic labels of both code pairs are explicit, nonzero and normalized. The selected spin-four and spin-five codes have nontrivial cubic moments

[F2]

The spin- $J$ block is $V_{J} = C^{2 J + 1}$ with orthonormal magnetic basis $e_{J, m}$ indexed by $m = - J, - J + 1, \dots, J$ . Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Proof

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

Goal: Establish the stated formula, convergence and qualifications.

1.1F1algebra

In the spin-four pair the first column uses labels $- 4, 2$ and the second uses $- 2, 4$ . Sorting their union gives $- 4, - 2, 2, 4$ , whose consecutive differences are two, four and two. In the spin-five pair the first column uses $- 2, 5$ and the second uses $- 5, 2$ . Sorting gives $- 5, - 2, 2, 5$ , whose consecutive differences are three, four and three.

2.1F1F2step 1.1algebra∎

Any difference of two distinct ordered labels is a sum of one or more consecutive positive gaps, so the minimum distinct-label separation equals the smallest consecutive gap. It is therefore two for spin four and three for spin five. A spin-J magnetic basis has $2 J + 1$ vectors, giving dimensions nine and eleven. This is a comparison of magnetic support and host dimension, not a claim of global optimality among codes.

Depends on

Used by · 1

Aydin–Barg rank-one correction and lower host angular momentum

Dependency tree · 2 levels

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