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.

Both selected codes have exact tensor-rank distance three

Statement

Define the within-block tensor-rank distance of a two-dimensional spin-J code to be the least integer $r \geq 0$ for which some component of a rank-r spherical tensor family from $V_{J}$ to itself has nonscalar compression to the code; use infinity if none exists. Also define its spin-polynomial distance as the least d for which some element of $P_{d}$ has nonscalar compression. If each of the two explicit codes in the cubic-moment lemma has the complete order-one pair-product condition, then both distances equal three for each code. These are explicit error-filtration conventions, not a claim that the original article or the independent Aydin–Barg paper defined a distance metric.

Facts & Assumptions

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

[F1]

Normalized lowering commutators of J_+ cubed give a rank-three family; rank-at-most-two families are quadratic spin polynomials. Low-rank tensor families are spin polynomials

[F2]

The pair-product hypothesis implies scalar compression of all quadratic spin polynomials and all tensor components of rank at most two. Pair-product correction detects quadratic spin errors

[F3]

Both code isometries have zero first moment and the stated unequal cubic diagonal moments. The selected spin-four and spin-five codes have nontrivial cubic moments

[F4]

The magnetic basis, ladder formulas and defining tensor-family equations use the finite-spin convention. Finite spin blocks, spherical tensors and Clebsch–Gordan phases

[F5]

The complete selected order-one error span and its pair-product correction condition are defined on a finite sum of spin blocks containing the input block, with every selected spin $J^{'}$ satisfying $∣ J^{'} - J ∣ \leq 1$ . Order-one absorption–emission tensor error span

Proof

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

Goal: Establish the stated formula, convergence and qualifications.

1.1F2F3F5algebra

By [F2], the assumed pair-product condition of [F5] gives scalar compression in degrees and tensor ranks zero through two for either code. By [F3], $J_{z}^{3}$ has nonscalar compression for both, so their spin-polynomial distances are exactly three. To identify a rank-three witness explicitly, put $Q_{3} = 5 J_{z}^{3} - [3 J (J + 1) - 1] J_{z}$ .

1.2F1F4algebra

The zero-weight component of the family in [F1] is a nonzero constant multiple of $(ad J_{-})^{3} J_{+}^{3}$ . This operator equals $- 24 Q_{3}$ . Indeed, on a magnetic basis vector of weight m put $p (m) = J (J + 1) - m (m + 1)$ and $f (m) = p (m) p (m + 1) p (m + 2)$ . Expanding the three commutators gives diagonal coefficient $f (m) - 3 f (m - 1) + 3 f (m - 2) - f (m - 3) = - 24 [5 m^{3} - (3 J (J + 1) - 1) m]$ by expanding the displayed cubic products. The ladder formulas in [F4] make the same products zero at any invalid endpoint. Thus equality holds on every basis vector.

2.1F1F2F3step 1.1step 1.2algebra∎

Since the first-moment compression is zero, [F3] gives compression of $Q_{3}$ equal to $diag (- 80, 80)$ in spin four and $diag (150, - 150)$ in spin five. Both are nonscalar. Step 1.2 therefore exhibits an undetected rank-three tensor component. With the lower bound of step 1.1, both codes have exact tensor-rank distance three, as well as the already proved polynomial distance three.

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