LemmaStatement: AI-generatedProof: AI-generated

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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.

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Signed diagonal binomial moments agree at order one

Statement

For the real arrays $α, β$ of the signed spin-four isometry, define $w_{j, a, b} = (\binom{6}{j}) / \sqrt{(\binom{8}{j + a}) (\binom{8}{j + b})}$ for integers $j, a, b$ with $0 \leq j \leq 6$ and $0 \leq a, b \leq 2$ . Then $D_{a b} = \sum_{j = 0}^{6} w_{j, a, b} (α_{j + a} α_{j + b} - β_{j + a} β_{j + b}) = 0$ for all nine pairs $(a, b) \in {0, 1, 2}^{2}$ . The common diagonal weighted norms at $a = 0, 1, 2$ are $5 / 14, 1 / 7, 5 / 14$ , respectively. Equivalently one may sum to $8$ , declaring each term with $j > 6$ to be zero before taking any quotient.

Statement references

Real coefficient arrays for the signed spin-four isometry

The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ|

Facts & Assumptions

[F1]

The real arrays have supports ${0, 6}$ and ${2, 8}$ , respectively, with $α_{0}^{2} = β_{8}^{2} = 1 / 3$ and $α_{6}^{2} = β_{2}^{2} = 2 / 3$ . Each individual support has its two indices separated by six. Real coefficient arrays for the signed spin-four isometry

[F2]

For natural numbers $n, k$ , $(\binom{n}{k})$ is the number of $k$ -element subsets of an $n$ -element set. In particular $(\binom{n}{0}) = (\binom{n}{n}) = 1$ , $(\binom{n}{1}) = n$ , and $(\binom{n}{k}) = 0$ for $k > n$ . The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ|

Proof

Given: The real arrays, integer index ranges and zero-term convention in the Statement.

Goal: Prove all nine diagonal-difference identities and the three common weighted norms.

1.1givenF1F2algebra

For each integer $0 \leq k \leq 8$ , an eight-element set has a $k$ -element subset, so $(\binom{8}{k}) > 0$ by [F2]. The index ranges give $0 \leq j + a, j + b \leq 8$ , hence all displayed weights in the sum to $6$ are defined. If $a \neq b$ , a nonzero product from one array would require two supported indices to differ by $b - a$ , whose nonzero absolute value is at most two. By [F1] its distinct supported indices differ by six. Thus no such product exists, and both weighted sums vanish for all $a \neq b$ .

1.2F2algebra

Count the needed binomial values from [F2]. Each two-element subset of an $n$ -element set occurs twice among the $n (n - 1)$ ordered pairs of distinct elements. Each three-element subset of a six-element set occurs six times among its $6 \cdot 5 \cdot 4$ ordered triples of distinct elements. Thus $(\binom{6}{2}) = 15$ , $(\binom{6}{3}) = 20$ and $(\binom{8}{2}) = 28$ . Complementation bijects $k$ -element subsets with $(n - k)$ -element subsets, so together with the boundary and singleton values of [F2] this gives $((\binom{6}{j}))_{j = 0}^{6} = (1, 6, 15, 20, 15, 6, 1)$ and $(\binom{8}{0}) = (\binom{8}{8}) = 1$ , $(\binom{8}{6}) = 28$ .

2.1givenF1F2step 1.1step 1.2algebra∎

For $a = b = 0$ , the alpha sum is $1 / 3 + (2 / 3) / 28 = 5 / 14$ , while the beta sum is $(2 / 3) 15 / 28 = 5 / 14$ . For $a = b = 1$ , the alpha sum is $(2 / 3) 6 / 28 = 1 / 7$ , and the beta sum is the same. For $a = b = 2$ , the alpha sum is $(2 / 3) 15 / 28 = 5 / 14$ , while the beta sum is $(2 / 3) / 28 + 1 / 3 = 5 / 14$ . These are all supported contributions by [F1], and the binomial values are those of step 1.2. Taking differences proves all remaining $D_{a a} = 0$ . If the sum is extended to $8$ , its additional terms are zero by the stated convention, which never evaluates a quotient outside the original range. Together with step 1.1 this proves every claim.

Depends on

Used by · 1

Finite signed-code KL product test

Dependency tree · 2 levels

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