ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge 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.

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The Spencer support set for rank label eleven in characteristic three

Example

In a field of characteristic $3$ with loop parameter $δ = 2$ , the recurrence quantum characteristic is $ℓ = 3$ . For rank label $N = 11$ , Spencer's support set is ${11, 5}$ .

Facts & Assumptions

Given: $char K = 3$ , $δ = 2 \in K$ , $ℓ = 3$ , and $N = 11$ .

[F1]

The recurrence is $[0] = 0$ , $[1] = 1$ , $[r + 1] = δ [r] - [r - 1]$ , and $ℓ$ is the least positive index with $[ℓ] = 0$ (Temperley–Lieb quantum integers and quantum characteristic).

[F2]

If $ℓ < \infty$ , the digits are those of $N + 1$ in the scales $s_{0} = 1$ , $s_{i} = ℓ P^{i - 1}$ , and the support set uses a positive top sign and arbitrary lower signs (Spencer quantum-characteristic digits and support set).

Verification

Given: $K$ has characteristic $3$ , $δ = 2$ , and $N = 11$ .

1.1F1givenalgebra

In $K$ , $[1] = 1$ , $[2] = 2$ , and $[3] = 2 \cdot 2 - 1 = 3 = 0$ , so the least positive vanishing index is $ℓ = 3$ . Here $P = 3$ , and the scales are $s_{0} = 1$ , $s_{1} = 3$ , $s_{2} = 9$ .

2.1F2givenstep 1.1algebra

The digits of $N + 1 = 12$ are $N_{0} = 0$ , $N_{1} = 1$ , $N_{2} = 1$ , because $12 = 0 \cdot 1 + 1 \cdot 3 + 1 \cdot 9$ . With highest index $2$ , the support values are $1 \cdot 9 \pm 1 \cdot 3 \pm 0 \cdot 1 - 1$ , namely $11$ and $5$ . Both are nonnegative, so ${Supp}_{3, 3} (11) = {11, 5}$ .

3.1F2step 2.1∎

This computation determines only the set in the definition; identifying its elements with actual composition factors requires the separate Spencer decomposition theorem and is not asserted by this example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources

Robert A. Spencer, The Modular Temperley-Lieb Algebra (standard reference, not scraped)
Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (standard reference, not scraped)