ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A separable grid tensor has TT rank one

Example

For nonzero vectors $u^{(k)} \in R^{n_{k}}$ , consider the separable grid tensor $A = u^{(1)} \otimes \dots \otimes u^{(d)} .$

Verification

Given: $d \geq 2$ and nonzero vectors $u^{(k)} \in R^{n_{k}}$ .

[F1] Finite tensor arrays and the Frobenius inner product: a cut unfolding groups the first $k$ indices into rows and the remaining indices into columns.

[F2] Tensor train cores, bond ranks, and canonical gauges: TT cores contract over their shared bond indices.

[F3] Exact TT ranks and root-sum-square truncation error: the exact minimal bond size at cut $k$ is $rank (A^{⟨ k ⟩})$ .

1.1F1givenconstructalgebra

At cut $k$ , the unfolding is the outer product of the vector with entries $\prod_{j = 1}^{k} u_{i_{j}}^{(j)}$ and the vector with entries $\prod_{j = k + 1}^{d} u_{i_{j}}^{(j)}$ . Both vectors are nonzero, so this matrix has rank one.

2.1F2F3step 1.1algebra∎

Set $G_{k} (1, i_{k}, 1) = u_{i_{k}}^{(k)}$ for each core. Their contraction is $\prod_{k} u_{i_{k}}^{(k)} = A_{i_{1}, \dots, i_{d}}$ , so this is an exact TT with every internal bond size one. By [F3] these sizes are minimal; full-rank TT-SVD therefore also returns this exact rank-one structure.

Depends on

Used by

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Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ivan V. Oseledets, Tensor-Train Decomposition, SIAM Journal on Scientific Computing 33(5) (2011), §2 (standard reference, not scraped)
Mi-Song Dupuy, Tensor trains for high-dimensional problems, full lecture notes, Chapter 1, Example 1.2.3 (2026) (standard reference, not scraped)