ExampleConstruction: AI-generatedVerification: 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.

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

Changing the reference matrix changes complexity, not the network

Example

Use the one-layer network with identity activation, $A_{1} = (1, - 1)^{⊤}$ , and input $x = 1$ . Its scores are $(1, - 1)$ for either choice of reference matrix. With reference $M_{1} = A_{1}$ , its spectral network complexity is $R_{A} = 0$ ; with reference $M_{1} = 0$ , it is $R_{A} = 2$ . The layer operator norm and every prediction stay fixed, while the reference-dependent complexity changes.

Example references

Spectral network complexity with reference matrices: defines the network score map, reference-dependent complexity, and induced operator norm.

Multiclass margins, ramp loss, and empirical margin loss: defines predictions by score maximization and the multiclass margin.

A simultaneous spectral margin bound for multiclass networks: uses reference matrices in its explicit complexity terms.

Facts & Assumptions

Given: The one-layer identity-activation network, its matrix, input, and the two stated reference matrices.

[F1]

The score map depends on the layer matrix and activation; the reference matrix enters $R_{A}$ through the transpose-difference mixed norm. Spectral network complexity with reference matrices

[F2]

The predicted label maximizes the score coordinates, and the margin compares the true-label score with the maximum competing score. Multiclass margins, ramp loss, and empirical margin loss

[F3]

The theorem's layer terms depend on both the layer matrix norms and their distances from the chosen reference matrices. A simultaneous spectral margin bound for multiclass networks

Verification

technique · direct

1.1F1F2givenalgebra

Since the matrix and activation are identical in both cases, the network computes $A_{1} x = (1, - 1)$ for either reference. Its operator norm is $\sqrt{2}$ , so the predicted class is class $1$ and the margin for label $1$ is $2$ .

2.1F1F3step 1.1algebra∎

If $M_{1} = A_{1}$ , the transpose-difference norm is zero and the defining formula gives $R_{A} = 0$ . If $M_{1} = 0$ , that norm is $2$ by summing the two scalar column norms of $(1, - 1)$ ; hence $R_{A} = \sqrt{2} (2 / \sqrt{2}) = 2$ . The calculations change the complexity input only; [F3] does not assert that the floored explicit theorem term vanishes when $R_{A} = 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

Bartlett, Foster, and Telgarsky, Spectrally-normalized margin bounds for neural networks, Definition (1.2) (standard reference, not scraped)