ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

FE-HMM energy error ledger with distinct components

Example

Suppose a verified FE-HMM ledger has ellipticity bounds $λ = 1$ , $Λ = 2$ , macro best-approximation bound $η_{H} \leq 1 / 10$ , micro tensor discrepancy bound $Δ_{m i c r o} \leq 1 / 20$ , sampling discrepancy bound $Δ_{r e s} \leq 1 / 20$ , and homogenized energy norm $∥ \nabla u_{0} ∥_{2} \leq 1$ . Then the macro theorem certifies $∥ \nabla (U_{H} - u_{0}) ∥_{2} \leq \frac{4}{9} .$ These are conditional certified inputs for the ledger; the example makes no claim that they arise for every coefficient, scale ratio, or mesh.

Facts & Assumptions

Given: The six rational bounds stated above, with the micro and sampling discrepancies measured against the exact sampled tensor and $A^{*}$ respectively.

[F1]

FE-HMM macro, micro, and sampling error separation: if $Δ = Δ_{m i c r o} + Δ_{r e s} < λ$ , then $∥ \nabla (U_{H} - u_{0}) ∥_{2} \leq \frac{λ + Λ}{λ - Δ} η_{H} + \frac{Δ}{λ - Δ} ∥ \nabla u_{0} ∥_{2} .$

[F2]

Energy identity for FE-HMM micro tensor error: the micro discrepancy can be bounded by the squared best periodic P1 gradient approximation error.

Proof

1.1F1algebra

Here $Δ \leq 1 / 20 + 1 / 20 = 1 / 10 < 1 = λ$ , so the theorem applies. Its two contributions are bounded by $\frac{1 + 2}{1 - 1 / 10} \cdot \frac{1}{10} = \frac{1}{3}, \frac{1 / 10}{1 - 1 / 10} \cdot 1 = \frac{1}{9} .$ Their sum is $1 / 3 + 1 / 9 = 4 / 9$ . The first contribution uses the macro best approximation; the second is the sum of the distinct micro and resonance discrepancy certificates.

2.1F2F1step 1.1algebra∎

The micro part may be justified by an upper bound $\max_{K} η_{K}^{2} \leq 1 / 20$ from [F2], while the resonance part is separately bounded by $1 / 20$ ; if every ratio is integer, the resonance part is exactly zero by the aligned-sampling lemma cited in [F1]. This illustrates the ledger’s distinct inputs without asserting an unconditional rate.

Depends on

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