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.

A two-domain heat trace balances outward fluxes

Statement

Take a one-dimensional trace space $Λ = R$ , two interior spaces $I_{i} = R$ , $Δ t = 2$ , $M_{1} = M_{2} = I_{2}$ , $A_{1} = diag (1 / 2, 3 / 2)$ , and $A_{2} = diag (1 / 2, 1 / 2)$ , with the second coordinate as trace. Let the condensed loads be $f_{1} = 1$ , $f_{2} = 5$ . The monolithic trace is $2$ and the outward fluxes are $3$ and $- 3$ . With $α = 1$ , $r_{1}^{0} = 2$ , tolerance $0.1$ , and budget $M = 2$ , the Robin–Robin algorithm returns the certified trace $2$ on its second pass.

Facts & Assumptions

Given: The finite-dimensional local blocks, trace coordinates, condensed loads, Robin parameter, incoming datum, tolerance, and pass budget in the Statement.

[F1]

Finite-dimensional backward-Euler diffusion transmission model: $B_{i} = Δ t^{- 1} M_{i} + A_{i}$ , $q_{i} = S_{i} λ_{i} - f_{i}$ , and the monolithic trace solves $(S_{1} + S_{2}) λ = f_{1} + f_{2}$ .

[F2]

The diffusion Schur complement is positive and minimizes energy: each positive-definite local block has a positive-definite Schur complement.

[F3]

Monolithic backward Euler transmission energy identity: when traces match and outward fluxes balance, paired interface work cancels in the monolithic energy identity.

[F4]

Finite-budget Robin-Robin transmission iteration: each pass performs the two Robin solves and exchanges and stops only when the stated residual test passes.

[F5]

Robin-Robin iteration contracts to the monolithic trace: a passing stop test certifies trace error at most $ε$ relative to the monolithic trace.

Verification

1.1givenF1F2algebra

The local backward-Euler blocks are $B_{1} = diag (1, 2)$ and $B_{2} = diag (1, 1)$ , which are positive definite. With zero off-diagonal blocks between the interior and trace coordinates, their Schur operators are $S_{1} = 2$ and $S_{2} = 1$ , so [F2] applies.

1.2F1F4algebra

On pass one, $λ_{1}^{1} = (2 + 1) / 3 = 1$ , $r_{2}^{1} = (1 - 2) 1 + 1 = 0$ , $λ_{2}^{1} = (0 + 5) / 2 = 2.5$ , and $r_{1}^{1} = (1 - 1) 2.5 + 5 = 5$ . Here $q_{1} = ∣ (1 - 2) / (1 + 2) ∣ = 1 / 3$ , $q_{2} = 0$ , and the stop threshold is $0.1 / (1 / 3) = 0.3$ ; the incoming-data change is $∣ 5 - 2 ∣ = 3$ , so the algorithm does not stop.

2.1F1F3step 1.1algebra

The monolithic scalar equation is $3 λ = 6$ , hence $λ_{*} = 2$ . Its fluxes are $q_{1} = 2 (2) - 1 = 3$ and $q_{2} = 1 (2) - 5 = - 3$ , so they balance and the interface work $λ_{*} (q_{1} + q_{2})$ vanishes as in [F3].

3.1F4F5step 1.2step 2.1algebra∎

On pass two, $λ_{1}^{2} = (5 + 1) / 3 = 2$ , $r_{2}^{2} = (1 - 2) 2 + 1 = - 1$ , $λ_{2}^{2} = (- 1 + 5) / 2 = 2$ , and $r_{1}^{2} = (1 - 1) 2 + 5 = 5$ . The incoming-data change is $0 \leq 0.3$ , so [F4] returns converged with trace $2$ ; [F5] certifies its error is at most $0.1$ relative to $λ_{*} = 2$ from step 2.1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Emil Engström and Eskil Hansen, Time-dependent Steklov–Poincaré operators and space-time Robin–Robin decomposition for the heat equation (2022) (standard reference, not scraped)