Alphabeta MathHow 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 solver-invalid FWI iterate
Example
Fix an FWI velocity-model solver with a declared finite admissibility predicate that checks that solver's coefficient, boundary, mesh, and time-step conditions, including its solver-specific Courant–Friedrichs–Lewy (CFL) condition. If an optimization method proposes a candidate velocity map with , the benchmark records the call as a solver/admissibility failure; its evaluation protocol specifies whether and how that failed trial contributes to any numerical aggregate. In its FWI DAPS experiment, InverseBench reports that a smaller learning rate gave numerically stable but data-ineffective outputs, while a slightly larger learning rate produced a noisy velocity map that violated the PDE solver's CFL condition and failed. The paper gives no exact violating iterate or universal CFL threshold; this example claims neither.
Example references
Facts & Assumptions
Given: A chosen FWI solver , its declared finite admissibility predicate, an optimizer-produced velocity map , the Boolean result , and the qualitative DAPS observation reported in the Example.
Benchmark risk, compute budgets, and uncertainty records: a failed solver/admissibility call is recorded with its status and is not silently omitted from results.
Finite-element PDE solver admissibility and certificate contract: an admissibility predicate tests the specified model's hypotheses, and any CFL condition is method-specific; the definition supplies no universal threshold.
Scope of InverseBench findings: InverseBench reports the FWI step-size/CFL behavior as an empirical observation for its named solver and methods.
Verification
Since , the proposed iterate fails the selected solver's declared admissibility test; it is therefore not an admissible input for that solver call.
By [F1], the benchmark records this call as failure and does not silently omit it. [F1] does not prescribe whether the failed call receives or contributes to a numerical score, so no particular scoring rule is asserted.
The stated smaller-step/larger-step DAPS behavior is the paper's reported FWI observation for its numerical PDE setup, as recorded by [F3]; no inference is made from it to other solvers or parameter ranges.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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