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.
Prior-matched and out-of-distribution cases
Example
For an operational within-source reference, InverseBench trains its multi-coil MRI diffusion prior on fastMRI knee images and evaluates on held-out images from the same named data collection. This is a dataset-family match, not a claim that the population training and test laws are equal. Its separate out-of-distribution diagnostics change the source: a black-hole-imaging prior is trained on simulated general-relativistic magnetohydrodynamic images, while one stress test uses digit images; in linear inverse scattering, the prior is trained on sources with one through six cells, while the stress test uses nine-cell sources. The paper reports outputs drawn toward black-hole structure in the digit case and more effective recovery when the scattering source is close to the trained prior. These are finite, method- and dataset-specific observations; they provide no risk bound or distribution-shift guarantee.
Example references
Training and test laws are separately specified benchmark fields
Risk depends on the named joint test law
Prior-support mismatch is distinct from operator or noise mismatch
InverseBench prior-shift findings are empirical and task-scoped
Zheng et al., InverseBench, §4.1, §4.3, Figure 5, and Appendix C
Facts & Assumptions
Given: The finite train/test splits, each prior's training-source family, and the paper's selected out-of-distribution stress tests and qualitative observations.
Benchmark risk, compute budgets, and uncertainty records: the training and test laws are separate inputs and need not be equal.
Inverse reconstruction risk under train and test laws: test risk is computed under the specified joint test law, and no comparison with training risk follows without an additional premise.
Operator, noise-law, and prior-support mismatch: prior-support mismatch is a distinct model property and by itself supplies no posterior or risk-robustness theorem.
Scope of InverseBench findings: InverseBench's results concern its named data, solver, methods, and tuning protocol.
Verification
The MRI training and held-out test splits come from the same named fastMRI knee collection, so they provide an operational within-source comparison; [F1] does not identify their underlying probability laws.
The black-hole digit and nine-cell scattering cases deliberately change the test-source family from the source used to train each prior. The reported output tendencies are finite observations under [F4], and [F2]–[F3] preclude promoting them to a general test-risk or robustness bound.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
21 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.