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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The rational residual-tube certificate checker terminates
Statement
For every finite input record, Finite checker for rational residual-tube certificates returns certified or inconclusive after finitely many operations. It performs no unbounded subdivision or search.
Facts & Assumptions
Given: A finite certificate record and the finite checker procedure in Finite checker for rational residual-tube certificates.
Finite checker for rational residual-tube certificates: all loops traverse supplied finite arrays, a finite acyclic DAG, finite panels and finite supplied subdivisions; all arithmetic operations are the finite interval operations of Outward-rounded interval, rectangle and ball operations.
Proof
Structural validation traverses finite input arrays and the finite graph. A topological traversal visits each acyclic node finitely often, and the derivative recursion creates a fixed finite number of derivative nodes per source node and coordinate. Therefore the syntax and derivative phases terminate.
The numerical phase visits each of finitely many panels and cells and executes a fixed finite list of interval operations per field, reconstruction, derivative, and margin expression. Each interval operation returns either a finite interval or one of its specified statuses in finitely many endpoint operations. Every comparison is rational and finite. The procedure therefore reaches a passing or failed check and returns one of its two statuses.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- R. Baker Kearfott, Interval Computations: Introduction, Uses, and Resources, §1 (standard reference, not scraped)
- S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, §§5.1–5.5 (standard reference, not scraped)