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.
The Reference Has Zero Specific Relative Entropy
Example
Take , the unit-intensity Poisson configuration reference. This gives the zero-information baseline for the A-page specific relative entropy.
Facts & Assumptions
Given: as probability laws on the configuration space.
Finite-region relative entropy is when (Relative Entropy of Probability Measures).
Specific relative entropy is the large-cube limit of normalized finite-region relative entropies (Specific Relative Entropy Density).
Poisson Reference Counts and Translation Invariance: under the declared AC hypothesis, is stationary and has expected unit-cell count one, so it belongs to the stationary finite-intensity domain of [F2].
Verification
Given: .
By [F3], satisfies the stationary finite-intensity hypotheses of [F2]. For every bounded cube , equality of the laws gives and the Radon–Nikodym derivative -almost everywhere. Thus .
Every normalized cube entropy is therefore zero, so its limit is by [F2]. Also -almost surely for every such .