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.
Finite DMC information density
Definition
Given an input law , let be its output law. The information density is when , and is otherwise. For block sequences define , with any sum containing equal to . On the support of the induced joint law, both numerator and denominator are positive, and the value is finite.
Depends on
Used by
- Conditional information variance and DMC dispersion Definition
- IID DMC information density converges to mutual information Lemma
- Strictly feasible DMC input laws achieve every lower mutual-information rate Lemma
- Finite-DMC dependence-testing achievability bound Theorem
- Nondegenerate normal approximation for finite unconstrained DMCs Theorem
- Random coding achieves every rate below finite DMC capacity Theorem
Dependency tree · two levels
13 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
- Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 19 §§19.1–19.2 (standard reference, not scraped)
- Robert Gallager, Principles of Digital Communication I, Chapter 8 §8.7.4, MIT OpenCourseWare (standard reference, not scraped)