Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated
How 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.

One-letter Gallager and expurgated exponent functions

Definition

Let W be a finite DMC from finite nonempty X to Y, with rates and logarithms in nats. For an input law P and ρ≥0, define E0(ρ,P):=−ln⁡∑y∈Y(∑x∈XP(x)W(y∣x)1/(1+ρ))1+ρ. Separately, for ρ≥1, define B(x,x′):=∑yW(y∣x)W(y∣x′) and E0x(ρ,P):=−ρln⁡∑x,x′P(x)P(x′)B(x,x′)1/ρ. For every a>0, 0a=0; thus zero channel transitions and zero Bhattacharyya coefficients contribute zero to these sums. These are one-letter functions at a fixed P; the global maxima and rate-dependent exponent functions are separate definitions.

Depends on

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Sources