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.
A perfectly correlated uniform pair has zero conditional entropy
Facts & Assumptions
Given: As stated in the item.
Mutual information of finite random variables: I(M;R)=H(M)+H(R)−H(M,R)=H(M)−H(M∣R).
Conditional entropy of finite memory given side information: conditional entropy is computed from the conditional laws on positive-probability register values.
The pair law assigns probability 1/2 to (0,0) and (1,1) and zero to the other two pairs.
Example
Let M,R∈{0,1} have joint probabilities p(0,0)=p(1,1)=1/2 and p(0,1)=p(1,0)=0. Then H(M)=H(R)=log2, H(M,R)=log2, H(M∣R)=0, and I(M;R)=log2.
Verification
Given: The finite joint probability law in [A1].
Both marginals are uniform, so their entropies are −2(1/2)log(1/2)=log2. The joint law has two positive entries of size 1/2, so H(M,R)=log2.
Conditional on either value of R, M is determined, so its conditional law has one probability 1 and one 0; therefore H(M∣R)=0. Substituting these values into [F1] gives I(M;R)=log2.