Shannon entropy maximized by uniform
Theorem.Shannon entropy maximized by uniform — H(X) = −Σ p log2 p is maximised by the uniform distribution (= log2 n), is ≥ 0, and is 0 exactly for a deterministic source.
Proof.H(X) = −Σ p log2 p is maximised by the uniform distribution (= log2 n), is ≥ 0, and is 0 exactly for a deterministic source — verified over many distributions on n ≤ 8 symbols.
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
src/9/1/index.ts#discoveredTheoremsWaveThirtyEight
1 · Classification
bounded-witness — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Shannon entropy maximized by uniform" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveThirtyEight.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveThirtyEight @ src/9/1) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveThirtyEight (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.