log2_half
plain-language theorem explainer
The identity log base two of one half equals negative one is established here. Information theorists computing entropy for binary sources cite this result when evaluating the Shannon entropy of a fair coin. The proof is a short term-mode calculation that unfolds the log2 definition and applies standard real logarithm rewrite rules for inverses and self-equality.
Claim. $log_2(1/2) = -1$
background
The Information.Compression module derives data compression limits from J-cost, where information carries a J-cost and lossless compression reaches the minimum J-cost given by entropy. Shannon's source coding theorem asserts that average code length is at least H(X) = -sum p(x) log2 p(x), with equality achievable for long sequences. This lemma supplies a basic logarithm value used in entropy calculations for simple sources; it rests on the H reparametrization from CostAlgebra, where H(x) = J(x) + 1 satisfies the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y).
proof idea
The term proof unfolds log2, applies simp to the reciprocal, rewrites via Real.logb_inv 2 2, then invokes Real.logb_self_eq_one with a norm_num witness that 1 < 2.
why it matters
This lemma is invoked by the fair_coin_one_bit theorem to conclude that fairCoinEntropy equals 1. It supplies an elementary computational step inside the source coding theorem section, reinforcing that entropy equals the minimum J-cost for faithful representation. Within Recognition Science it anchors the information-theoretic bound without reference to the phi-ladder or the eight-tick octave.
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