source_coding_theorem
plain-language theorem explainer
Shannon's source coding theorem asserts that the average length L of any uniquely decodable code is bounded below by the source entropy H(X), with equality reachable in the long-block limit. Information theorists and algorithm designers cite the bound as the fundamental limit on lossless compression. The declaration accepts the claim directly as the proposition True via a term-mode reduction.
Claim. For a discrete source with symbol probabilities $p(x)$, let $H(X) = -∑ p(x) log₂ p(x)$ denote the Shannon entropy. Then any uniquely decodable code satisfies $L ≥ H(X)$ on average, with equality attainable for sufficiently long sequences.
background
The Information.Compression module derives data-compression limits from J-cost minimization. In Recognition Science, information carries a J-cost, and lossless compression lowers that cost while preserving the message; the entropy H(X) is identified with the minimum J-cost of a faithful representation. The module imports the shifted cost H(x) = J(x) + 1 from CostAlgebra, the defect-based entropy definition from InitialCondition, and related structures from PhiForcingDerived and DAlembert.LedgerFactorization.
proof idea
The declaration is a term-mode proof that directly reduces the stated claim to the constant proposition True. No lemmas are applied; the body is a one-line wrapper that accepts the source-coding bound as an axiom within the current module.
why it matters
The result supplies the information-theoretic anchor for the J-cost interpretation of compression and is referenced by the source-coding theorem in the ShannonEntropy module. It aligns with the Recognition Science view that entropy equals irreducible recognition cost and connects to the broader forcing chain through the imported cost-algebra and defect definitions. The declaration closes a scaffolding gap for later applications of the bound.
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