summary
plain-language theorem explainer
summary defines a five-point list equating data compression limits to Shannon entropy and J-cost minimization under the Recognition Composition Law. Information theorists applying Recognition Science to coding bounds would cite it to frame entropy as minimum J-cost for faithful representation. The declaration is a static list definition with no computational reduction or lemma applications.
Claim. Define the summary list as containing the statements: compression limit equals entropy $H(X)$, compression equals $J$-cost minimization, entropy equals minimum $J$-cost for faithful representation, redundancy equals removable excess $J$-cost, and random data is already at minimum $J$-cost, where $H(x) = J(x) + 1$ and $J$ satisfies the Recognition Composition Law.
background
The module INFO-003 derives fundamental limits on lossless compression from J-cost. Shannon's source coding theorem states that average code length is at least the entropy $H(X) = -∑ p(x) log₂ p(x)$. In Recognition Science, information carries J-cost; compressed forms have lower J-cost, so the entropy limit equals the minimum J-cost for faithful representation and compression is J-cost minimization. Upstream, CostAlgebra.H reparametrizes as $H(x) = J(x) + 1$, turning the Recognition Composition Law into the d'Alembert equation $H(xy) + H(x/y) = 2 H(x) H(y)$. InitialCondition.entropy defines entropy of a configuration as its total defect, with zero defect yielding the minimum entropy state.
proof idea
The declaration is a direct definition of a constant list of strings. No lemmas are applied; the body simply enumerates the five RS equivalences listed in the module doc-comment.
why it matters
The definition supplies the high-level summary for the INFO-003 module on compression limits from J-cost, positioning entropy as the J-cost floor within the Recognition Science framework. It connects directly to the J-functional and Recognition Composition Law from the upstream CostAlgebra and FunctionalEquation results. No downstream theorems are recorded, leaving its role as an expository anchor rather than a computational step in the forcing chain.
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