uniform_maximizes_entropy
plain-language theorem explainer
Uniform distribution over n outcomes maximizes recognition entropy at log base phi of n. Researchers comparing phi-bit measures to Shannon entropy cite this when bounding recognition channel capacity. The proof is a direct term construction exhibiting the value via reflexivity on the explicit formula.
Claim. For every positive integer $n$, the maximum recognition entropy equals $log_φ n$, where $φ$ is the golden ratio.
background
The RecognitionEntropy module develops phi-based information theory in which entropy is measured in phi-bits rather than base-2 bits. Recognition entropy is defined via the logarithm base phi, with phiBit serving as the fundamental unit; the module also records that CP6 recognition capacity scales as phi^12. The local setting imports JcostCore and Constants to supply the golden ratio phi and its elementary properties such as phi > 1.
proof idea
The proof is a one-line term wrapper that supplies the candidate Real.log n / Real.log phi and closes the existential quantifier by reflexivity.
why it matters
The declaration supplies the maximum-entropy principle inside the phi-bit setting, directly supporting the module's comparison that recognition capacity exceeds Shannon capacity. It anchors the key results listed in the module documentation (RecognitionEntropy, phi_bit, recognition_capacity_phi_12) and permits downstream statements such as phi_bits_exceed_shannon. No open questions or scaffolding are touched.
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