thermodynamic_entropy_connection
plain-language theorem explainer
The declaration asserts that thermodynamic entropy equals Boltzmann's constant times Shannon entropy under an appropriate scaling and interpretation in terms of defect distributions. Physicists connecting information measures to classical thermodynamics would cite this link when tracing entropy to J-cost minimization. The proof reduces immediately to the trivial proposition without invoking any lemmas.
Claim. $S_ {thermo} = k_B S_{Shannon}$ for appropriate interpretation of the entropies in terms of total defect and probability ratios.
background
The Information.ShannonEntropy module derives Shannon entropy from the J-cost structure on probability distributions. J-cost is the recognition effort J(x) = (x + x^{-1})/2 - 1 applied to ratios; total J-cost over a distribution yields the entropy expression. Upstream results supply the necessary pieces: entropy of a configuration equals its total defect (zero defect gives minimum entropy), cost functions arise from multiplicative recognizers and observer forcing events, and k_B is fixed at 1.380649e-23 with the Landauer bound stated as positive for T > 0.
proof idea
The proof is a one-line term that applies trivial directly to the asserted proposition. No upstream lemmas from InitialCondition, MultiplicativeRecognizerL4, or ComputationLimitsStructure are called inside the body; the connection is placed after the module's derivation of shannonEntropy from probJCost and entropy_is_expected_surprisal.
why it matters
The theorem supplies the explicit bridge from the module's information results to thermodynamic entropy, supporting the claim that physical entropy originates in recognition costs. It sits inside the INFO-001 derivation path that extracts Shannon entropy from J-cost over distributions and aligns with the Recognition Composition Law. No downstream uses are recorded, leaving open the explicit scaling for concrete systems such as black-hole entropy.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.