IndisputableMonolith.Information.Shannon_Entropy_Max_RS
Module packaging the Recognition Science reading of Shannon's maximum-entropy principle: a nonnegative domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate that the max-entropy claim holds in RS-native units. Information theorists working in the monolith would cite the certificate and the threshold lemmas. The file is mostly definitions plus elementary nonnegativity and positivity facts, not a deep uniqueness proof.
claimIn RS-native units one defines a domain cost $C$ from the Recognition $J$-cost, proves $C\ge 0$, fixes a canonical positive threshold $\theta>0$, and packages an inhabited certificate that Shannon entropy is maximized subject to the RS cost constraint at that threshold.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified forcing chain. The Cost import supplies that $J$-calculus; Constants supplies the RS tick $\tau_0=1$ so information quantities sit in dimensionless RS units ($c=1$, $\hbar=\varphi^{-5}$).
Classical max-entropy says the distribution of largest Shannon entropy under linear moment constraints is exponential. Here the constraint is phrased through a domain cost assembled from $J$, so the same variational idea is stated inside the monolith's cost language rather than ad hoc Lagrange multipliers.
Sibling names indicate the local API: domainCost and its evaluation/nonnegativity lemmas, canonicalThreshold with positivity, and a certificate type ShannonEntropyMaxCert with an inhabited instance.
proof idea
Definition-and-certificate module, not a long derivation. Domain cost is introduced from the imported $J$-cost, then nonnegativity is recorded. A canonical threshold constant is fixed and shown positive. The max-entropy claim is wrapped as a certificate structure with an inhabitation proof, so downstream code can assume the RS max-entropy package without reopening the classical argument. No deep analytic entropy proof lives here; the work is interface assembly on top of Cost and Constants.
why it matters in Recognition Science
Places Shannon max-entropy inside the Information domain of the monolith so later RS arguments can cite a single certificate rather than raw classical statements. It ties information bounds to the same $J$-cost that forces $\varphi$, the eight-tick octave, and $D=3$ upstream. With no recorded downstream users yet, the module is a leaf API: ready for ledger, measurement, or channel-capacity developments that need a nonnegative domain cost and a positive threshold in RS units. It does not itself close a forcing-chain step (T0–T8); it exports the information-side cost language those steps already assume.
scope and limits
- Does not prove uniqueness of the max-entropy distribution from first RS principles.
- Does not derive Shannon entropy from the Recognition Composition Law.
- Does not numerically fix $\alpha$ or particle masses; only cost/threshold interface.
- Does not supply channel-capacity or coding theorems beyond the certificate shell.
- Does not depend on eight-tick or $D=3$ lemmas inside this file.