rs_entropy

Module F-011 builds temperature and the canonical ensemble from the ledger's J-cost structure and finite observer resolution. The module already had entropy as total defect, time as tick count, and dynamics as variational minimization; temperature was missing because $T=\partial E/\partial S$ needs a named entropy and energy.

A configuration is an $N$-tuple of positive real ratios (ledger entries). The total defect is $\sum_i J(x_i)$, with $J$ the unique cost from the forcing chain (T5: $J(x)=(x+x^{-1})/2-1$). That sum is non-negative and vanishes only at the unity configuration.

RS energy is the companion log-charge $\sum_i \log(x_i)$, conserved under the variational dynamics. Entropy here is the extensive defect count; energy is the conserved charge that pairs with it in the thermodynamic identities.

One-line definition: RS entropy is identified with total_defect on the same configuration. No further algebra; non-negativity and the zero-at-unity characterization are proved separately by wrapping the corresponding total-defect lemmas.

This is Step 1 of the F-011 derivation: entropy as defect count, the bridge from InitialCondition into thermodynamics. Downstream, Boltzmann weight is $\exp(-S(c))$, entropy splits as observer plus system defect over a subsystem partition, and thermal equilibrium equates temperature (hence equal $\sigma/N$) across baths.

Gravity reuses the same name for black-hole entropy on the RS horizon cell count (e.g. entropy quadruples when mass doubles). In the broader framework it is the extensive cost that pairs with the conserved log-charge, so temperature can be read as a Lagrange multiplier for unseen entries under coarse-graining.