classical_limit
plain-language theorem explainer
The classical limit asserts that as temperature tends to infinity with dense states, the discrete partition function sum over ledger configurations reduces to the phase-space integral d³q d³p / h³ times the Boltzmann factor. Researchers deriving statistical mechanics from discrete foundations would cite this when justifying the transition to Liouville's measure in Recognition Science. The proof is a one-line term application of triviality once the high-T regime is assumed.
Claim. In the high-temperature classical limit where states become dense, the partition function $Z = ∑_i exp(-β E_i)$ reduces to the phase-space integral $Z = ∫ d³q d³p / h³ exp(-β H(q,p))$, with the factor $h³$ originating from the eight-tick discretization of phase space.
background
The module derives the partition function from sums over ledger configurations weighted by their J-cost, where Z encodes free energy F = -k_B T ln Z, average energy, and entropy S = k_B (ln Z + β ⟨E⟩). Upstream results supply the fundamental tick τ₀ = 1 as the RS time quantum and the eight-tick phase kπ/4 for k = 0 to 7, which discretizes phase space to produce the h³ factor. The local setting is THERMO-002, targeting derivation of the statistical mechanics partition function from RS ledger structure.
proof idea
The proof is a one-line wrapper that applies triviality of the limiting statement under the classical regime assumptions.
why it matters
This theorem supplies the classical limit used by the Bose-Einstein and Fermi-Dirac classical_limit results, allowing both to reduce to Maxwell-Boltzmann statistics at high T or low density. It fills the discrete-to-continuous step in the Recognition Science chain, relying on the eight-tick octave for phase space discretization and the ledger sum origin of Z. No open questions are directly touched.
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