twoLevelPartition
plain-language theorem explainer
The two-level partition function for a system with ground-state energy zero and excited-state energy epsilon at positive temperature T is defined as one plus the exponential of minus beta times T times epsilon. Researchers modeling spin systems or the Ising model within the Recognition Science ledger framework would cite this as the elementary case. The definition is a direct one-line algebraic expression using the module's beta function.
Claim. For a two-level system with energies $0$ and $ε$, the partition function at temperature $T > 0$ is $Z = 1 + e^{-β(T) ε}$, where $β(T)$ is the inverse temperature.
background
The module derives the statistical mechanics partition function as a sum over ledger configurations weighted by J-cost, with $Z = Σ_i exp(-β E_i)$. This encodes free energy, average energy, and entropy. The two-level case is the simplest example with $E_0 = 0$ and $E_1 = ε$, serving as the basis for Ising and spin models. Upstream results include the definition of T as fundamental periods and the beta function for inverse temperature in the same module.
proof idea
This is a direct definition that expands to the expression 1 + exp(-beta T hT * epsilon). No additional lemmas are invoked beyond the module-local beta definition.
why it matters
This definition supplies the elementary case for the two-level theorems twoLevel_at_zero and twoLevel_gt_one. It realizes the THERMO-002 target that Z emerges from RS ledger structure, feeding calculations of free energy and entropy. It connects to the J-cost weighting and phi-ladder mass formulas in the broader Recognition Science chain.
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