probability
plain-language theorem explainer
The probability definition assigns to energy level i the normalized Boltzmann factor at inverse temperature beta. Researchers deriving statistical mechanics from recognition cost functionals cite it when computing state occupancies under fixed total J-cost. The definition is a direct quotient of the level-specific Boltzmann factor by the system's partition function.
Claim. The probability $P_i$ that the system occupies energy level $i$ at inverse temperature $β$ is given by the Boltzmann factor for that level divided by the partition function $Z$ over all levels, so $P_i = e^{-β E_i}/Z$.
background
A System is a non-empty list of energy levels, each carrying a recognition cost J. The module derives the Boltzmann distribution by weighting states according to their J-cost, with β acting as the Lagrange multiplier that enforces the total cost constraint. Upstream results define the Boltzmann factor as the exponential of the negative cost scaled by β, matching the standard form exp(-E/kT) but with J-cost as the primitive; this matches the module's core insight that cost-optimal allocation yields the exponential distribution.
proof idea
This is a direct definition that extracts the energy level at index i, computes its Boltzmann factor at the given β, and normalizes by the partition function of the entire system.
why it matters
This definition supplies the explicit probability measure for the Boltzmann distribution derived from J-cost, which feeds into audit summaries, coincidence bounds, and resolution-time structures. It closes the derivation step from the J-cost functional to observable probabilities, supporting the paper proposition on statistical mechanics from Recognition Science. It connects to the framework where J-uniqueness underpins the cost functional and the eight-tick octave governs discrete level spacing.
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