module module moderate

`IndisputableMonolith.Thermodynamics.PartitionFunction`

The PartitionFunction module defines the partition function and derived thermodynamic quantities for discrete systems in Recognition Science, expressing energies via the J-cost from the Cost module. Researchers working on statistical mechanics within the RS framework would cite it when connecting ledger sums to free energy, entropy, and heat capacity. The module consists entirely of definitions and basic positivity statements with no complex proofs.

claim$k_B$ denotes the Boltzmann constant, $Z = $ partitionFunction for a DiscreteSystem with energies drawn from energyFromJCost, $F = $ freeEnergy $ = -k_B T / k_B T$ wait no, $F = -k_B T$ ln $Z$, $S = $ entropy, $C = $ heatCapacity.

background

The module imports Constants, where the fundamental RS time quantum satisfies τ₀ = 1 tick, and Cost, which supplies the J-cost function used to assign energies. It introduces DiscreteSystem as a finite collection of states equipped with ledgerProperties, then defines partitionFunction as the sum over states of exp(-beta * energyFromJCost). Derived objects include freeEnergy, averageEnergy, entropy, and heatCapacity, all expressed in RS-native units with beta = 1/(k_B T).

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

This module supplies the thermodynamic primitives that feed into equilibrium derivations and links microscopic J-cost to macroscopic potentials. It closes the gap between the Cost module and higher-level applications of the Recognition Composition Law and phi-ladder mass formulas.

scope and limits

Does not derive the partition function from first principles beyond ledger sums.
Does not address continuous systems or quantum field theory.
Does not compute numerical values for specific physical systems.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (21)

def k_B L34
def beta L37
structure DiscreteSystem L42
def partitionFunction L55
theorem partition_function_positive L59
def freeEnergy L75
def averageEnergy L79
def entropy L85
def heatCapacity L89
theorem partition_from_ledger_sum L107
def ledgerProperties L118
def energyFromJCost L133
theorem temperature_controls_fluctuations L140
def twoLevelPartition L153
theorem twoLevel_gt_one L157
theorem twoLevel_at_zero L164
def harmonicOscillatorPartition L177
theorem classical_limit L188
theorem quantum_statistics_from_8tick L203
def implications L216
structure PartitionFunctionFalsifier L229