Thermodynamics Jcost Boltzmann

The Boltzmann bridge

In statistical mechanics, the Boltzmann distribution assigns probabilities to states based on their energy: a state with energy E has probability proportional to exp(-E/T), where T is temperature. The J-cost Boltzmann bridge in Recognition Science takes this classical form and replaces energy with a specific cost function J(x) = (x + 1/x)/2 - 1. This cost measures the recognition effort of a system in state x, and the bridge treats that effort exactly as physics treats energy.

The framework's machine-checked library of formal theorems proves five properties of the resulting Gibbs weight w(x) = exp(-J(x)/T). First, the weight at x=1 equals 1, and this is the maximum possible weight for any positive x, because J(1)=0 is the unique minimum of the cost. Second, the weight is symmetric: w(x) = w(1/x), since the cost itself is symmetric under reciprocals. Third, higher cost always means lower weight: if J(x) > J(y), then w(x) < w(y). Fourth, the ground state x=1 strictly dominates any excited state x≠1. Fifth, the free energy F = -T ln(Z) is nonpositive whenever the ground state is in the state space, since the partition function Z is at least 1.

The classical Boltzmann distribution has a long history: Ludwig Boltzmann formulated the statistical interpretation of entropy in the 1870s, and Josiah Willard Gibbs generalized it into the canonical ensemble. The framework's contribution is not a new statistical mechanics but a new identification: it proves these five thermodynamic properties follow from the forced form of J, not from any arbitrary choice of energy function. The bridge is a theorem in the machine-checked library, with zero axioms beyond the standard three and zero unproven assumptions.

In Recognition Science, this bridge is the thermodynamic basis for natural selection. The framework models a biological system as a recognition system, a discrete record of events where each state carries a cost. At low temperature, the weight ratio between the ground state and any other state diverges, which models strong selection: the cheapest state overwhelmingly dominates. At high temperature, the weight differences shrink, modeling weak selection or drift. This is the framework's plain-language statement: the mathematics of Boltzmann statistics, applied to a forced cost function, makes the dominance of the cheapest state a theorem rather than an assumption.

The bridge does not claim that real biological populations always reach the ground state. It establishes a formal connection: if a system's dynamics follow this Gibbs weight, then the ground state dominates and selection pressure increases as temperature drops. The practical consequence is that the framework can treat evolutionary outcomes as thermodynamic equilibria, with the cost function J providing the energy surface. This is a definitional choice about what to model, and the theorems then apply to that model.

/-- Gibbs weight at any positive x is at most 1 (= weight at x=1).
    Since J(x) >= 0 for x > 0, we have -J(x)/T <= 0, so exp(-J/T) <= exp(0) = 1. -/
theorem weight_maximized_at_one (sys : RecognitionSystem) (x : ℝ) (hx : 0 < x) :
    gibbs_weight sys x ≤ gibbs_weight sys 1 := by
  rw [gibbs_weight_one]
  unfold gibbs_weight
  have hJ : 0 ≤ Jcost x := Jcost_nonneg hx
  have hT : 0 < sys.TR := sys.TR_pos
  have h_neg : -Jcost x / sys.TR ≤ 0 := by
    apply div_nonpos_of_nonpos_of_nonneg
    · linarith
    · linarith
  calc exp (-Jcost x / sys.TR)
      ≤ exp 0 := exp_le_exp.mpr h_neg
    _ = 1 := exp_zero

/-- Gibbs weight is symmetric: w(x, T) = w(1/x, T) for x > 0.
    This follows from J(x) = J(1/x). -/
theorem weight_symmetric (sys : RecognitionSystem) (x : ℝ) (hx : 0 < x) :
    gibbs_weight sys x = gibbs_weight sys x⁻¹ := by
  unfold gibbs_weight
  rw [Jcost_symm hx]

/-- Higher J-cost means lower Gibbs weight: if J(x) > J(y) then w(x) < w(y).
    This is the thermodynamic basis for natural selection. -/
theorem higher_cost_lower_weight (sys : RecognitionSystem) (x y : ℝ)
    (h : Jcost x > Jcost y) :
    gibbs_weight sys x < gibbs_weight sys y := by
  unfold gibbs_weight
  apply exp_lt_exp.mpr
  have hT := sys.TR_pos
  exact div_lt_div_of_pos_right (by linarith) hT

/-- The weight ratio between x=1 (ground state) and x > 0, x ≠ 1 is > 1.
    The ground state always has strictly higher probability than any excited state. -/
theorem ground_state_dominates (sys : RecognitionSystem) (x : ℝ)
    (hx : 0 < x) (hx_ne : x ≠ 1) :
    gibbs_weight sys x < gibbs_weight sys 1 := by
  rw [gibbs_weight_one]
  unfold gibbs_weight
  have hJ : 0 < Jcost x := low_temp_selection x hx hx_ne
  have hT := sys.TR_pos
  have h_neg : -Jcost x / sys.TR < 0 := by
    exact div_neg_of_neg_of_pos (by linarith) hT
  calc exp (-Jcost x / sys.TR) < exp 0 := exp_lt_exp.mpr h_neg
    _ = 1 := exp_zero

/-- Free energy is nonpositive for any nonempty state space.
    F = -T * ln(Z) and Z >= 1 (from the ground state alone). -/
theorem free_energy_nonpos {Ω : Type*} [Fintype Ω] [Nonempty Ω]
    (sys : RecognitionSystem) (X : Ω → ℝ)
    (h_ground : ∃ ω, X ω = 1) :
    free_energy_from_Z sys X ≤ 0 := by
  unfold free_energy_from_Z
  have hT := sys.TR_pos
  have hZ := partition_function_pos sys X
  suffices h : 1 ≤ partition_function sys X by
    have h_log : 0 ≤ log (partition_function sys X) :=
      log_nonneg (by linarith)
    nlinarith
  unfold partition_function
  obtain ⟨ω₀, hω₀⟩ := h_ground
  calc (1 : ℝ) = gibbs_weight sys 1 := (gibbs_weight_one sys).symm
    _ = gibbs_weight sys (X ω₀) := by rw [hω₀]
    _ ≤ ∑ ω, gibbs_weight sys (X ω) :=
        Finset.single_le_sum (fun ω _ => (gibbs_weight_pos sys (X ω)).le)
          (Finset.mem_univ ω₀)

What this page does not claim

Real biological populations always reach the ground state. The framework derives the Boltzmann distribution from first principles; it identifies the cost function within an existing statistical mechanics. The bridge applies to systems without a well-defined cost function.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Thermodynamics/JCostBoltzmann.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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