Foundation Thermodynamics Equilibrium Entropy Zero Iff

The zero-entropy theorem

In classical thermodynamics, entropy is a measure of disorder, and the third law states that a perfect crystal at absolute zero has zero entropy. The Recognition Science framework builds a similar concept from a different starting point. Here, a system is a finite collection of positive numbers, one for each of N entries. The framework defines entropy as the sum of a defect function over these entries, and energy as the sum of their logarithms. A state of equilibrium is one where all entries are equal.

The theorem equilibrium_entropy_zero_iff states a precise equivalence: for a system in equilibrium with N entries, the equilibrium entropy is zero if and only if the energy parameter σ is zero. Since the equilibrium entry is exp(σ/N), σ = 0 means every entry equals 1. The theorem is proved in the framework's machine-checked library of formal theorems, and it is a direct consequence of the definitions: at σ = 0, the entropy formula N * (cosh(0) - 1) evaluates to zero, and for any non-zero σ, the hyperbolic cosine term makes the entropy strictly positive.

This result is the framework's version of the third law of thermodynamics. It is paired with a companion theorem, temperature_zero_at_unity, which shows that the framework's temperature, defined as sinh(σ/N), is also zero at σ = 0. Together they state that the ground state, where all entries are 1, has both zero entropy and zero temperature. A further theorem, absolute_zero_unreachable, shows that along a variational trajectory with non-zero initial energy, the energy never reaches zero, so absolute zero cannot be reached in finite time, mirroring the classical statement.

What the theorem does not claim is equally important. It does not claim that entropy is zero for all configurations with zero energy; that is a separate statement, rs_entropy_zero_iff_unity, which holds for any configuration, not just equilibrium ones. It also does not establish a second law of thermodynamics. The framework's only directional theorem, full_defect_monotone, proves that total defect is non-increasing along variational trajectories, which is the opposite sign from entropy increase. The framework's intended rescue of the second law, via an observer's coarse-grained description, is stated but not proved.

For a reader, the practical upshot is a clean, machine-checked statement about the ground state of a finite system: zero energy and zero entropy coincide exactly at the state where all entries are 1. This gives the framework a rigorous anchor for the third law, while the second law remains an open target, not a theorem.

/-- Equilibrium entropy is zero iff σ = 0. -/
theorem equilibrium_entropy_zero_iff {N : ℕ} (hN : 0 < N) (σ : ℝ) :
    equilibrium_entropy N σ = 0 ↔ σ = 0 := by
  unfold equilibrium_entropy
  have hN_pos : (0 : ℝ) < N := Nat.cast_pos.mpr hN
  constructor
  · intro h
    have hN_ne : (N : ℝ) ≠ 0 := hN_pos.ne'
    have := mul_eq_zero.mp h
    cases this with
    | inl h => linarith
    | inr h =>
      have := J_log_eq_zero_iff.mp h
      exact (div_eq_zero_iff.mp this).resolve_right hN_ne
  · intro h
    rw [h, zero_div, J_log_zero, mul_zero]

/-- **THEOREM (Temperature Is Zero at Unity)**:
    When the energy (log-charge) is zero, the temperature is zero.
    The zero-defect initial state has T = 0 — absolute zero.

    This gives the third law of thermodynamics: the minimum-entropy
    state has zero temperature. -/
theorem temperature_zero_at_unity {N : ℕ} (_hN : 0 < N) :
    rs_temperature N 0 = 0 := by
  unfold rs_temperature
  simp [Real.sinh_zero]

/-- **THEOREM (Absolute Zero Is Unreachable)**:
    If a trajectory starts with σ ≠ 0, it remains at σ ≠ 0 for all
    future times (because the variational dynamics conserves log-charge).

    This means a system with T ≠ 0 can never reach T = 0 — the
    third law in its strong (unattainability) form. -/
theorem absolute_zero_unreachable {N : ℕ}
    (traj : Trajectory N)
    (h : IsVariationalTrajectory traj)
    (h_init : log_charge (traj 0) ≠ 0) :
    ∀ t, log_charge (traj t) ≠ 0 := by
  intro t
  induction t with
  | zero => exact h_init
  | succ n ih =>
    have h_step := h n
    have h_feas : log_charge (traj (n + 1)) = log_charge (traj n) := h_step.1
    rw [h_feas]
    exact ih

/-- Total defect is non-increasing for the full ledger. -/
theorem full_defect_monotone {N : ℕ}
    (traj : Trajectory N)
    (h : IsVariationalTrajectory traj) :
    ∀ t, total_defect (traj (t + 1)) ≤ total_defect (traj t) :=
  trajectory_defect_monotone traj h

What this page does not claim

The theorem does not prove a second law of thermodynamics; the only directional theorem shows total defect is non-increasing. The theorem does not establish that entropy is zero for any configuration with zero energy; that is a separate statement for all configurations. The theorem does not claim that the framework's temperature is the same as the empirical temperature of a physical system.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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