Holography Gibbs Uniqueness Equilibrium Forces Gibbs Form

The forced form

The Gibbs state is the probability distribution over a system's possible states that appears throughout statistical mechanics. For a system with energy levels E_i at inverse temperature β, the Gibbs state assigns to state i the probability exp(−βE_i)/Z, where Z is the partition function that makes the probabilities sum to one. This exponential form is usually introduced as a postulate or derived from a maximization of entropy subject to a fixed average energy. The framework's result concerns a different route to the same destination.

In Recognition Science, the declaration equilibrium_forces_gibbs_form proves that the exponential form is not merely assumed but forced. The framework models a reference state as a ledger, a discrete record of probabilities assigned to each possible state. The key premise is not that the reference has the exponential form, but that it minimizes a free energy functional, the quantity β times average energy minus entropy. The theorem states that any such minimizer must equal the Gibbs state, record by record. This is a proved result in the framework's machine-checked library of formal theorems, with no framework-specific axioms.

The proof proceeds through a variational principle. For any probability distribution p over the states, the free energy F(p) = β⟨E⟩_p − S(p) is bounded below by −log Z, where S is the Shannon entropy. The bound is attained exactly at the normalized Gibbs state, and nowhere else. The equality condition for the Gibbs inequality, which says that entropy equals cross-entropy only when the two distributions are identical, upgrades the usual tangent bound log x ≤ x − 1 to its strict form, forcing every slack term to vanish. The exponential form emerges as the unique solution of the minimization problem, not as an input.

In Recognition Science, this result sharpens an earlier assembly. Previously, the framework posited that the reference state had the exponential form as a model premise. Now the premise is strictly weaker: the reference minimizes free energy. The exponential form is derived from that premise, and the same entropy bound follows. The remaining physics burden is why the reference state minimizes free energy at the deficit-free period β = 2π/κ, a stationarity statement that the framework treats as a target for its KMS lane.

What the declaration does not claim is equally precise. It does not establish that any physical system actually reaches the Gibbs state; it establishes a conditional: if a reference minimizes free energy, then it has the Gibbs form. The theorem does not derive the value of β or the energy levels; those remain inputs. The variational principle itself is unconditional mathematics, but the identification of the reference state with a physical equilibrium is a model premise, not a proved result.

/-- **The exponential form is forced.** Any free-energy-minimizing reference IS the
normalized Gibbs state, record by record. The form assumption in `GibbsReference` is
hereby a theorem downstream of the variational premise. -/
theorem equilibrium_forces_gibbs_form {n : ℕ} {q : Fin n → ℝ} {beta : ℝ}
    {En : Fin n → ℝ} (h : EquilibriumReference q beta En) :
    q = gibbsState beta En := by
  obtain ⟨hq0, hq1, hF⟩ := h
  have hn : 0 < n := by
    rcases Nat.eq_zero_or_pos n with rfl | hn
    · exact absurd hq1 (by simp)
    · exact hn
  exact (free_energy_eq_iff hn q beta En hq0 hq1).mp hF

/-- **Uniqueness of the minimizer.** `F(p) = −log Z` iff `p` IS the Gibbs state. -/
theorem free_energy_eq_iff {n : ℕ} (hn : 0 < n) (p : Fin n → ℝ)
    (beta : ℝ) (En : Fin n → ℝ)
    (hp : ∀ i, 0 ≤ p i) (hp1 : ∑ i, p i = 1) :
    freeEnergyFunctional beta En p = -Real.log (partitionFunction beta En)
      ↔ p = gibbsState beta En := by
  have hcross := crossEntropy_gibbsState hn p beta En hp1
  have hiff := gibbs_inequality_eq_iff p (gibbsState beta En) hp
    (gibbsState_pos hn beta En) hp1 (le_of_eq (gibbsState_normalized hn beta En))
  constructor
  · intro hF
    apply hiff.mp
    unfold freeEnergyFunctional at hF
    linarith
  · intro hpg
    have hS := hiff.mpr hpg
    unfold freeEnergyFunctional
    linarith

/-- **The exponential form is forced.** Any free-energy-minimizing reference IS the
normalized Gibbs state, record by record. The form assumption in `GibbsReference` is
hereby a theorem downstream of the variational premise. -/
theorem equilibrium_forces_gibbs_form {n : ℕ} {q : Fin n → ℝ} {beta : ℝ}
    {En : Fin n → ℝ} (h : EquilibriumReference q beta En) :
    q = gibbsState beta En := by
  obtain ⟨hq0, hq1, hF⟩ := h
  have hn : 0 < n := by
    rcases Nat.eq_zero_or_pos n with rfl | hn
    · exact absurd hq1 (by simp)
    · exact hn
  exact (free_energy_eq_iff hn q beta En hq0 hq1).mp hF

What this page does not claim

The theorem does not prove that any physical system reaches the Gibbs state. The theorem does not derive the value of the inverse temperature β or the energy levels E_i. The identification of the free-energy minimizer with a physical equilibrium is a model premise, not a proved result.

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/Holography/GibbsUniqueness.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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