Holography Gibbs Uniqueness Equilibrium Reference Nonvacuous

The nonvacuous equilibrium

In statistical mechanics, a Gibbs state is the probability distribution that minimizes free energy at a fixed temperature. The machine-checked library of formal theorems proves that this minimizer always exists: for any finite set of energy levels and any inverse temperature, the Gibbs state is a genuine probability distribution and it attains the minimum free energy. This is the theorem equilibriumReference_nonvacuous in plain language: the variational principle is not empty, there is always a state that achieves the bound.

The deeper result is that the exponential form is forced. The library proves that any reference state which minimizes free energy must equal the Gibbs state, record by record. The exponential form is no longer posited as a model assumption; it is derived as the unique solution of the variational problem. This is a theorem, not a definitional choice, and it holds unconditionally for finite state spaces.

What the declaration does not claim is the physical identification. The theorem assumes that the reference state minimizes free energy at a specific inverse temperature, the deficit-free period. Why the seam closure state should be that minimizer is a separate question, a stationarity or second-law statement that remains open. The theorem also does not by itself establish the Bekenstein bound; the full assembly requires additional premises about the partition function and the horizon rate.

/-- The premise is non-vacuous: the Gibbs state itself satisfies it. -/
theorem equilibriumReference_nonvacuous {n : ℕ} (hn : 0 < n)
    (beta : ℝ) (En : Fin n → ℝ) :
    EquilibriumReference (gibbsState beta En) beta En :=
  ⟨fun i => (gibbsState_pos hn beta En i).le,
   gibbsState_normalized hn beta En,
   free_energy_gibbsState hn beta En⟩

/-- **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

/-- **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 establish that any physical system actually reaches the free-energy minimum. The theorem does not prove the Bekenstein bound; it only provides one premise for the conditional assembly. The theorem does not identify the inverse temperature with the horizon rate; that is a separate model premise.

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.

Derived articles

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