Foundation Ground State Dynamics Zero Charge Equilibrium Is Unity

The neutral equilibrium

A ledger, a discrete record of events, assigns a positive number to each of its N entries. The framework's dynamics say that a configuration at equilibrium, one that does not change under the update rule, must coincide with the uniform minimizer of its conserved sector. This is a proved theorem in the machine-checked library of formal theorems: any equilibrium equals the uniform configuration determined by its own log-charge, a scalar that measures the total logarithmic deviation of the entries.

When that log-charge is zero, the statement sharpens. The theorem zero_charge_equilibrium_is_unity proves that in a zero-charge sector, the unique equilibrium is the unity configuration, the state where every entry equals 1. The proof is a short chain: an equilibrium equals the uniform configuration at its log-charge, a zero log-charge makes that uniform configuration the unity configuration, and the result follows by definitional equality. The theorem requires only that N is positive and that the configuration is an equilibrium with zero charge; it does not require any additional regularity or boundary condition.

The same result applies to a one-channel ratio observable. A ratio r, packaged as a Configuration 1 with entry r, has log-charge equal to the natural logarithm of r. If such a ratio is stable, an equilibrium, and lies in the neutral sector, then r must equal 1. The theorem stable_zero_charge_ratio_eq_one proves this directly from the unity result. In plain terms: a stable ratio that carries no net charge cannot drift from one.

What the declaration does not claim is as important as what it proves. It does not assert that every configuration reaches equilibrium, only that if a configuration is an equilibrium in a zero-charge sector, it is the unity configuration. It does not claim that the unity configuration is stable in any dynamical sense beyond being an equilibrium; the theorem says nothing about whether small perturbations return to unity. It also does not claim that the log-charge itself is conserved by the dynamics, only that the equilibrium characterization holds for configurations whose log-charge is zero.

/-- Any equilibrium coincides with the uniform minimizer of its conserved sector. -/
theorem equilibrium_entries_eq_uniform {N : ℕ} (hN : 0 < N)
    (c : Configuration N) (hEq : IsEquilibrium c) :
    c.entries = (uniform_config hN (log_charge c)).entries := by
  exact variational_step_unique hN c c (uniform_config hN (log_charge c))
    hEq (uniform_is_variational_successor hN c)

/-- The zero-charge equilibrium is the unity configuration. -/
theorem zero_charge_equilibrium_is_unity {N : ℕ} (hN : 0 < N)
    (c : Configuration N) (hEq : IsEquilibrium c)
    (hCharge : log_charge c = 0) :
    c.entries = (unity_config N hN).entries := by
  calc
    c.entries = (uniform_config hN (log_charge c)).entries :=
      equilibrium_entries_eq_uniform hN c hEq
    _ = (uniform_config hN 0).entries := by rw [hCharge]
    _ = (unity_config N hN).entries := by
      funext i
      simp [uniform_config, unity_config]

/-- Stable one-channel ratios in the neutral sector are forced to unity. -/
theorem stable_zero_charge_ratio_eq_one (r : ℝ) (hr : 0 < r)
    (hEq : IsEquilibrium (ratioConfig r hr))
    (hCharge : log_charge (ratioConfig r hr) = 0) :
    r = 1 := by
  have hEntries :
      (ratioConfig r hr).entries = (unity_config 1 (by norm_num)).entries :=
    zero_charge_equilibrium_is_unity (N := 1) (by norm_num) (ratioConfig r hr) hEq hCharge
  have h0 := congrFun hEntries ⟨0, by simp⟩
  simpa [ratioConfig, unity_config] using h0

What this page does not claim

The theorem does not prove that every configuration converges to equilibrium. The theorem does not establish dynamical stability of the unity configuration under perturbations. The theorem does not assert conservation of log-charge under the update rule.

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

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