Foundation Ground State Dynamics Equilibrium Entries Eq Uniform

The Equilibrium Ledger

In the Recognition Science framework, a ledger is a discrete record of events, and a configuration is a list of positive entries recording how those events are weighted. The declaration equilibrium_entries_eq_uniform is a machine-checked theorem stating that any equilibrium configuration, one that sits at a stable rest point of the variational dynamics, must equal the uniform configuration for its sector. The sector is fixed by the logarithm of the total charge, a conserved quantity that the dynamics cannot change. The proof is short: it applies the unique variational step rule to show that an equilibrium is also the unique minimizer of its conserved sector, and the uniform configuration is that minimizer.

The theorem's force appears in a special case. When the total charge is zero, the uniform configuration has every entry equal to 1. The corollary zero_charge_equilibrium_is_unity states this directly: in a zero-charge sector, the unique equilibrium is the unity configuration, where every entry is 1. The framework then packages a single ratio r as a one-channel configuration and applies the corollary. The result, stable_zero_charge_ratio_eq_one, says that if a positive ratio r is an equilibrium and its log-charge is zero, then r = 1. In plain terms: in a neutral sector, stability forces the ratio to be exactly one, not merely close to it.

This is a theorem about the framework's own variational ledger, not a statement about any physical system. It does not claim that all equilibria in every sector are unity; only the zero-charge ones are. It does not claim that the dynamics always reaches an equilibrium, only that if a configuration is one, it has the uniform form. The theorem also does not derive the golden ratio or any other constant; it is a local stability result that feeds into the larger forcing chain but stands on its own as a statement about the ledger's rest points.

/-- 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 claim that equilibria in nonzero-charge sectors are unity. The theorem does not claim that the dynamics always converges to an equilibrium. The theorem does not derive the golden ratio or any other constant.

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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