Foundation Ground State Dynamics Stable Zero Charge Ratio Eq One

The stable ratio

A ratio compares two amounts. In many physical and economic models, a system settles into a state where that ratio stops changing. The Recognition Science framework studies such states through a discrete record of events called a ledger, where each entry is a positive real number. The theorem stable_zero_charge_ratio_eq_one states a precise condition under which the only stable ratio is 1, meaning the two amounts must be equal.

The framework defines a configuration as a finite list of positive numbers. A configuration is an equilibrium when it is a minimizer of a variational cost, a point where the system has no tendency to move. The theorem considers a one-channel ratio, a configuration with a single entry r. It assumes two things: that this configuration is an equilibrium, and that its log-charge is zero. The log-charge is the sum of the natural logarithms of the entries, a measure of the total multiplicative size of the system.

Under these two assumptions, the theorem proves r = 1. The proof is short and direct. It first uses a general result, equilibrium_entries_eq_uniform, which says any equilibrium coincides with the uniform configuration of its conserved sector. A uniform configuration has all entries equal. Applying this to the one-channel ratio, the equilibrium must be uniform, so its single entry is some value. The zero-charge assumption then forces that value to be 1, because the log of 1 is 0, and the log-charge of a uniform configuration with entry r is the log of r.

The result is a theorem in the framework's machine-checked library of formal theorems. It is not a statement about the physical world by itself. It is a statement about the framework's own definitions: given the framework's notion of equilibrium and charge, a stable zero-charge ratio must be unity. The theorem does not say that all real-world ratios are 1. It does not say that a zero-charge sector always exists. It establishes a conditional fact within the framework's model.

The consequence is that within the framework, balance is not a choice. If a system is in equilibrium and its conserved charge is zero, the only stable ratio is equality. This is a structural result, not an empirical one. It shows that the framework's definitions are consistent and that a particular kind of stability is uniquely determined.

/-- 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

/-- 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)

What this page does not claim

The theorem does not claim that all real-world ratios are 1. The theorem does not claim that a zero-charge sector always exists. The theorem does not claim that the framework's definitions correspond to any particular 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/GroundStateDynamics.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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