Foundation Ground State Dynamics

The stable ground state

Foundation ground state dynamics is the study of what a recognition system does when nothing pushes on it. The framework models a system as a ledger, a discrete record of events, whose state is a configuration of positive numbers, one per channel. Dynamics enters through a variational rule: the next state is chosen to lower a conserved quantity, much as a ball rolls downhill until it reaches a valley floor. The first theorem of the module states that any equilibrium, any state that persists unchanged, coincides exactly with the uniform minimizer of its conserved sector. In plain words, a system at rest has no choice but to sit at the lowest point of its own configuration space.

The second theorem sharpens this for a special sector. Each configuration carries a quantity called its log-charge, a measure of how far its entries sit from unity on a logarithmic scale. When that charge is zero, the sector is neutral, and the unique equilibrium is the unity configuration, the state where every channel holds the number 1. The proof is short and direct: the equilibrium must equal the uniform minimizer, and in a zero-charge sector the uniform minimizer is exactly the all-ones state. This is not a numerical coincidence; it is a forced equality, derived from the variational rule and the definition of charge.

The module then packages this result for the simplest possible system, a single channel holding one ratio r. A one-channel ratio is just a positive real number, and its log-charge is the ordinary natural logarithm of r. In the neutral sector, where log r equals zero, the stability condition forces r to equal 1. The theorem stable_zero_charge_ratio_eq_one states this cleanly: a stable one-channel ratio in the neutral sector is unity. This is the dynamic counterpart of the static cost result; where the cost function forces the golden ratio as the unique self-similar scaling, the dynamic rule forces the neutral ground state to be the identity.

What this changes is the picture of what "rest" means in the framework. Rest is not a default or a choice; it is a consequence of the variational rule, and in a neutral sector it is unique. The all-ones configuration is not merely a convenient baseline; it is the only state that survives the requirement of stability. This gives the framework a concrete anchor point: when a system carries no net charge, its ground state is the unity configuration, and any deviation from unity requires a nonzero charge to sustain it.

/-- 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 module does not derive the value of the golden ratio or any other constant. The module does not claim that all physical systems reach their equilibrium. The module does not address what happens outside the zero-charge sector.

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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