Cosmology Recognition Equilibrium Cost Phi Eq Zero Iff

The zero-cost condition

The declaration cost_phi_eq_zero_iff is a theorem in the machine-checked library of formal theorems. It states that for any two real numbers a and b, the recognition cost Jcost(phi^(a-b)) equals zero if and only if a equals b. Here phi is the golden ratio, approximately 1.618, and Jcost is the cost function from the framework's forcing chain, defined as J(x) = (x + 1/x)/2 - 1. The theorem is a corollary of two earlier results: jcost_eq_zero_iff, which says the cost is zero only at ratio one, and phi_rpow_eq_one_iff, which says a power of phi equals one only when the exponent is zero.

In plain terms, the theorem pins down the single configuration where the recognition cost vanishes. The cost is a measure of imbalance between two quantities. The theorem says that imbalance is zero exactly when the two quantities are equal. This is not a statement about what the cost is, but about when it is zero. It is a sharp condition, not an approximation.

Within the framework, this result is part of a larger convergence story. The library also proves that a dynamics which repeatedly resolves pairs by sending them to their mean conserves the total level sum, lowers the variance by exactly half the squared difference, and never increases the variance. The cost is nonnegative, and the total cost over a graph is zero if and only if the field is constant on every edge, a state the library calls consensus. The cost_phi_eq_zero_iff theorem is the local version of that global consensus condition.

The theorem does not claim that the cost function is derived from first principles here; that is a separate result. It does not claim that the dynamics always reaches consensus in finite time; the library proves convergence properties, not a rate. It does not claim that the golden ratio is the only base that works; the theorem uses phi because the framework forces it, but the algebraic identity holds for any positive base not equal to one.

/-- Two regions carry no forced distinction exactly when their levels are equal. -/
theorem cost_phi_eq_zero_iff (a b : ℝ) :
    Jcost ((Constants.phi : ℝ) ^ (a - b)) = 0 ↔ a = b := by
  rw [jcost_eq_zero_iff (Real.rpow_pos_of_pos Constants.phi_pos _),
      phi_rpow_eq_one_iff, sub_eq_zero]

/-- The recognition cost is nonnegative. -/
theorem jcost_nonneg {x : ℝ} (hx : 0 < x) : 0 ≤ Jcost x := by
  rcases eq_or_ne x 1 with h | h
  · subst h; norm_num [Jcost]
  · exact le_of_lt (RefineTrigger.jcost_pos hx h)

/-- The recognition cost vanishes exactly at ratio one. -/
theorem jcost_eq_zero_iff {x : ℝ} (hx : 0 < x) : Jcost x = 0 ↔ x = 1 := by
  constructor
  · intro h0
    by_contra hne
    have hp : 0 < Jcost x := RefineTrigger.jcost_pos hx hne
    rw [h0] at hp
    exact lt_irrefl 0 hp
  · intro h; subst h; norm_num [Jcost]

/-- **The recognition ground state is consensus.** The total recognition cost vanishes if
and only if the level field is constant on every coupled pair. So the zero-cost
configuration the descent converges to is exactly graph consensus. -/
theorem totalCost_eq_zero_iff {n : ℕ} (x : Fin n → ℝ) (edges : Finset (Fin n × Fin n)) :
    totalCost x edges = 0 ↔ ∀ e ∈ edges, x e.1 = x e.2 := by
  rw [totalCost, Finset.sum_eq_zero_iff_of_nonneg
        (fun _ _ => jcost_nonneg (Real.rpow_pos_of_pos Constants.phi_pos _))]
  constructor
  · intro h e he; exact (cost_phi_eq_zero_iff _ _).mp (h e he)
  · intro h e he; exact (cost_phi_eq_zero_iff _ _).mpr (h e he)

/-- **Sigma is conserved.** Resolving a pair leaves the total level unchanged. -/
theorem pairResolve_levelSum {n : ℕ} (x : Fin n → ℝ) {i j : Fin n} (h : i ≠ j) :
    levelSum (pairResolve x i j) = levelSum x := by
  unfold levelSum
  have hagree : ∀ k, k ≠ i → k ≠ j → pairResolve x i j k = x k :=
    fun k hi hj => pairResolve_other x hi hj
  have hsp := sum_split_pair (pairResolve x i j) x h hagree
  rw [pairResolve_at_i, pairResolve_at_j] at hsp
  -- hsp : (∑ resolved) - (∑ x) = ((xi+xj)/2 + (xi+xj)/2) - (x i + x j)
  have hzero : ((x i + x j) / 2 + (x i + x j) / 2) - (x i + x j) = 0 := by ring
  rw [hzero] at hsp
  linarith [hsp]

/-- **The variance Lyapunov law.** Each forced resolution lowers the level variance by
exactly `(x i - x j)^2 / 2`. The decrement is the recognition gap that was resolved, so
the descent is strict until every coupled pair is equal. -/
theorem variance_pairResolve {n : ℕ} (x : Fin n → ℝ) {i j : Fin n} (h : i ≠ j) :
    variance (pairResolve x i j) = variance x - (x i - x j) ^ 2 / 2 := by
  unfold variance
  rw [meanLevel_pairResolve x h, varAround_pairResolve x h (meanLevel x)]

What this page does not claim

The theorem does not prove that the cost function is the only possible one. The theorem does not assert that the dynamics reaches consensus in finite time. The theorem does not claim that the golden ratio is the only base for which the identity holds.

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

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