Cosmology Recognition Equilibrium Var Around Pair Resolve

The pair resolution

In mathematics, a spread is a measure of how far a set of numbers is from a chosen center. The simplest spread is the sum of squared distances from that center. If the center is the average of the numbers, that sum is the variance, a central quantity in statistics and least squares. The declaration varAround_pairResolve concerns a particular operation on two numbers and what that operation does to such a sum.

The operation is called a pair resolution. Given two numbers, it replaces both by their average, their midpoint. The declaration proves an exact identity: after resolving a pair, the sum of squared distances from any fixed center decreases by exactly half the squared difference of the two original numbers. In symbols, if the two numbers are a and b, the drop is (a - b)² / 2. The center need not be the mean; the identity holds for any chosen center, which is what makes the result general.

This identity is a theorem in a machine-checked library of formal theorems. It is not a numerical approximation or a heuristic. The library proves it from the definition of the resolution and the algebraic properties of squares and sums. The proof is complete and axiom-clean, resting only on the standard logical axioms of the underlying type theory.

In Recognition Science, this theorem becomes a tool for a cosmology. The framework models a universe as a discrete record of events, a ledger, where each tick one forced recognition event occurs: a coupled pair of regions is resolved to their mean. The theorem shows that this dynamics has a Lyapunov function, a quantity that strictly decreases with each tick. The variance of the recognition levels drops by exactly the squared difference of the resolved pair, so the spread never grows. The dynamics is a strict descent until every coupled pair is equal, which is the consensus state where the recognition cost is zero.

The theorem does not claim that the total edge demand, the sum of costs over all coupled pairs, is monotone. It does not claim that the dynamics reaches consensus in finite time, only that it descends. It does not claim that the center of the spread is preserved, only that the spread itself decreases. The theorem is a local statement about one pair resolution, and its global consequences for the whole ledger are separate theorems in the library.

/-- **The exact variance drop (around any reference).** Resolving a pair lowers the spread
by exactly `(x i - x j)^2 / 2`, independent of the reference point. -/
theorem varAround_pairResolve {n : ℕ} (x : Fin n → ℝ) {i j : Fin n} (h : i ≠ j) (c : ℝ) :
    varAround (pairResolve x i j) c = varAround x c - (x i - x j) ^ 2 / 2 := by
  unfold varAround
  have hagree : ∀ k, k ≠ i → k ≠ j →
      (pairResolve x i j k - c) ^ 2 = (x k - c) ^ 2 :=
    fun k hi hj => by rw [pairResolve_other x hi hj]
  have hsplit := sum_split_pair (fun k => (pairResolve x i j k - c) ^ 2)
    (fun k => (x k - c) ^ 2) h hagree
  simp only [pairResolve_at_i, pairResolve_at_j] at hsplit
  -- hsplit : (∑ resolved sq) - (∑ x sq) = (2 * ((xi+xj)/2 - c)^2) - ((xi-c)^2 + (xj-c)^2)
  have hid : (((x i + x j) / 2 - c) ^ 2 + ((x i + x j) / 2 - c) ^ 2)
      - ((x i - c) ^ 2 + (x j - c) ^ 2) = -((x i - x j) ^ 2 / 2) := by ring
  rw [hid] at hsplit
  linarith [hsplit]

/-- **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)]

/-- The spread never grows under a forced resolution. -/
theorem variance_nonincreasing {n : ℕ} (x : Fin n → ℝ) {i j : Fin n} (h : i ≠ j) :
    variance (pairResolve x i j) ≤ variance x := by
  rw [variance_pairResolve x h]
  nlinarith [sq_nonneg (x i - x j)]

/-- **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)

What this page does not claim

The total edge demand is monotone under pair resolution. The dynamics reaches consensus in finite time. The center of the spread is preserved by the resolution.

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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