Cosmology Recognition Equilibrium Mean Level Pair Resolve

Averaging two levels

In the Recognition Science account, a recognition event is a forced adjustment between two coupled regions. The operation called pair resolve, a discrete record of a forced adjustment, replaces the two levels of a pair with their arithmetic mean. The theorem meanLevel_pairResolve states that this operation leaves the mean level of the entire field unchanged. If the field holds levels 2 and 4 on two sites and 3 on a third, the mean is 3; after resolving the pair, the levels become 3, 3, 3, and the mean is still 3.

The same theorem is the keystone of a larger convergence fact. Each resolution also lowers the variance, the sum of squared deviations from the mean, by exactly half the squared difference of the resolved pair. For levels 2 and 4, the drop is (2-4)^2 / 2 = 2. Repeating resolutions on a fixed graph of couplings drives the variance to zero, which means every coupled pair becomes equal. The recognition cost, a measure of imbalance, is nonnegative and reaches zero only at that consensus state. Thus the dynamics provably relaxes to a uniform configuration, and the mean level is the conserved quantity that pins down which uniform configuration.

This is a formal result in the framework's machine-checked library of formal theorems, not a numerical simulation. The theorem is stated for a finite field of real-valued levels on a graph, with the pair resolution defined exactly as replacing two entries by their average. The proof is axiom-clean, using only the three standard axioms of the underlying type theory. The result is a theorem about the defined operation, not an empirical claim about any physical system.

The theorem does not claim that the dynamics always reaches consensus in finite time, nor that the total edge demand, a different aggregate, is monotone. It does not claim that the mean level is conserved under any other operation, such as a birth of new levels. The result is specific to the pair-resolve move and its exact variance decrement.

What the theorem changes is the picture of the framework's ground state. The ground state is not a special configuration chosen by hand; it is the unique configuration forced by the dynamics, and it is characterized by consensus on every edge. The mean level is the single number that survives the entire descent, making it the natural label for the equilibrium state.

theorem meanLevel_pairResolve {n : ℕ} (x : Fin n → ℝ) {i j : Fin n} (h : i ≠ j) :
    meanLevel (pairResolve x i j) = meanLevel x := by
  unfold meanLevel; rw [pairResolve_levelSum x h]

/-- **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 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)

/-- The forced recognition resolution of one coupled pair: send both endpoints to their
mean. This is the sigma = 0, J-minimal move the forward dynamics posts each tick. -/
noncomputable def pairResolve {n : ℕ} (x : Fin n → ℝ) (i j : Fin n) : Fin n → ℝ :=
  fun k => if k = i ∨ k = j then (x i + x j) / 2 else x k

What this page does not claim

The theorem does not claim the dynamics reaches consensus in finite time. The theorem does not claim the total edge demand is monotone. The theorem does not claim the mean level is conserved under births of new levels.

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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