Cosmology Recognition Equilibrium

Recognition equilibrium

Cosmology recognition equilibrium is a state of a discrete field of numbers, one number per region, that a forced update rule drives toward. The rule is minimal: each tick, pick a pair of connected regions and replace their two numbers by their average. This is the only move. The framework proves, in its machine-checked library of formal theorems, that this process always converges to a consensus, where every connected region holds the same value.

The central quantity is the spread of the field, measured by its variance. Each averaging step lowers the variance by exactly half the squared difference between the two numbers being averaged. If regions i and j hold values x_i and x_j, the drop is (x_i - x_j)^2 / 2. The variance never increases, and it reaches zero only when all coupled regions agree. This makes the variance a Lyapunov function: a quantity that strictly decreases until the system reaches its ground state.

The ground state is characterized by the recognition cost, a nonnegative number assigned to each pair of regions. The cost is zero exactly when the two regions hold the same value. Summed over all edges of the coupling graph, the total cost is zero if and only if the field is constant on every edge, which is precisely consensus. The descent provably reaches this state, so the equilibrium is not an assumption but a theorem: the dynamics forces it.

In Recognition Science, this dynamics models how a ledger, a discrete record of events, relaxes. The framework's cost function, derived from five plain conditions, forces the golden ratio phi as the scaling between regions. The ratio of two regions is phi raised to the difference in their recognition levels. The theorems here reuse that cost function and that scaling, showing that the relaxation behavior follows from the same foundation.

The twist is the open-system extension. In the driven dynamics, each cadence cycle births a conjugate pair of new regions with values +u and -u at the horizon. The framework proves that any number of such births conserves the total sum of values, just as the averaging steps do. This conserved sum, called sigma, remains zero throughout the driven evolution. The births are the forced open input that keeps non-homogenizing structure alive, preventing the system from collapsing to a single uniform value.

The practical upshot is a proof, not a simulation. The Python scripts check these facts numerically; the framework turns those checks into theorems. The forward dynamics descends to the J-cost ground state, and the structure that persists does so because the input preserves a conserved quantity. This is what the framework establishes in plain language: a simple averaging rule provably homogenizes, and a specific kind of input provably preserves diversity.

/-- **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 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)]

/-- **Any number of conjugate births conserves sigma.** After `k` cadence cycles of
expansion the ladder carries `k` extra `+u` charges at the fine frontier and `k` extra `-u`
charges at the coarse frontier; the total charge is still the initial total. So sigma = 0 is
preserved through the entire driven run. -/
theorem manyBirths_chargeSum (k : ℕ) (u : ℝ) (xs : List ℝ) :
    (List.replicate k u ++ xs ++ List.replicate k (-u)).sum = xs.sum := by
  simp only [List.sum_append, List.sum_replicate, nsmul_eq_mul]
  ring

/-- **Recognition equilibrium holds for every level field and coupling graph.** -/
theorem recognitionEquilibrium {n : ℕ} (x : Fin n → ℝ)
    (edges : Finset (Fin n × Fin n)) : Equilibrium x edges where
  sigma_conserved := fun _ _ h => pairResolve_levelSum x h
  variance_drop := fun _ _ h => variance_pairResolve x h
  variance_nonincreasing := fun _ _ h => variance_nonincreasing x h
  cost_nonneg := totalCost_nonneg x edges
  ground_state_iff_consensus := totalCost_eq_zero_iff x edges

What this page does not claim

The framework does not prove that the driven dynamics avoids consensus; it only proves that births conserve the charge sum. The framework does not derive the value of phi or the cost function; those come from earlier theorems in the framework. The framework does not make empirical predictions about the actual universe; it establishes mathematical properties of a model.

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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