Cosmology Recognition Equilibrium Total Cost Eq Zero Iff

The ground state of consensus

In the Recognition Science framework, a recognition event is a discrete record of one part of a system taking note of another, and the cost of that event is a forced number, not a chosen one. The framework's central result derives a specific cost function J(x) = (x + 1/x)/2 - 1, which is zero only when the two parts are in exact ratio one. The theorem totalCost_eq_zero_iff extends this from a single pair to an entire network: it proves that the total cost, summed over all coupled edges of a graph, is zero if and only if every edge connects two regions with equal recognition levels. In plainer words, the only configuration that costs nothing is one where every linked pair agrees completely.

The theorem is part of a larger machine-checked library of formal theorems, and it is proved with no gaps and no special axioms beyond the three standard logical ones. The proof runs through a chain of exact facts. First, the single-pair cost J(x) is nonnegative, and it is zero exactly when x equals 1. Second, the ratio between two regions is written as phi raised to the power of the difference in their levels, where phi is the golden ratio, about 1.618. Since phi to any nonzero power is never 1, the cost for a pair vanishes exactly when the two levels are equal. Summing this over all edges gives the network result: the total cost is a sum of nonnegative terms, so it is zero precisely when every term is zero, which happens precisely when every edge joins equal levels.

The theorem's significance is that it identifies the ground state of the framework's forward dynamics. The dynamics evolves a field of recognition levels by repeatedly resolving coupled pairs, sending both endpoints to their mean. The library proves that this resolution conserves the total level sum and decreases the variance by exactly half the squared difference. So the spread never grows, and the system provably descends toward the state where every coupled pair is equal. The theorem closes the loop: that consensus state is exactly the zero-cost ground state, and no other configuration achieves zero cost.

What the theorem does not claim is just as important. It does not say that the dynamics always reaches this ground state in finite time, nor that the total edge demand, as opposed to the variance, is monotone along the descent. It also does not address what happens in the driven, open-system extension where new conjugate pairs are born at the horizon; there, the conserved charge sum keeps the system from homogenizing, and the equilibrium theorem does not apply. The theorem is a statement about closed-system consensus, nothing more.

/-- **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 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]

/-- **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 claim the dynamics reaches the zero-cost state in finite time. The theorem does not claim the total edge demand is monotone along the descent. The theorem does not apply to the driven open-system extension with conjugate births.

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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