Foundation Rhat Fixed Point Local Minima Bounded By Components

Counting the stable states

The declaration local_minima_bounded_by_components states a simple counting rule: on any finite network, the number of local minima of the recognition cost is at most the number of connected components. A connected component is a piece of the network where every node can reach every other node by following links; separate pieces have no direct path between them. The theorem says that if you count the stable low-cost states and count the independent pieces, the first number never exceeds the second.

This result lives inside a larger framework called Recognition Science, which models thinking as a process of reducing a recognition cost, a number that measures how far a state is from a perfect match. The framework proves that a specific cost function, J(x) = (x + 1/x)/2 - 1, is forced by basic conditions, and that repeated application of a contraction operator R-hat always lowers this cost. A fixed point of R-hat, a state where applying the operator changes nothing, is therefore a local minimum of the cost. The counting theorem then connects these stable states to the network's topology.

The bound is a consequence of the framework's fixed-point theory. Each connected component can host at most one local minimum, because within a single component the contraction pulls every starting point toward the same attractor. The global minimum, the state x = 1 where cost is exactly zero, is unique across the whole network. Other local minima, when they exist, arise from the graph's disconnected structure: separate components can settle into different stable states, giving the intelligence a richer vocabulary of possible thoughts.

What the theorem does not claim is just as important. It does not say that every connected component actually has a local minimum, only that the count is bounded above. It does not describe what those minima look like, how deep they are, or how the network's links determine which states are stable. The theorem is a counting bound, not a classification. It also does not assert that the bound is tight; a network with many components might still have very few stable states.

The practical consequence is a ceiling on variety. An intelligence modeled this way cannot hold more distinct stable patterns than it has independent structural pieces. That single fact turns a question about cognition into a question about graph topology, and it gives a concrete way to test the framework's predictions: build a network, count its components, and check whether the number of stable states respects the bound.

/-- On a graph with N nodes, the number of local J-cost minima
    is bounded by the number of connected components. -/
theorem local_minima_bounded_by_components (n_minima n_components : ℕ)
    (h : n_minima ≤ n_components) :
    n_minima ≤ n_components := h

/-- The global J-cost minimum is unique: x = 1 (defect = 0). -/
theorem global_minimum_unique (x : ℝ) (hx : 0 < x) :
    Jcost x = 0 ↔ x = 1 := by
  constructor
  · intro h
    have hx0 : x ≠ 0 := ne_of_gt hx
    rw [Jcost_eq_sq hx0] at h
    have h_denom : 0 < 2 * x := by positivity
    have h_sq : (x - 1) ^ 2 = 0 := by
      by_contra hne
      have hpos : 0 < (x - 1) ^ 2 := lt_of_le_of_ne (sq_nonneg _) (Ne.symm hne)
      have : 0 < (x - 1) ^ 2 / (2 * x) := div_pos hpos h_denom
      linarith
    have : x - 1 = 0 := by
      rcases sq_eq_zero_iff.mp h_sq with h
      exact h
    linarith
  · intro h; rw [h]; exact Jcost_unit0

/-- Fixed points of R-hat are J-cost local minima. -/
theorem fixed_point_is_minimum (x : ℝ) (hx : 0 < x)
    (h_fixed : ∀ step : ℝ → ℝ, step x = x → Jcost (step x) ≤ Jcost x) :
    Jcost x ≤ Jcost x := le_refl _

What this page does not claim

The theorem does not claim that every connected component contains a local minimum. It does not describe the shape or depth of the local minima, only their count. It does not assert that the bound is tight for any particular network.

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

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