Foundation Rhat From Jcost Gradient Jcost Lyapunov Unique Fixed Point

The unique resting point

A process that repeatedly adjusts a positive number, always moving it toward a target value of 1. The rule might take a big step or a small one, but each step must make the number closer to 1 in a specific sense. The question is whether such a process can get stuck somewhere else. The answer, under the framework's conditions, is no: any rule that always reduces the cost function J(x) = (x + 1/x)/2 - 1 and keeps 1 as a fixed point must have 1 as its only fixed point. This is the content of the theorem jcost_lyapunov_unique_fixed_point.

The theorem is a statement about a cost function, a measure of how far a number is from an ideal state. The framework's cost function has a unique minimum at x = 1, where it equals zero. The theorem considers any smooth function f that maps positive numbers to positive numbers, satisfies f(1) = 1, and strictly decreases the cost at every step away from 1. Under these conditions, the proof shows that if f(y) = y for some positive y, then y must be 1. The argument is a simple contradiction: if y were a fixed point other than 1, then applying f would both leave y unchanged and strictly decrease its cost, which is impossible.

This uniqueness result is a structural fact about the framework's dynamics. It does not say that the midpoint map x ↦ (x + 1)/2 is the only possible update rule. It says that among all rules that decrease the cost and fix 1, the only place any of them can stop is at 1. The midpoint map is one such rule, and the framework's library proves that it decreases the cost and fixes 1, but the uniqueness theorem applies to the whole class of cost-decreasing rules. The theorem is a statement about the shape of the cost function, not about a specific algorithm.

The result matters because it gives the framework a stable foundation. If the cost function is the measure of how far a recognition event is from perfect, then the unique resting point of any cost-reducing process is the perfect state. The theorem does not claim that any particular process will reach that state, only that no process satisfying the conditions can settle anywhere else. This is a guarantee about the shape of the cost function, not about the path taken across it.

In Recognition Science, this theorem is part of the derivation of the recognition operator from the cost function. The framework models a recognition event as a discrete record of a comparison, and the cost of that recognition is forced by the five conditions that lead to J. The uniqueness theorem shows that the cost function has a single basin of attraction, which the framework identifies with the emergence of a preferred update rule. The theorem is a formal result in the framework's machine-checked library of formal theorems, with no unproved assumptions beyond the standard axioms of the underlying logic.

/-- The unique fixed point of any J-cost-decreasing map with J as Lyapunov
    function is x = 1. -/
theorem jcost_lyapunov_unique_fixed_point {f : ℝ → ℝ}
    (hfixed : f 1 = 1)
    (hdecreasing : ∀ x : ℝ, 0 < x → x ≠ 1 → Jcost (f x) < Jcost x) :
    ∀ y : ℝ, 0 < y → f y = y → y = 1 := by
  intro y hy hfy
  by_contra hne
  exact absurd (hdecreasing y hy hne) (by rw [hfy]; exact lt_irrefl _)

What this page does not claim

The theorem does not claim that the midpoint map is the only update rule that decreases the cost. The theorem does not claim that any particular process will actually reach the fixed point at 1. The theorem does not claim anything about the behavior of the cost function outside the positive real numbers.

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

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