Foundation Existence Uniqueness From Cost Jcost Isolated From Zero

One zero, and only one

The cost function J(x) = (x + 1/x)/2 - 1 measures how far a positive number x sits from 1. When x equals 1, the cost is exactly 0. The question is whether any other positive number could also cost nothing. The theorem jcost_isolated_from_zero answers this directly: for any positive x that is not 1, the cost is strictly greater than 0. In plainer terms, the only way to pay zero cost is to be at the value 1; every other positive number costs something positive.

This is a uniqueness statement, and it is stronger than merely saying the cost has a minimum. A minimum could in principle be shared by several points. The theorem rules that out: the set of points with zero cost contains exactly one member. The proof runs through the functional form itself, not through any extra assumptions. Because J(x) is symmetric under swapping x and 1/x, and because the cost is positive whenever x differs from 1, the zero set collapses to the single point {1}. The machine-checked library of formal theorems records this as a theorem with no unproved axioms.

In Recognition Science, this fact carries a specific meaning. The framework models reality as a discrete record of recognition events, and the cost of recognition is forced by the functional equation. The theorem says that the state of zero cost, which the framework identifies with existence, is not plural: there cannot be two distinct positive values both costing nothing. The uniqueness is a property of the cost function alone, derived before any physical content is added.

What the theorem does not claim is just as important. It does not say that the value 1 is the only positive number that exists; it says only that 1 is the only positive number with zero cost under this specific function. It does not assert that the cost function itself is the unique function satisfying the five conditions; that is a separate theorem. And it does not claim that the cost is positive for all x; the statement is restricted to positive x, and the cost at x = 1 is exactly zero by definition.

/-- Away from 1, J-cost is strictly positive (isolation). -/
theorem jcost_isolated_from_zero {x : ℝ} (hx : 0 < x) (hne : x ≠ 1) :
    0 < Jcost x := Jcost_pos_of_ne_one x hx hne

/-- The cost-zero set in ℝ+ has cardinality 1 (in the sense that any two
    members are equal). -/
theorem cost_zero_set_has_one_member {x y : ℝ}
    (hx : 0 < x) (hy : 0 < y)
    (hJx : Jcost x = 0) (hJy : Jcost y = 0) :
    x = y := by
  rw [(cost_zero_set_singleton x hx).mp hJx,
      (cost_zero_set_singleton y hy).mp hJy]

/-- The cost-zero set is exactly {1}. -/
theorem cost_zero_set_singleton :
    ∀ x : ℝ, 0 < x → (Jcost x = 0 ↔ x = 1) := by
  intro x hx
  constructor
  · intro h
    by_contra hne
    exact absurd h (ne_of_gt (Jcost_pos_of_ne_one x hx hne))
  · rintro rfl; exact Jcost_unit0

What this page does not claim

This theorem does not claim that 1 is the only positive number that exists, only that it is the only one with zero cost. This theorem does not establish that the cost function J is the unique function satisfying the five conditions. This theorem does not apply to non-positive numbers; the statement is restricted to x greater than 0.

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

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