Foundation Existence Uniqueness From Cost Cost Zero Set Has One Member

The single zero-cost point

A cost function assigns a price to every comparison. In the Recognition Science framework, the cost of recognizing one positive number as another is written J(x), where x is the ratio of the two numbers. The framework's central result, proved in a machine-checked library of formal theorems, is that this cost must take the form J(x) = (x + 1/x)/2 - 1. The question this page answers is simpler: which positive numbers, if any, have a cost of exactly zero? The answer, proved as the theorem cost_zero_set_has_one_member, is that only the number 1 does.

That conclusion is the uniqueness half of a larger existence claim. The framework's earlier work showed that J(x) = 0 if and only if x = 1, which already says the zero-cost set contains 1 and nothing else. The companion theorem makes the uniqueness explicit: if two positive numbers x and y both have J-cost zero, then x = y. The proof is short and runs through the earlier singleton result. A supporting theorem, jcost_isolated_from_zero, adds that any positive number other than 1 has strictly positive cost, so the zero is not merely unique but isolated: there is no cluster of near-zero-cost points around it.

The framework reads this as a structural fact about recognition itself. Existence is not plural: there cannot be two distinct cost minima on the positive real line. The cost function has one and only one place where it touches zero, and that place is the neutral point where the two numbers being compared are equal. The isolation property matters for stability: small perturbations away from 1 do not produce nearly free recognitions, they produce recognitions with a cost bounded away from zero.

What the theorem does not claim is equally important. It does not say that recognition events are impossible at other ratios, only that they are never free. It does not assign physical meaning to the number 1 beyond being the ratio of a number to itself; it is a statement about the cost function's shape, not about which objects exist in the world. The theorem also does not derive the cost function itself. It assumes the form J(x) = (x + 1/x)/2 - 1 and proves facts about that form. The derivation of the form from first principles is a separate, earlier result in the framework.

/-- 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

/-- 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]

/-- 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

What this page does not claim

The theorem does not derive the cost function J; it assumes the form and proves properties of it. The theorem does not say recognition events are impossible at other ratios, only that they are never free. The theorem does not assign physical existence to the number 1 beyond being the ratio of a number to itself.

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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Derived articles

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