Foundation Existence Uniqueness From Cost Existence Uniqueness Cert

The uniqueness certificate

The cost function J(x) = (x + 1/x)/2 - 1 has a simple property: it is zero only when x equals 1. For any positive number x, J(x) = 0 if and only if x = 1. This is not an approximation or a numerical observation; it is a theorem proved in a machine-checked library of formal theorems. The library verifies every step of the proof, so the result carries no hidden assumptions beyond the standard logical foundations of mathematics.

The same library also proves a stronger statement: there cannot be two distinct positive numbers where the cost is zero. If J(x) = 0 and J(y) = 0 for positive x and y, then x = y. This is the uniqueness half of the claim. The cost function also has a symmetry property: J(x) = J(1/x), meaning it treats a number and its reciprocal the same way. And it is isolated from zero: any positive number other than 1 has strictly positive cost, so the zero at x = 1 is not part of a flat region or a cluster of nearby zeros.

These four facts are packaged together in a single declaration called ExistenceUniquenessCert, which is a certificate: a structured object that bundles the zero-iff-one statement, the uniqueness statement, the symmetry statement, and the isolation statement into one unit. The declaration is not itself a theorem; it is a definition of what a complete existence-and-uniqueness certificate must contain. The library then provides a concrete instance of this certificate, built from the individual theorems, showing that the cost function genuinely satisfies all four requirements.

In Recognition Science, this certificate is used to support the idea that a certain recognition event has a unique fixed point. The framework models recognition as a discrete record of events, and the cost function measures the price of a recognition event. The certificate shows that within this model, there is exactly one point where the cost is zero, which the framework identifies with the unit of recognition. This is a mathematical fact about the cost function, not a physical measurement.

What the certificate does not claim is just as important. It does not say that the cost function is the only possible cost function, or that the universe actually uses this particular function. It does not claim that x = 1 corresponds to any specific physical object or event. It only establishes a property of a defined mathematical object. The certificate is a piece of pure mathematics, and any physical interpretation is a separate step that the framework makes elsewhere, not a consequence of this declaration alone.

/-- 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]

/-- J-cost is symmetric in log-ratio sense. -/
theorem jcost_log_symmetric {x : ℝ} (hx : 0 < x) :
    Jcost x = Jcost x⁻¹ := Jcost_symm hx

/-- 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 certificate does not claim that J is the only possible cost function. The certificate does not claim that x = 1 corresponds to any specific physical object. The certificate does not claim that the universe actually uses this cost function.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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