Foundation Existence Uniqueness From Cost Cost Zero Set Singleton

The single zero-cost point

The recognition framework, a system that models physical structure as the cost of storing a discrete record of events, defines a cost function J(x) on positive real numbers. The theorem cost_zero_set_singleton proves, in the machine-checked library of formal theorems, that the set of points where this cost is zero is exactly the singleton {1}. In plain language: for any positive number x, J(x) = 0 if and only if x = 1. The proof is short: it uses the already-proved fact that J(x) is strictly positive whenever x is not 1, and that J(1) = 0 by definition.

The result is stronger than a mere observation. A companion theorem, cost_zero_set_has_one_member, states that any two positive numbers with zero cost must be equal; together these establish uniqueness. The framework's own documentation frames this as answering a claim from an earlier work: existence is not plural. There cannot be two distinct cost minima on the positive reals. The theorem also implies an isolation property: for any positive distance δ, the minimum of J(x) on the set of points at least δ away from 1 is strictly positive. In other words, the zero-cost point is not just unique, it is separated from all other points by a positive cost gap.

This uniqueness is a purely formal consequence of the cost function's algebraic form. It does not depend on any physical measurement or on the framework's later derivations of constants like the golden ratio. The theorem is proved with no axioms beyond the standard logical ones, according to the pack. The cost function itself is defined by J(x) = (x + 1/x)/2 - 1, and the theorem is a direct check of that formula.

What the theorem does not claim is equally important. It does not say that the number 1 is the only possible state of the universe, nor that the cost function is the only possible cost function. It does not claim that the zero-cost point is physically realized; it only says that if a positive number has zero cost, that number must be 1. The theorem is a statement about the cost function's shape, not about which values actually occur in nature. The framework's later steps, which connect this cost to physical constants, are separate claims with their own evidence.

/-- 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

/-- 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

The theorem does not claim that the number 1 is the only possible state of the universe. The theorem does not claim that the cost function J(x) is the only possible cost function. The theorem does not claim that the zero-cost point is physically realized in nature.

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.

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