Foundation Cost Axioms J Tendsto At Top As X To Zero

The cost of nothing

In the Recognition Science framework, a ledger (a discrete record of events) assigns every positive ratio x a cost J(x) = (x + 1/x)/2 - 1. This cost function measures how far a configuration sits from perfect balance at x = 1, where J(1) = 0. The framework's central theorem proves that any cost function satisfying three primitive axioms, normalization, composition, and calibration, must equal this J. Those axioms are not arbitrary; they encode an economic inevitability about how recognition events must be priced.

The specific declaration J_tendsto_atTop_as_x_to_zero establishes a precise limit statement: as x approaches zero from above, J(x) tends to positive infinity. In plainer terms, the cost of being at a ratio arbitrarily close to nothing grows without any upper bound. The machine-checked proof shows that for any real number M, no matter how large, there exists a small positive epsilon such that every x between 0 and epsilon has J(x) greater than M. This is not an approximation or a numerical observation; it is a theorem in the framework's machine-checked library of formal theorems.

This unboundedness result carries a striking consequence. The framework defines "x exists" to mean that x is positive and J(x) = 0. Since J(x) = 0 only when x = 1, the only positive ratio that exists is unity itself. The limit statement then yields a derived meta-principle: nothing cannot recognize itself, because approaching nothingness costs infinity. The cost of nonexistence is not merely high; it is unbounded, so no finite ledger can ever record it.

What the declaration does not claim is equally important. It does not assert that the cost function J is the only possible cost function; that uniqueness requires additional regularity conditions beyond the three primitive axioms. It does not claim that the limit statement applies at x = 0 itself, where J is undefined. And it does not claim that the meta-principle about nothingness is a physical law; it is a derived statement within the framework's formal system, not an empirical observation about the universe.

/-- The canonical cost functional:
  J(x) = ½(x + x⁻¹) - 1

This is the **unique** solution to the three axioms (proven below). -/
noncomputable def J (x : ℝ) : ℝ := (x + x⁻¹) / 2 - 1

/-- J equals zero exactly at x = 1. -/
theorem J_eq_zero_iff {x : ℝ} (hx : 0 < x) : J x = 0 ↔ x = 1 := by
  constructor
  · intro hJ
    simp only [J] at hJ
    -- (x + 1/x)/2 - 1 = 0  ⟹  x + 1/x = 2  ⟹  x² - 2x + 1 = 0  ⟹  x = 1
    have h1 : x + x⁻¹ = 2 := by linarith
    have hx0 : x ≠ 0 := hx.ne'
    have h2 : x^2 + 1 = 2 * x := by
      field_simp at h1
      linarith
    have h3 : (x - 1)^2 = 0 := by ring_nf; linarith
    have h4 : x - 1 = 0 := by nlinarith [sq_nonneg (x - 1)]
    linarith
  · intro hx1
    simp [J, hx1]

/-- As x → 0⁺, J(x) → +∞.

This is the **core economic principle**: approaching "nothing" costs infinity.
This is why existence is inevitable—non-existence is infinitely expensive. -/
theorem J_tendsto_atTop_as_x_to_zero :
    Filter.Tendsto J (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop := by
  rw [Filter.tendsto_atTop]
  intro M
  obtain ⟨ε, hε_pos, hε⟩ := J_arbitrarily_large_near_zero M
  -- We need {x : M ≤ J x} ∈ nhdsWithin 0 (Ioi 0)
  rw [Filter.Eventually, mem_nhdsWithin_iff_exists_mem_nhds_inter]
  use Set.Iio ε
  refine ⟨Iio_mem_nhds hε_pos, ?_⟩
  intro x ⟨hx_lt, hx_pos⟩
  exact le_of_lt (hε x hx_pos hx_lt)

/-- **Law of Existence**: A ratio x "exists" (is realizable) iff J(x) = 0.

In the RS framework, existence corresponds to being at a cost minimum.
The only minimum is at x = 1 (perfect balance/golden ratio fixed point). -/
def Exists (x : ℝ) : Prop := 0 < x ∧ J x = 0

/-- Unity is the unique existent. -/
theorem unity_is_unique_existent : ∀ x : ℝ, Exists x ↔ x = 1 := by
  intro x
  by_cases hx : 0 < x
  · exact law_of_existence hx
  · simp only [Exists]
    constructor
    · intro ⟨hpos, _⟩; exact absurd hpos hx
    · intro heq; subst heq; exact ⟨one_pos, by simp [J]⟩

/-- **Meta-Principle (Derived)**: "Nothing cannot recognize itself."

In the cost framework, "Nothing" corresponds to the limit x → 0.
Recognition requires a finite cost, but J(0) → ∞, so recognition
of "Nothing" by "Nothing" would require infinite cost—impossible.

This makes MP a **derived theorem**, not a primitive axiom. -/
theorem mp_from_cost :
    ∀ M : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → J x > M := by
  exact J_arbitrarily_large_near_zero

What this page does not claim

The declaration does not prove that J is the only possible cost function without additional regularity assumptions. The declaration does not apply at x = 0 itself, where J is undefined. The meta-principle about nothingness is a formal derivation, not an empirical law of physics.

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/CostAxioms.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: