Foundation Law Of Existence Defect Tendsto At Top At Zero

The defect near zero

The real numbers have a natural way to measure how far a positive number is from 1: the expression (x + 1/x)/2 - 1. This quantity is zero exactly when x equals 1, and it grows as x moves away from 1 in either direction. The framework calls this quantity the defect, a measure of how much a positive number fails to be the special value 1.

A proved theorem in the machine-checked library states that as x approaches 0 from above, the defect grows without bound. In the language of limits, the defect tends to infinity as x tends to 0 from the positive side. The proof is a direct calculation: when x is very small, the term 1/x dominates the expression, so the whole quantity becomes arbitrarily large.

This behavior has a striking consequence within the framework. The framework defines existence for a positive real number as having defect exactly zero. Since the defect is zero only at x = 1, and since it explodes near zero, the framework proves that no positive number can have a small but nonzero defect. A number is either exactly 1, with defect zero, or it is far from 1, with defect bounded away from zero. There is no middle ground where something is almost existent.

In Recognition Science, this theorem supports the broader claim that existence is an all-or-nothing property. The framework models existence as a discrete ledger, a record where each entry either fully exists or does not. The theorem about the defect near zero is what rules out a continuum of partial existence. It is a formal guarantee that the ledger has no fuzzy entries.

The theorem does not say anything about what happens for negative numbers, since the defect is only defined for positive inputs. It also does not claim that the defect is the only possible measure of existence, or that the framework's choice of this particular expression is forced by the theorem alone. The theorem is a statement about a specific function, not about the metaphysics of existence.

/-- As x → 0⁺, defect(x) → +∞.

Technical proof: J(x) = (x + 1/x)/2 - 1 ≥ 1/(2x) - 1 → +∞ as x → 0⁺. -/
theorem defect_tendsto_atTop_at_zero :
    Filter.Tendsto defect (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop := by
  unfold defect J
  -- The proof uses that 1/x → +∞ as x → 0⁺, and (x + 1/x)/2 - 1 ≥ 1/(2x) - 1
  have hinv : Filter.Tendsto (fun x : ℝ => x⁻¹) (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop :=
    tendsto_inv_nhdsGT_zero
  rw [Filter.tendsto_atTop]
  intro r
  rw [Filter.tendsto_atTop] at hinv
  have hev := hinv (2 * (r + 2))
  -- On nhdsWithin 0 (Ioi 0), x is positive. Combine with x⁻¹ ≥ 2(r+2)
  have hpos : ∀ᶠ x in nhdsWithin (0 : ℝ) (Set.Ioi 0), 0 < x := eventually_mem_nhdsWithin
  apply Filter.Eventually.mono (hev.and hpos)
  intro x ⟨hinvx, hx0⟩
  have h1 : (x + x⁻¹) / 2 - 1 ≥ x⁻¹ / 2 - 1 := by linarith
  have h2 : x⁻¹ / 2 - 1 ≥ (2 * (r + 2)) / 2 - 1 := by linarith
  linarith

/-- **Existence Predicate**: x exists in the RS framework iff x > 0 and defect(x) = 0. -/
structure Exists (x : ℝ) : Prop where
  pos : 0 < x
  defect_zero : defect x = 0

/-- **Defect Zero Characterization**: defect(x) = 0 ⟺ x = 1 (for x > 0). -/
theorem defect_zero_iff_one {x : ℝ} (hx : 0 < x) : defect x = 0 ↔ x = 1 := by
  simp only [defect, J]
  constructor
  · intro h
    have hx0 : x ≠ 0 := hx.ne'
    -- (x + 1/x)/2 - 1 = 0 implies (x + 1/x) = 2
    have h1 : x + x⁻¹ = 2 := by linarith
    -- Multiply by x: x² + 1 = 2x, so (x-1)² = 0
    have h2 : x * (x + x⁻¹) = x * 2 := by rw [h1]
    have h3 : x^2 + 1 = 2 * x := by field_simp at h2; linarith
    nlinarith [sq_nonneg (x - 1)]
  · intro h; simp [h]

/-- As x → 0⁺, defect(x) → +∞.

Technical proof: J(x) = (x + 1/x)/2 - 1 ≥ 1/(2x) - 1 → +∞ as x → 0⁺. -/
theorem defect_tendsto_atTop_at_zero :
    Filter.Tendsto defect (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop := by
  unfold defect J
  -- The proof uses that 1/x → +∞ as x → 0⁺, and (x + 1/x)/2 - 1 ≥ 1/(2x) - 1
  have hinv : Filter.Tendsto (fun x : ℝ => x⁻¹) (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop :=
    tendsto_inv_nhdsGT_zero
  rw [Filter.tendsto_atTop]
  intro r
  rw [Filter.tendsto_atTop] at hinv
  have hev := hinv (2 * (r + 2))
  -- On nhdsWithin 0 (Ioi 0), x is positive. Combine with x⁻¹ ≥ 2(r+2)
  have hpos : ∀ᶠ x in nhdsWithin (0 : ℝ) (Set.Ioi 0), 0 < x := eventually_mem_nhdsWithin
  apply Filter.Eventually.mono (hev.and hpos)
  intro x ⟨hinvx, hx0⟩
  have h1 : (x + x⁻¹) / 2 - 1 ≥ x⁻¹ / 2 - 1 := by linarith
  have h2 : x⁻¹ / 2 - 1 ≥ (2 * (r + 2)) / 2 - 1 := by linarith
  linarith

What this page does not claim

The theorem does not apply to negative numbers, as the defect is only defined for positive inputs. The theorem does not prove that the defect is the only possible measure of existence. The theorem does not make a metaphysical claim about what existence is, only a statement about a specific 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/LawOfExistence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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