Foundation Cost Axioms J Arbitrarily Large Near Zero

The cost of approaching nothing

The cost function J(x) = (x + x⁻¹)/2 − 1 measures the price of being at ratio x relative to unity. It is the unique function satisfying the framework's three primitive axioms: normalization (unity costs zero), a composition law, and a calibration condition. The theorem J_arbitrarily_large_near_zero states that for any real number M, there exists a positive ε such that for every x with 0 < x < ε, the inequality M < J(x) holds. In plain terms, as x approaches zero from the positive side, J(x) grows without bound; no matter how large a threshold you pick, you can find ratios small enough that their cost exceeds it.

This is not an asymptotic estimate or a numerical observation. It is a proved theorem in the framework's machine-checked library of formal theorems, derived directly from the definition of J. The proof is short: since J(x) = (x + x⁻¹)/2 − 1 and the term x⁻¹ dominates as x tends to zero, the function must blow up. The theorem is used to establish a stronger statement, nothing_costs_infinity, which says there is no finite upper bound C such that J(x) ≤ C for all positive x. That in turn supports the framework's derived meta-principle that approaching nothingness carries infinite cost, a statement about the structure of the cost function itself, not about any physical process.

What the theorem does not claim is equally precise. It does not assert that J(0) is defined or equals infinity; the function is only defined for positive x, and the theorem concerns the behavior as x approaches zero, not the value at zero. It does not say that the cost becomes infinite for any particular positive ratio, only that it exceeds any chosen bound for sufficiently small ratios. The theorem is a statement about the real numbers and the function J, not about existence, logic, or physics. Those connections are made by other declarations in the framework, and they rest on this theorem as one ingredient among several.

In the broader framework, this unboundedness result is a load-bearing piece. It gives content to the idea that the cost of recognition is forced, not chosen: the cost function cannot be tamed near zero, no matter what finite resources or thresholds one considers. The theorem also feeds directly into the law of existence, which states that for positive x, existence is equivalent to x = 1. Since J(1) = 0 and J(x) > 0 for all other positive x, the unique point of zero cost is unity, and the blow-up near zero ensures that nothingness is not a cheap alternative to balance. This is what the theorem changes: it rules out any finite-cost escape as x shrinks, leaving unity as the only zero-cost configuration.

/-- For any bound M, there exists ε > 0 such that J(x) > M for all 0 < x < ε.

    Direct proof: Choose ε = 1/(2(M + 2)). For 0 < x < ε, we have
    x⁻¹ > 2(M + 2), so J(x) ≥ x⁻¹/2 - 1 > M + 2 - 1 = M + 1 > M. -/
theorem J_arbitrarily_large_near_zero (M : ℝ) :
    ∃ ε > 0, ∀ x, 0 < x → x < ε → M < J x := by
  -- Choose ε = 1/(2(max M 0 + 2))
  let M' := max M 0 + 2
  have hM'_pos : M' > 0 := by positivity
  refine ⟨1 / (2 * M'), by positivity, ?_⟩
  intro x hx_pos hx_small
  simp only [J]
  -- Key computation: for 0 < x < 1/(2*M'), we have J(x) > M
  -- J(x) = (x + 1/x)/2 - 1 ≥ 1/(2x) - 1
  -- Since x < 1/(2*M'), we have 1/x > 2*M', so 1/(2x) > M', thus J(x) > M' - 1 > M
  have hx_ne : x ≠ 0 := hx_pos.ne'
  have h2M'_pos : 2 * M' > 0 := by positivity
  -- From x < 1/(2*M'), get 2*M' < 1/x
  have h_key : 2 * M' * x < 1 := by
    calc 2 * M' * x = x * (2 * M') := by ring
      _ < (1 / (2 * M')) * (2 * M') := mul_lt_mul_of_pos_right hx_small h2M'_pos
      _ = 1 := by field_simp
  have h_inv : 2 * M' < 1 / x := by
    rw [div_eq_mul_inv, lt_mul_inv_iff₀ hx_pos]
    exact h_key
  -- Now J(x) ≥ (1/x)/2 - 1 > M' - 1 > M
  have hJ_lower : (x + x⁻¹) / 2 - 1 > (1/x) / 2 - 1 := by
    -- (x + 1/x)/2 - 1 > (1/x)/2 - 1 ⟺ x/2 > 0, which follows from x > 0
    rw [one_div]
    have hx_half_pos : x / 2 > 0 := by linarith
    linarith
  have hJ_bound : (1/x) / 2 - 1 > M' - 1 := by
    -- From h_inv: 2*M' < 1/x, so M' < (1/x)/2
    nlinarith [h_inv]
  have hM_lt : M < M' - 1 := by simp only [M']; linarith [le_max_left M 0]
  linarith

/-- Alternative formulation: No finite-cost state can approach Nothing. -/
theorem nothing_costs_infinity :
    ¬∃ C : ℝ, ∀ x, 0 < x → J x ≤ C := by
  push_neg
  intro C
  obtain ⟨ε, hε, hJ⟩ := J_arbitrarily_large_near_zero C
  use ε / 2
  constructor
  · linarith
  · exact hJ (ε / 2) (by linarith) (by linarith)

/-- The Law of Existence: x exists ⟺ x = 1. -/
theorem law_of_existence {x : ℝ} (hx : 0 < x) : Exists x ↔ x = 1 := by
  simp only [Exists, J_eq_zero_iff hx, and_iff_right hx]

What this page does not claim

The theorem does not define J(0) or assign it a value; it concerns the limit behavior as x approaches zero. The theorem does not state that any particular positive ratio has infinite cost, only that costs exceed any finite bound for sufficiently small ratios. The theorem does not by itself establish any physical or logical consequence; those connections are made by other declarations in the framework.

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)

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