Foundation Cost First Existence Non Existence Has Positive Cost

The cost of not existing

The declaration non_existence_has_positive_cost states a simple inequality: for any positive real number x that is not equal to 1, the recognition cost J(x) is strictly greater than zero. In plain words, it says that being different from the unique cost minimum is never free. The statement is a theorem in the framework's machine-checked library of formal theorems, proved from the definition of recognition cost and the prior result that J(x) = 0 if and only if x = 1 for positive x.

The theorem is part of a larger cost-first existence principle. The framework defines a pattern x as existing in the recognition sense exactly when J(x) = 0, which the library proves is equivalent to x = 1. This makes existence a selected property rather than a primitive one: among all positive values, only the cost minimum counts as existing. The declaration under discussion adds the other half of the picture: every non-minimal positive value pays a strictly positive cost, so non-existence in this sense is never costless.

The theorem also connects to a divergence result. The library proves that the cost function is unbounded as x approaches zero from above: there is no finite constant C such that J(ε) ≤ C for all positive ε. This means the cost of approaching "nothing" in the positive direction grows without limit. Together, the two results paint a consistent picture: the cost minimum is the only free state, and moving away from it, in either direction, carries a price.

What the declaration does not claim is important. It does not say that non-existence is impossible or forbidden; it only says it is costly. It also does not assert anything about negative values, since the theorem's hypothesis requires x > 0. The statement is purely about the cost function's behavior, not about what patterns actually exist in the physical world. It is a structural claim about the mathematics of recognition cost, not an empirical prediction about cosmology or ontology.

/-- Non-existence costs more than zero. -/
theorem non_existence_has_positive_cost {x : ℝ} (hx : 0 < x) (hne : x ≠ 1) :
    0 < Jcost x :=
  Jcost_pos_of_ne_one x hx hne

/-- RSExists iff x = 1 (the unique J-cost minimiser). -/
theorem rsExists_iff_one {x : ℝ} (hx : 0 < x) :
    RSExists x ↔ x = 1 := by
  unfold RSExists
  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 unique "nothing" reference: cost is unbounded on (0,∞). -/
theorem divergence_at_zero_direction :
    ¬ ∃ (C : ℝ), ∀ (ε : ℝ), 0 < ε → Jcost ε ≤ C := by
  intro ⟨C, hC⟩
  -- Pick ε = 1/(2*(|C|+2)); then J(ε) > |C|+1 > C
  -- Actual proof: pick ε = 1/4, then J(1/4) = (1/4-1)²/(2·1/4) = (9/16)/(1/2) = 9/8
  -- That only bounds J away from C when C < 9/8.
  -- For large C, pick ε = 1/(C+2):
  -- J(1/(C+2)) = (1/(C+2)-1)²/(2/(C+2)) = (C+1)²/(C+2)²·(C+2)/2 = (C+1)²/(2(C+2))
  -- For C ≥ 0: (C+1)²/(2(C+2)) > C ↔ (C+1)² > 2C(C+2) = 2C²+4C ↔ C²+2C+1 > 2C²+4C ↔ 0 > C²+2C-1
  -- This fails for C ≥ 1. Need a better choice. Use ε = 1/(2C+4):
  -- J(1/(2C+4)) = (1/(2C+4)-1)²/(2/(2C+4)) = ((2C+3)/(2C+4))²·(2C+4)/2 = (2C+3)²/(2(2C+4))
  -- Compare with C: (2C+3)²/(2(2C+4)) > C ↔ (2C+3)² > 2C(2C+4) = 4C²+8C
  -- = 4C²+12C+9 > 4C²+8C ↔ 4C+9 > 0, which holds for C > -9/4.
  -- For C ≤ -3, J(ε) ≥ 0 > C since C < 0. Done by cases.
  -- Use J(1) = 0 to handle C < 0, and a direct computation for C ≥ 0
  by_cases hC_neg : C < 0
  · linarith [hC 1 one_pos, Jcost_unit0]
  push_neg at hC_neg  -- C ≥ 0
  -- J(1/(2C+4)) = (2C+3)²/(2(2C+4)) > C for C ≥ 0
  have h2C4 : (0 : ℝ) < 2 * C + 4 := by linarith
  have hε := hC (1 / (2 * C + 4)) (div_pos one_pos h2C4)
  have hJval : Jcost (1 / (2 * C + 4)) = (2 * C + 3) ^ 2 / (2 * (2 * C + 4)) := by
    rw [Jcost_eq_sq (by positivity)]
    field_simp
    ring
  rw [hJval] at hε
  have hnum : 0 ≤ (2 * C + 3) ^ 2 := sq_nonneg _
  -- (2C+3)²/(2(2C+4)) ≤ C ↔ (2C+3)² ≤ 2C(2C+4) = 4C²+8C
  -- But (2C+3)² = 4C²+12C+9 > 4C²+8C = 2C(2C+4) for C ≥ 0 (since 4C+9 > 0)
  have hrewrite : (2 * C + 3) ^ 2 / (2 * (2 * C + 4)) ≤ C ↔
      (2 * C + 3) ^ 2 ≤ C * (2 * (2 * C + 4)) := by
    rw [div_le_iff₀ (by positivity : (0 : ℝ) < 2 * (2 * C + 4))]
  rw [hrewrite] at hε
  nlinarith [sq_nonneg (2 * C + 3)]

What this page does not claim

The theorem does not claim that non-existence is impossible, only that it carries a positive cost. The theorem says nothing about negative values of x, since its hypothesis requires x > 0. The declaration makes no empirical claim about what patterns actually exist in the physical universe.

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/CostFirstExistence.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: