Foundation Cost First Existence Cost First Existence Cert

The cost certificate

The declaration CostFirstExistenceCert is a machine-checked bundle of three theorems about the recognition cost function J. In plain terms, it certifies that a positive number counts as existing in the recognition sense exactly when J(x) = 0, that any other positive value carries strictly positive cost, and that cost grows without bound as the input approaches zero. The first two facts together imply that the only positive value with zero cost is 1, the unique minimum of the cost function.

The cost function itself is defined as J(x) = (x + 1/x)/2 - 1 for positive real x. It is a forced consequence of five plain conditions on how recognition costs must behave, proved in the framework's machine-checked library of formal theorems. The certificate packages the three consequences as a single structure, so that any later theorem can invoke the whole bundle at once rather than reproving each part.

In Recognition Science, the certificate supports a selection principle: stable configurations are those with minimum recognition cost. The framework models existence as a cost-minimisation condition, so the certificate is what makes that modelling precise. It does not claim that the physical universe began with a cost calculation, nor that the pre-Big-Bang era is a geometric space. Those remain interpretive statements in the framework's literature, not formal theorems.

The certificate also does not claim that the cost function is the only possible one, nor that cost-minimisation alone explains all physical law. It establishes a structural fact about the cost function, not a complete theory of existence. The three component theorems are each proved with no unproved assumptions in the framework's library.

/-- Recognition existence: `x` exists iff J(x) = 0. -/
def RSExists (x : ℝ) : Prop := Jcost x = 0

/-- 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

/-- 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

/-- 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)]

/-- 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

What this page does not claim

The certificate does not claim that the universe began with a cost calculation. It does not claim that cost-minimisation alone explains all physical law. It does not claim that the pre-Big-Bang era is a geometric space.

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

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