Foundation Cost First Existence Rsexists

Cost-first existence

In the Recognition Science framework, the declaration RSExists is a formal definition of what it means for a positive real number to "exist" in the recognition sense. The definition is simple: a positive number x exists if and only if its recognition cost J(x) equals zero. Since the cost function J is already proved to have a unique minimum at x = 1, the definition immediately implies that the only positive number that exists is 1. This is not an empirical claim about the physical universe; it is a structural choice within the framework, made so that existence is tied to the unique point where recognition cost vanishes.

The definition is accompanied by two proved theorems. The first, rsExists_iff_one, states that for any positive real x, RSExists x is logically equivalent to x = 1. The proof uses the already-established fact that J(x) is strictly positive whenever x is not 1. The second theorem, non_existence_has_positive_cost, states directly that if x is positive and not equal to 1, then J(x) is greater than zero. Together these theorems make the definition precise: existence is exactly the zero-cost point, and everything else carries a positive cost. A third theorem, divergence_at_zero_direction, shows that the cost function is unbounded as x approaches zero from above, meaning there is no finite upper bound on J(ε) for arbitrarily small positive ε. This is presented as the framework's formal way of saying that "nothing" is not a stable reference point.

In the framework's own account, this definition is the core of a "cost-first selection principle": stable configurations are those with minimum recognition cost, and laws are said to emerge from the constraint that the universe minimises J. The pre-Big-Bang era is described as a pre-geometric setting where this selection happens. These are interpretive statements within the framework, not theorems about physics. The machine-checked library of formal theorems establishes only the logical equivalence between existence and the value 1, and the positivity of cost elsewhere. It does not prove that the physical universe actually minimises J, nor that any particular law of physics follows from this definition alone.

What the declaration does not claim is as important as what it proves. RSExists does not assert that the number 1 is the only real number, nor that physical objects are somehow unreal. It does not say that the cost function J is the actual energy of the universe, nor that the framework has derived the fine-structure constant or any specific coupling from this definition. The declaration is a formal anchor: it gives a precise meaning to "existence" within the framework, and it proves the logical consequences of that meaning. Whether that meaning corresponds to physical reality is a separate question, one the framework itself treats as a matter of interpretation, not as a proved result.

/-- 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)]

What this page does not claim

RSExists does not claim that only the number 1 is real in the physical universe. The declaration does not derive any specific law of physics from the cost minimum alone. The framework does not claim that the cost function J is the actual energy of the 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

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