Cost T5 Cost Uniqueness On Pos

The uniqueness theorem

The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a ratio x between two quantities. For x close to 1, meaning the quantities nearly match, the cost is tiny and grows quadratically: J(1 + ε) is about ε²/2. For x far from 1, the cost grows roughly linearly, so doubling a large ratio doubles the cost. The function is symmetric under inversion, J(x) = J(1/x), so swapping the two quantities leaves the cost unchanged. It vanishes only at x = 1, where the quantities are identical.

The theorem T5_cost_uniqueness_on_pos, proved in the machine-checked library of formal theorems, states that any function F on positive real numbers satisfying the framework's five plain conditions must equal J everywhere. The conditions are reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The proof works by showing F agrees with J on all exponentials, then extending that agreement to every positive number. The name "on pos" refers to the domain: all positive real numbers, excluding zero and negatives.

In Recognition Science, recognition (a discrete record of events) incurs a cost (a forced price, not a chosen one). The theorem says that any candidate cost function meeting the five conditions is identical to J. This is a uniqueness result, not an existence result: the framework's other theorems establish that J itself satisfies the conditions. The uniqueness is what makes J special: no alternative formula can pass the same tests.

What the theorem does not claim is as important as what it proves. It does not say that the five conditions are the only possible axioms for a cost function; other axiom sets could yield other functions. It does not claim that J is the unique function satisfying any subset of the conditions; dropping the composition law, for instance, would allow many functions. The theorem also does not assert that J is the actual cost used by any physical process; that is a modeling claim, not a theorem. The uniqueness is conditional on the five stated conditions, and the proof is airtight only within those conditions.

The practical consequence is that the framework's cost function is not an arbitrary choice. Any theory of recognition that accepts the five plain conditions is forced to use J, no matter how it is derived. This makes J a structural constant of the framework, like a fixed exchange rate between ratios and prices. The theorem also provides a check: if a proposed cost function differs from J anywhere on the positive reals, it must violate at least one of the five conditions.

theorem T5_cost_uniqueness_on_pos {F : ℝ → ℝ} [JensenSketch F] :
  ∀ {x : ℝ}, 0 < x → F x = Jcost x :=
by
  intro x hx
  have hAgree : AgreesOnExp F := by
    intro t
    exact le_antisymm (JensenSketch.axis_upper (F:=F) t) (JensenSketch.axis_lower (F:=F) t)
  exact (agree_on_exp_extends (F:=F) hAgree) hx

noncomputable def Jcost (x : ℝ) : ℝ := (x + x⁻¹) / 2 - 1

lemma Jcost_symm {x : ℝ} (hx : 0 < x) : Jcost x = Jcost x⁻¹ := by
  have hx0 : x ≠ 0 := ne_of_gt hx
  rw [Jcost_eq_sq hx0, Jcost_eq_sq (inv_ne_zero hx0)]
  field_simp [hx0]
  ring

/-- J(x) = 0 iff x = 1, for positive x. -/
lemma Jcost_eq_zero_iff (x : ℝ) (hx : 0 < x) : Jcost x = 0 ↔ x = 1 := by
  constructor
  · intro h
    by_contra h1
    exact absurd h (ne_of_gt (Jcost_pos_of_ne_one x hx h1))
  · intro h
    rw [h]
    exact Jcost_unit0

lemma Jcost_one_plus_eps_quadratic (ε : ℝ) (hε : |ε| ≤ (1 : ℝ) / 2) :
    ∃ (c : ℝ), Jcost (1 + ε) = ε ^ 2 / 2 + c * ε ^ 3 ∧ |c| ≤ 2 := by
  classical
  have hbounds := abs_le.mp hε
  have hpos : 0 < 1 + ε := by
    have : -(1 : ℝ) / 2 ≤ ε := by simpa [neg_div] using hbounds.1
    linarith
  have hne : 1 + ε ≠ 0 := ne_of_gt hpos
  have hcalc : Jcost (1 + ε) = ε ^ 2 / (2 * (1 + ε)) := by
    simpa [pow_two, add_comm, add_left_comm, add_assoc, sub_eq_add_neg]
      using (Jcost_eq_sq hne)
  let c : ℝ := -1 / (2 * (1 + ε))
  have h_eq :
      Jcost (1 + ε) = ε ^ 2 / 2 + c * ε ^ 3 := by
    have : ε ^ 2 / (2 * (1 + ε)) = ε ^ 2 / 2 + (-1 / (2 * (1 + ε))) * ε ^ 3 := by
      field_simp [hne]
      ring
    simpa [hcalc, c] using this
  have hden_pos : 0 < 2 * (1 + ε) := by nlinarith [hpos]
  have habs : |c| = 1 / (2 * (1 + ε)) := by
    simp [c, div_eq_mul_inv, abs_mul, abs_inv, abs_of_pos hpos]
  -- Use 1/(2(1+ε)) ≤ 1 from (1+ε) ≥ 1/2
  have hone_le : (1 : ℝ) ≤ 2 * (1 + ε) := by
    have : (1 / 2 : ℝ) ≤ 1 + ε := by linarith
    simpa [two_mul] using mul_le_mul_of_nonneg_left this (by norm_num : (0 : ℝ) ≤ 2)
  have hdiv_le_one : 1 / (2 * (1 + ε)) ≤ 1 := by
    have hpos1 : 0 < (1 : ℝ) := by norm_num
    simpa [one_div] using one_div_le_one_div_of_le hpos1 hone_le
  have hbound : |c| ≤ 2 := by
    have h1 : |c| ≤ 1 := by simpa [habs] using hdiv_le_one
    have h12 : (1 : ℝ) ≤ 2 := by norm_num
    exact le_trans h1 h12
  exact ⟨c, h_eq, hbound⟩

What this page does not claim

The five conditions are the only possible axioms for a cost function. J is the unique function satisfying any subset of the five conditions. J is the actual cost used by any physical process. The theorem proves existence of a cost function satisfying the conditions.

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/Cost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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