Cost Calibration

Calibration

In the Recognition Science framework, the ledger (a discrete record of recognition events) assigns a cost to every comparison between two quantities. The cost function J(x) measures how expensive it is to recognize a ratio x between two things. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. But a uniqueness theorem only pins down the shape of the function up to a scale; calibration is the condition that fixes that scale uniquely.

Calibration is the fourth of the five conditions. It states that the second derivative of the cost function, measured at the point where the two quantities are equal, must equal 1. In plain terms, this means the cost curve has a specific, fixed curvature at its minimum. Without this condition, you could stretch or compress the cost function vertically and still satisfy the other four conditions. Calibration removes that freedom, so the cost of recognition is not just shaped a certain way, it has a definite size.

The module proves this calibration property for the specific cost function J. The proof works by switching to logarithmic coordinates, where the cost function takes a simpler form. In these coordinates, J becomes Jlog(t) = cosh(t) - 1, where t is the logarithm of the ratio x. The hyperbolic cosine function cosh(t) has the property that its second derivative at zero is exactly 1. The module proves this step by step: it shows that the first derivative of Jlog is sinh(t), that the second derivative is cosh(t), and that cosh(0) = 1. A final theorem packages this result as the UnitCurvature class, which states the calibration condition for any function.

What calibration establishes, in the end, is that the cost function has a natural unit of measurement. The curvature at the identity point is not an arbitrary number; it is exactly 1. This completes the characterization of J: the five conditions, including calibration, force the cost function to be exactly J(x) = (x + 1/x)/2 - 1, with no free parameters left over. The scale is not chosen by hand; it is forced by the requirement that the curvature at the point of equality be exactly one.

/-- The calibration theorem: second derivative at zero equals 1 -/
theorem Jlog_second_deriv_at_zero : deriv (deriv Jlog) 0 = 1 := by
  rw [deriv2_Jlog]
  exact cosh_zero

/-- Jlog equals cosh t - 1 -/
lemma Jlog_eq_cosh (t : ℝ) : Jlog t = Real.cosh t - 1 := Jlog_as_cosh t

/-- First derivative of Jlog is sinh -/
lemma deriv_Jlog (t : ℝ) : deriv Jlog t = sinh t := by
  exact (hasDerivAt_Jlog_new t).deriv

/-- Second derivative of Jlog is cosh -/
lemma deriv2_Jlog (t : ℝ) : deriv (deriv Jlog) t = cosh t := by
  have h1 : deriv Jlog = sinh := by
    funext s; exact deriv_Jlog s
  rw [h1]
  exact (hasDerivAt_sinh t).deriv

/-- Calibration for Jcost in log-coordinates: second derivative at zero is 1. -/
theorem Jcost_comp_exp_second_deriv_at_zero :
    deriv (deriv (fun t : ℝ => Jcost (Real.exp t))) 0 = 1 := by
  -- Jcost ∘ exp = Jlog by definition
  have h : (fun t : ℝ => Jcost (Real.exp t)) = Jlog := rfl
  rw [h]
  exact Jlog_second_deriv_at_zero

What this page does not claim

This module does not prove that the five conditions are necessary, only that the calibration condition holds for the specific cost function J. Calibration does not determine the value of any physical constant; it only fixes the scale of the cost function.

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

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