Cost Calibration Jlog Second Deriv At Zero

The unit curvature fact

The declaration Jlog_second_deriv_at_zero is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that for the function Jlog(t) = cosh(t) - 1, the second derivative at t = 0 equals exactly 1. In plain terms, this means the curve of the function bends upward at a specific, unit rate at its lowest point. This is not an approximation or a numerical observation; it is a proved result, derived from the definition of Jlog and standard facts about hyperbolic functions.

This fact matters because Jlog is the logarithmic form of the framework's central cost function, the cost, a measure of the price of a recognition event. The theorem establishes what the framework calls the unit normalization axiom: the scale of the cost is fixed so that its curvature at the identity is exactly one. Without this, the cost function could be rescaled arbitrarily, and the framework's derived constants, such as the golden ratio and the fine-structure constant, would not be pinned down. The theorem is the calibration step that makes those later derivations meaningful.

The framework proves more than just this single value. It shows that the first derivative of Jlog is sinh(t) and the second derivative is cosh(t), from which the value at zero follows immediately because cosh(0) = 1. It also packages this result as a general condition called UnitCurvature, which any function must satisfy to play the role of the cost. The theorem is a small but load-bearing piece of the larger proof that the cost function is unique.

What the theorem does not claim is also important. It does not say that the second derivative is 1 everywhere; it is only true at t = 0. It does not prove that Jlog itself is the only function with this property, only that this particular function has it. And it does not, by itself, derive any physical constant; it only fixes the scale of the cost function, leaving the derivation of constants to later theorems in the chain.

/-- 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

/-- 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

/-- Alternative formulation: Jlog has unit curvature at the identity -/
theorem Jlog_unit_curvature : deriv (deriv Jlog) 0 = 1 :=
  Jlog_second_deriv_at_zero

What this page does not claim

The second derivative of Jlog is not 1 everywhere, only at t = 0. Jlog is not proved to be the only function with unit curvature at zero. This theorem alone does not derive 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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