Cost Calibration Jlog Unit Curvature

Unit curvature at the identity

In calculus, the second derivative of a function measures its curvature: how sharply the graph bends. The declaration Jlog_unit_curvature proves, in the framework's machine-checked library of formal theorems, that the curvature of a particular cost function Jlog at its zero point is exactly 1. This is the unit normalization axiom, and it is what pins down the size of the framework's fundamental unit of cost.

The function Jlog is defined in log-coordinates, where the framework's cost function Jcost is composed with the exponential function. In these coordinates, Jlog equals cosh t minus 1, the hyperbolic cosine shifted down by one. The first derivative is sinh t, and the second derivative is cosh t. At t equals zero, cosh 0 is 1, so the second derivative at zero is 1. This is the entire content of the theorem: a clean, checkable fact about a standard function from analysis.

The theorem's role is to complete the characterization of J, the framework's unique cost function. The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The calibration axiom, this unit curvature condition, is one of those five conditions. It is the one that fixes the scale: without it, the cost function could be rescaled by any positive factor and still satisfy the other conditions. The theorem shows that the framework's chosen scale is not arbitrary but is forced by the requirement that the curvature at the identity be 1.

In plain terms, this means the framework's unit of cost is not a free parameter. It is determined by a geometric condition on the cost function itself. The theorem does not claim that this unit corresponds to any physical unit like the joule or the electronvolt; it only establishes the mathematical scale within the framework. It also does not claim that the cost function J is the only function with this property; the theorem is a statement about Jlog specifically, not a uniqueness result.

The consequence is that the framework's constants, such as the reduced Planck constant hbar = phi^-5 and the gravitational constant G = phi^5/pi, are not fitted numbers but are derived from this forced scale. The unit curvature theorem is the anchor that makes those derivations possible.

/-- 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

What this page does not claim

This theorem does not claim that Jlog is the only function with unit curvature at zero. It does not claim that the unit of cost corresponds to any physical unit such as the joule. It does not claim that the framework's constants are measured values; they are derived within the framework.

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)

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.

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