Cost Calibration Jcost Comp Exp Second Deriv At Zero

The unit curvature calibration

In the Recognition Science framework, the central object is a cost, a function that assigns a nonnegative number to any ratio between two recognized states, with cost zero when the states match. The framework derives this function's form from a handful of forced conditions, and the result is J(x) = (x + 1/x)/2 - 1. A key question is how to fix the scale of this cost, since multiplying the whole function by any constant would preserve its shape but change every numerical output. The declaration Jcost_comp_exp_second_deriv_at_zero answers that question: it proves that when the cost is written in logarithmic coordinates, its second derivative at the point where the ratio is one, the identity, is exactly 1.

To see what this means, consider the function Jlog(t) = J(e^t), which is simply the cost expressed in log-coordinates. The theorem establishes that the second derivative of this function at t = 0 is 1. In plain terms, the curvature of the cost function at its balance point is fixed to a specific, unit value. This is not a convention chosen for convenience; the framework proves it follows from the same forced conditions that determine the cost's overall shape. The theorem is the formal anchor for what the framework calls the unit normalization axiom, the rule that fixes the scale uniquely and completes the characterization of J.

The proof itself is a short chain of standard calculus facts. The framework's library shows that Jlog equals cosh(t) - 1, that its first derivative is sinh(t), and that its second derivative is cosh(t). Since cosh(0) = 1, the second derivative at zero is 1. The declaration Jcost_comp_exp_second_deriv_at_zero packages this result for the original cost function J, confirming that the same calibration holds when the cost is composed with the exponential function.

In Recognition Science, this calibration is load-bearing. It fixes the scale of the cost, which in turn fixes the scale of derived quantities such as the golden ratio and the framework's constants. The unit curvature result is a theorem, proved in the machine-checked library of formal theorems, not a definitional choice or a hypothesis. It is one of the five plain conditions that together force the unique form of the cost function.

What this declaration does not claim is equally important. It does not assert that the cost function itself has unit curvature in its original coordinates; the statement is specifically about the log-coordinate version. It does not derive the value of any physical constant or the fine-structure constant. It establishes a calibration fact about the framework's cost function, nothing more. The theorem is a precise, narrow result: the second derivative of Jlog at zero equals 1, and that is the whole of it.

/-- 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

/-- 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

What this page does not claim

The cost function J itself has unit curvature at the identity in its original coordinates. This theorem derives the value of any physical constant or coupling such as the fine-structure constant. The unit curvature result is a definitional choice or a hypothesis rather than a proved theorem.

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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