Explain the Lean theorem `Jcost_is_calibrated` in module `IndisputableMonolith.CostUniqueness`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

1. Plain English Meaning

The theorem Jcost_is_calibrated states that the fundamental cost function $J(x)$ has a normalized curvature of 1 at its minimum, when evaluated in logarithmic coordinates. Specifically, if we substitute $x = e^t$, the second derivative of $J(e^t)$ with respect to $t$, evaluated at $t = 0$ (which corresponds to $x = 1$), is exactly 1.

2. Significance in Recognition Science

In the RS framework, calibration is a definitional MODEL choice that sets the absolute scale of the cost function. While structural properties like reciprocal symmetry restrict the functional form of the cost, they allow arbitrary scalar multiples. The calibration condition fixes this scaling degree of freedom. It serves as a necessary boundary condition for the T5 uniqueness forcing chain, proving that any admissible cost function must exactly equal $J(x) = \frac{x + x^{-1}}{2} - 1$.

3. Reading the Formal Statement

theorem Jcost_is_calibrated : FunctionalEquation.IsCalibrated Jcost

The type FunctionalEquation.IsCalibrated Jcost represents the target mathematical property. The proof begins with the change tactic, which exposes the underlying definitional equality in Lean: deriv (deriv (fun t : ℝ => Jcost (Real.exp t))) 0 = 1 Here, fun t : ℝ => Jcost (Real.exp t) is Lean's syntax for the function $t \mapsto J(e^t)$. deriv is the mathematical derivative. Applying deriv twice and evaluating at 0 yields 1.

4. Visible Dependencies and Certificates

The theorem relies entirely on the external calculation IndisputableMonolith.CPM.LawOfExistence.RS.Jcost_log_second_deriv_normalized.

Within the supplied module, this theorem acts as a critical input:

• It satisfies the hCalib requirement in unique_cost_on_pos_from_rcl, the axiom-free proof that any function matching these properties equals Jcost.
• It is bundled into Jcost_regularity_cert, a formalized certificate grouping Jcost's continuity, strict convexity, and calibration properties.

5. What this Declaration Does Not Prove

• The calculus derivation: This declaration does not perform the algebraic differentiation of $\cosh(t) - 1$; it merely points to the imported module where the calculus is executed.
• Cost uniqueness: This theorem only establishes one local geometric property at $t=0$. It does not prove that Jcost is the only calibrated function, which requires the full d'Alembert differential equation machinery of the T5 theorem.