IndisputableMonolith.Cost.Calibration
The Cost.Calibration module supplies the closed-form identification of the logarithmic J-cost with cosh t minus one, plus its first and second derivatives at zero. Researchers working on unit curvature or the phi-ladder mass formulas cite these identities to anchor analytic computations. The module consists of direct algebraic verifications and derivative lemmas built on the base Cost definitions.
claim$J_{\log}(t) = \cosh t - 1$, with $J_{\log}'(0) = 0$ and $J_{\log}''(0) = 1$.
background
Recognition Science encodes recognition defects via the J-cost function satisfying the Recognition Composition Law. The upstream Cost module introduces the base J definition and its functional equation. This calibration module works in logarithmic coordinates to convert the multiplicative J into an additive form that matches hyperbolic functions.
The setting supports the T5 J-uniqueness step and subsequent curvature calculations. Jlog is the explicit map that lets the framework use standard hyperbolic identities for the eight-tick octave and spatial dimension derivations.
proof idea
This is a definition module with supporting lemmas. The core identity follows by direct substitution of the exponential argument into the J definition, after which the listed derivative lemmas apply Mathlib hasDerivAt and second-derivative rules at the origin.
why it matters in Recognition Science
The module supplies the analytic calibration required by the parent Cost theorems and the T5 J-uniqueness landmark in the forcing chain. It enables the unit-curvature and Berry-threshold results that appear in downstream mass-formula and dimension derivations.