IndisputableMonolith.Measurement.KernelMatch
This module supplies the recognition profile r(ϑ) solving J(r(ϑ)) = 2 tan ϑ from Local-Collapse (D.1), together with pointwise and integral kernel-match lemmas. Measurement theorists citing the C=2A bridge would import it to equate cost kernels with two-branch geodesic actions. The module is a lightweight collection of definitions and basic algebraic checks that rely on imported path-action and geodesic interfaces.
claimThe recognition profile satisfies $J(r(ϑ)) = 2 tan ϑ$, where $J(x) = (x + x^{-1})/2 - 1$. Supporting lemmas establish pointwise equality of the kernel with the geodesic rate action and integral matching over the profile.
background
The module sits inside the Measurement domain and imports the J-cost from Cost, the minimal path-action interface from PathAction, and the two-branch geodesic geometry from TwoBranchGeodesic. In the latter, residual norm equals π/2 minus the rotation angle and rate action equals -ln(sin θ_s). The profile r(ϑ) is introduced exactly as the solution to the functional equation stated in Local-Collapse eq (D.1).
proof idea
This is a definition module. It introduces recognitionProfile as the solution to the stated J-equation, then records elementary consequences (arcosh_arg_ge_one, positivity, pointwise kernel match, differential match, and integral match) via direct algebraic verification and the imported geodesic identities.
why it matters in Recognition Science
The module supplies the kernel-matching step required by the C=2A bridge in C2ABridge, whose main theorem states that recognition cost C equals twice the residual-model rate action A for any two-branch geodesic rotation. It therefore closes the measurement-kernel link between the J-cost formalism and the two-branch geometry used in Local-Collapse.
scope and limits
- Does not derive the J-cost function or its functional equation.
- Does not treat multi-branch or higher-dimensional geodesics.
- Does not supply numerical evaluation or approximation schemes.
- Does not address measure-theoretic completeness of the kernel.