IndisputableMonolith.Measurement.KernelMatch
KernelMatch supplies the recognition profile r(ϑ) solving J(r(ϑ)) = 2 tan ϑ from Local-Collapse eq (D.1), together with pointwise, differential, and integral kernel-match lemmas. Measurement theorists bridging recognition cost to geodesic action cite it when establishing C = 2A. The module assembles algebraic identities on top of the imported Cost, PathAction, and TwoBranchGeodesic interfaces.
claimThe recognition profile satisfies $J(r(ϑ)) = 2 tan ϑ$, where $J$ is the J-cost function; this profile yields pointwise, differential, and integral kernel matches between recognition cost and two-branch geodesic action.
background
The module sits in the Measurement domain. It imports Cost for the J-cost function, PathAction for a lightweight interface to recognition paths and their action/weights, and TwoBranchGeodesic for the two-branch rotation geometry whose residual norm is ||R|| = π/2 - θ_s and whose rate action is A = -ln(sin θ_s). The central definition is the recognition profile drawn from equation (D.1) of Local-Collapse.
proof idea
This is a definition module, no proofs. It introduces recognitionProfile as the solution to the stated J-equation and records the three kernel-match statements as direct consequences of that equation.
why it matters in Recognition Science
KernelMatch supplies the profile and matching lemmas that feed the central theorem of C2ABridge establishing C = 2A exactly for any two-branch geodesic rotation. It therefore closes the algebraic step required by the measurement bridge in the Recognition Science framework.
scope and limits
- Does not supply measure-theoretic lemmas on piecewise additivity or domain shifts.
- Does not derive the profile equation from first principles.
- Does not compute explicit numerical values or closed forms for r(ϑ).
- Does not address multi-branch or higher-dimensional geodesics.