IndisputableMonolith.Measurement.C2ABridge
The C2ABridge module constructs the recognition path from two-branch geodesic rotation by applying the recognition profile to the rotation angle. Researchers deriving the Born rule from the J-cost functional cite this bridge. It chains the pointwise kernel identity from KernelMatch with the geodesic action definition from TwoBranchGeodesic to obtain the integral relation C = 2A.
claimThe module establishes the bridge $C = 2A$, where $C = 2A$ is the integrated recognition cost along the path constructed from the rotation profile $r(ϑ)$ and $A = -ln(sin θ_s)$ is the rate action of the two-branch geodesic.
background
This module sits in the Measurement domain and imports the PathAction interface for recognition paths and weights, the TwoBranchGeodesic formalization of rotation geometry (residual norm ||R|| = π/2 - θ_s, action A = -ln(sin θ_s)), and the KernelMatch module. KernelMatch proves that for the profile r(ϑ) = (1 + 2 tan ϑ) + √((1 + 2 tan ϑ)^2 - 1) one has J(r(ϑ)) = 2 tan ϑ pointwise, enabling the integral identity C = 2A. The setting combines the recognition composition law with the two-branch measurement geometry from Local-Collapse §3 and Appendix A.
proof idea
The module defines pathFromRotation to build the path from geodesic parameters, then applies the integral identity from KernelMatch together with the cotangent integral lemma to obtain measurement_bridge_C_eq_2A. It further defines weight_bridge, weight_equals_born, and amplitude_modulus_bridge to connect the cost to the amplitude.
why it matters in Recognition Science
This module supplies the C = 2A relation required by the BornRule module to derive P(I) = |α_I|² from the recognition cost J and the amplitude bridge 𝒜 = exp(-C/2)·exp(iφ). It completes the step from geodesic geometry to the cost functional in the chain leading to the Born rule.
scope and limits
- Does not include heavy measure-theoretic lemmas on piecewise additivity of paths.
- Does not derive the Born rule itself.
- Does not specify numerical values for constants such as alpha.
- Does not address extensions beyond the two-branch geodesic.