IndisputableMonolith.Measurement.TwoBranchGeodesic
The TwoBranchGeodesic module defines the two-branch rotation geometry for quantum measurement paths from initial angle θ_s to π/2. It supplies the explicit objects required by the C=2A equivalence and kernel-matching arguments. The structure rests on the lightweight PathAction interface for recognition costs and weights. Researchers deriving the central measurement bridge cite this module for the geodesic construction.
claimTwo-branch geodesic rotation from angle $θ_s$ to $π/2$ satisfying the recognition path action interface.
background
This module sits in the Measurement domain and imports the PathAction interface. That interface supplies a lightweight setup for recognition paths together with their action and weights; heavy measure-theoretic results on piecewise additivity and domain shifts are deliberately omitted. The module introduces the central object TwoBranchRotation together with supporting definitions for residual action, rate action, amplitude squares, and normalization invariants.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the geometric foundation for the C=2A bridge proved in C2ABridge, which states that C equals 2A exactly for any two-branch geodesic rotation. It is also imported by the lightweight export C2ABridgeLight and by KernelMatch, whose key lemma establishes the pointwise identity J(r(ϑ)) = 2 tan ϑ for the profile r(ϑ) = (1 + 2 tan ϑ) + √((1 + 2 tan ϑ)² − 1). The construction therefore closes the two-branch case inside the recognition path action framework.
scope and limits
- Does not contain measure-theoretic lemmas on piecewise additivity or domain shifts.
- Does not prove the C = 2A equivalence itself.
- Limits attention to two-branch geodesics; multi-branch cases are excluded.
- Does not address higher-dimensional or non-geodesic rotations.