IndisputableMonolith.Measurement.PathAction
Recognition paths are formalized here as time-parameterized positive rate functions together with associated action, weight, and amplitude maps. Measurement modules deriving Born's rule and the C=2A bridge import these objects to connect the J-cost to amplitudes. The module is definitional only, importing the Cost primitives and exposing RecognitionPath plus pathAction, pathWeight, and pathAmplitude.
claimA recognition path is a map $r:ℝ→ℝ_{>0}$. The path action is the functional $A[r]=∫J(r(t))dt$, the path weight is $w[r]=exp(-A[r])$, and the path amplitude is $α[r]=√w[r]·exp(iφ[r])$.
background
The module sits in the Measurement domain and imports only the Cost module, which supplies the J-cost functional. RecognitionPath is introduced exactly as a time-parameterized positive rate function. Sibling definitions pathAction, pathWeight, pathAmplitude, pathWeight_pos, and amplitude_mod_sq_eq_weight then equip each such path with its integrated cost, exponential weight, and complex amplitude.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The definitions feed four downstream modules. BornRule uses them to derive P(I)=|α_I|² from J and the bridge 𝒜=exp(-C/2)·exp(iφ). C2ABridge relies on them for the exact equivalence C=2A on two-branch geodesics. KernelMatch uses them for the pointwise identity J(r(ϑ))=2 tan ϑ. TwoBranchGeodesic uses them for the residual-norm and rate-action formulas.
scope and limits
- Does not derive Born's rule or any probability statement.
- Does not prove the C=2A identity or the kernel match.
- Does not introduce the two-branch geodesic geometry.
- Does not reference the forcing chain T0-T8 or the phi-ladder.
- Does not contain numerical evaluations or constant bounds.