Sensitivity Hierarchy for Branch-Locus Displacement in the Reduced Geometry of Reheating

This paper studies the local sensitivity hierarchy that governs branch-locus displacement in the reduced geometry of reheating. In the spatially extended moving-boundary framework, the reduced observables Di(R∗)=(uIV,E,Δθ)D^i(R_\ast)=(u_{\rm IV},E,\Delta\theta)Di(R∗)=(uIV,E,Δθ) define a local map from effective reheating-history parameters to onset, weighted access, and accumulated signed phase displacement. When this reduced map loses local rank, the inverse problem develops a fold structure, and the corresponding branch-locus displacement is controlled not by the total reduced-data perturbation but by its projection onto the fold-normal covector in reduced-data space. Within the power-law history class A(u)=A0e−αuA(u)=A_0e^{-\alpha u}A(u)=A0e−αu, ω(u)=ω0e−βu\omega(u)=\omega_0e^{-\beta u}ω(u)=ω0e−βu, the paper extracts the local fold normal from the reduced Jacobian of the map (A0,α,ω0,β)↦(uIV,E,Δθ)(A_0,\alpha,\omega_0,\beta)\mapsto (u_{\rm IV},E,\Delta\theta)(A0,α,ω0,β)↦(uIV,E,Δθ) and classifies the lever ranking into onset-dominated, efficiency-dominated, phase-dominated, and mixed sectors. It then applies this sensitivity basis to the metric–Palatini contrast, showing that the relevant local discriminator is the fold-normal projection of the reduced-data difference rather than its raw magnitude. The paper also introduces a framework for minimal non-power-law deformations, viewed as a finite-dimensional unfolding of the power-law core, and proposes a stability classification of fold sectors into robust, marginal, and transition categories. A full numerical survey of the deformation-induced stability boundaries is left for future work. Overall, the results show that branch loci in reheating-imprinted phase textures are characterized not only by their existence and location but also by a local sensitivity hierarchy of reduced directions that determines which observable most efficiently drives branch-locus displacement once branching is present. Version 2 clarifies the reduced inverse-geometry framework for branch-locus displacement in reheating-imprinted phase data. The revision replaces strict local-bijectivity language with full-row-rank regularity for the reduced map, corrects the interpretation of the smallest-singular-value maps, fixes the formulation-channel fractions in Table 1, and unifies the sensitivity-direction terminology throughout. The result is a cleaner and more consistent formulation of how the effective transverse sensitivity direction controls branch-locus displacement and metric–Palatini discrimination.