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) Dⁱ (R_) = (u ₈ₕ, E, ) 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) =A₀e^- uA (u) =A0e−αu, ω (u) =ω0e−βu (u) =₀e^- uω (u) =ω0e−βu, the paper extracts the local fold normal from the reduced Jacobian of the map (A0, α, ω0, β) ↦ (uIV, E, Δθ) (A₀, , ₀, ) (u ₈ₕ, E, ) (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.
Hiroyuki Shioiri (Fri,) studied this question.