Sensitivity Hierarchy and Observable Horizons in Reduced Reheating Geometry

Sensitivity Hierarchy and Observable Horizons in Reduced Reheating Geometry This paper formulates reduced reheating dynamics as a local inverse-geometry problem on the observable triplet D = (u_IV, E, Δθ), where u_IV denotes the onset of the phase-diagnostic window, E is an integrated efficiency measure, and Δθ is the accumulated signed phase displacement. The central object is the reduced Jacobian J^i_a = ∂D^i/∂Θ^a, which determines how local deformations in reheating parameters are transmitted into reduced observable space. On this basis, the paper introduces a quantitative interface-displacement sensitivity S_a = n_i J^i_a, measuring how efficiently each parameter direction shifts a nearby branch interface in reduced data space. This framework yields a sensitivity hierarchy that separates dominant, weak, and tangent deformation directions near a branch interface. Some parameter directions strongly move the local branch structure, while others are nearly ineffective at linear order. The paper then defines an observable horizon from the smallest singular value of the reduced Jacobian, or equivalently from the smallest Fisher eigenvalue, thereby distinguishing genuine geometric rank loss from noise-relative loss of practical observability. In this way, branch fragility and observational accessibility are treated as coupled but distinct structures within the same reduced geometry. Using a minimal local reheating history class A(u) = A0 e^{-αu}, ω(u) = ω0 e^{-βu}, the paper develops explicit diagnostics for branch-interface displacement, weak transmitted directions, and the onset of practical invisibility. It also introduces the branch-resolution zone: a regime in which the branch structure is geometrically present but progressively harder to resolve because the weakest observable channel approaches the noise threshold. A further result is a formulation-sensitive comparison between metric and Palatini reheating. Even when the same reduced observables are used, the corresponding reduced Jacobians, sensitivity hierarchies, and observable-horizon profiles can differ. In a minimal Palatini proxy with an effectively steepened amplitude decay, the dominant-direction boundary and the horizon contours shift relative to the metric case, showing that branch fragility and observational cost can be formulation-selective already at reduced level. Overall, the paper provides a reduced-geometry bridge between reheating dynamics, local inverse problems, and observable-horizon concepts. It clarifies that a reduced signature may appear smooth or weakly structured not because the underlying dynamics are simple, but because the relevant branch structure lies beyond the current observable horizon. V2: This version sharpens the inverse-geometry formulation of reduced reheating observables by clarifying the definition of the local branch interface, weakening the interface-normal identification to a numerical proxy in the one-soft-mode regime, and making the observable-horizon discussion mathematically consistent on the transmitted subspace. The text also refines the metric–Palatini comparison, weakens overly strong grid-level claims, and improves figure captions and notation throughout. V3: We formulate reduced reheating dynamics as a local inverse-geometry problem on the observable triplet D=(uIV,E,Δθ)D=(u_{\rm IV},E,\Delta\theta)D=(uIV,E,Δθ). A sensitivity hierarchy is defined via interface-displacement responses, while an observable horizon is introduced from the weakest transmitted singular channel (or smallest nonzero Fisher eigenvalue), separating geometric structure from noise-limited accessibility. Branch interfaces are treated as classification-induced structures, with numerical results based on weak-channel proxy geometry. A minimal metric–Palatini proxy shows formulation-dependent shifts in both sensitivity hierarchy and horizon structure.