Compatible Evolution and Stability in Axiomatic Second–Variation Geometry

This paper develops an axiomatic framework for the analysis of compatible evolution on second-order response geometries. The primitive response datum is an ordered bilinear form written as Bₓ = gₓ + omegaₓ. Its symmetric sector gₓ determines a readable Hilbert geometry, while its alternating sector omegaₓ is represented by a skew-adjoint response operator Aₓ. A central distinction of the paper is that response geometry does not, by itself, determine an evolution law. Dynamics must be supplied independently through a compatible generator Lₓ acting on the readable Hilbert realization. The metric decomposition Lₓ = Dₓ + Kₓ separates the symmetric, energy-changing sector Dₓ from the skew-adjoint, metrically neutral sector Kₓ. Along a compatible evolution, the readable quadratic energy satisfies dEₓ/ds = gₓ (xi, Dₓ xi). Accordingly, neither the antisymmetric response operator Aₓ nor the skew evolution sector Kₓ contributes directly to the diagonal quadratic energy derivative. Stability, criticality, and amplification are governed by the symmetric sector of the compatible generator rather than by the magnitude of the antisymmetric response alone. The principal dynamical quantity is the numerical abscissa omegag (Lₓ). Its sign distinguishes three local regimes: omegag (Lₓ) 0: existence of an instantaneously amplifying direction and finite-parameter norm amplification. The paper proves local dissipative stability under strict coercivity, establishes persistence of that stability under controlled higher-order nonlinear remainders, and gives a precise numerical-range criterion for transient amplification. It also separates spectral stability, asymptotic semigroup decay, non-normality, and transient growth. Critical reductions, slow response, metastable persistence, polynomial decay, and branching are obtained only under the additional spectral, invariant-subspace, and nonlinear hypotheses required for those conclusions. Representative finite-dimensional and operator-theoretic profiles illustrate the distinction between response geometry and compatible evolution. Technical appendices collect the required semigroup theory, numerical-range estimates, nonlinear remainder criteria, and a complete notation and dependency table. This version is a structural revision of the earlier stability formulation. The earlier work began from the broader idea that second variation could provide a unified basis for explaining stability phenomena. The present revision preserves that starting insight while refining its mathematical scope. The explicit decomposition Bₓ = gₓ + omegaₓ makes it possible to distinguish response geometry from evolution and to identify precisely which additional structure is required for dynamical conclusions. The revision therefore removes the identification of second-order response with dynamics, replaces norm-based antisymmetric distortion criteria with the numerical abscissa of the compatible generator, and restricts dynamical conclusions to the analytic hypotheses under which they are justified. The resulting logical structure is: response geometry compatible evolution analytic control-> conditional stability, criticality, or amplification. All principal constructions and conclusions are invariant under admissible unitary re-description.