Axiomatic Second-Variation Geometry: Admissible Second-Order Response Forms and Symmetric–Antisymmetric Structure

This is Version 2 of “Axiomatic Second-Variation Geometry: Admissible Second-Order Response Forms and Symmetric–Antisymmetric Structure.” This version substantially reconstructs the framework of the first version. The central clarification is that the basic object of the theory is not the ordinary Hessian of a scalar functional, but an admissible ordered second-order response formB_x : V_x × V_x → R.Accordingly, the decompositionB_x = g_x + ω_xis formulated as a decomposition of an admissible response form, not as a decomposition of an ordinary scalar Hessian. The ordinary Hessian case is recovered as the symmetric specialization ω_x = 0. The paper develops the symmetric sector g_x as the readable metric component of the theory. It produces the quotient Hilbert realizationH_x = V_x / Z_x,where Z_x is the symmetric null space. The antisymmetric sector ω_x is then represented, when admissible, by a skew-adjoint response operator A_x on this readable Hilbert realization. A main correction and strengthening of this version is the treatment of the antisymmetric sector. Pure antisymmetry is not interpreted as scalar curvature or negative Hessian curvature, sinceω_x(ξ,ξ)=0.Instead, the scalar content of the antisymmetric response appears only after an admissible quadratic projection:ΔS_x(ξ)=ḡ_x(J_x A_x ξ, J_x A_x ξ) ≥ 0.Under strict compatibility of the lift J_x, the zero set of this projected response is exactly the reversible kernel:ΔS_x(ξ)=0 ⇔ ξ ∈ ker A_x. The paper also proves stability under admissible re-description, separates coercive readable geometry from compact or localized response, and gives schematic realizations in arithmetic, Dirichlet–spectral, and Boolean/complexity settings. These realizations are presented as structural templates rather than as direct claims of number-theoretic or complexity-theoretic results. Overall, Version 2 clarifies the boundary between ordinary Hessian geometry and admissible second-order response geometry. It preserves the standard symmetry of scalar Hessians while allowing genuinely non-Hessian ordered response data to be represented through an antisymmetric sector, a reversible kernel, and a non-negative projected response.