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 formBₓ: Vₓ × Vₓ → R. Accordingly, the decompositionBₓ = gₓ + ωₓis 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 ωₓ = 0. The paper develops the symmetric sector gₓ as the readable metric component of the theory. It produces the quotient Hilbert realizationHₓ = Vₓ / Zₓ, where Zₓ is the symmetric null space. The antisymmetric sector ωₓ is then represented, when admissible, by a skew-adjoint response operator Aₓ 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, sinceωₓ (ξ, ξ) =0. Instead, the scalar content of the antisymmetric response appears only after an admissible quadratic projection: ΔSₓ (ξ) =ḡₓ (Jₓ Aₓ ξ, Jₓ Aₓ ξ) ≥ 0. Under strict compatibility of the lift Jₓ, the zero set of this projected response is exactly the reversible kernel: ΔSₓ (ξ) =0 ⇔ ξ ∈ ker Aₓ. 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.