Universal Variation Theory': Readable Phases of a Closed Admissible Structure

This version substantially revises and repositions Universal Variation Theory' as an axiomatic theory of closed admissible readability. Rather than extending the original UVT by adding functional-analytic axioms to a scalar second-variation framework, the present formulation takes a closed admissible structure U as its basic object and identifies mathematics, EVG, logic, information, thermodynamics, and description as compatible readable phases of that structure. Mathematics is treated as the formal-readable phase, not as an exterior foundation; EVG is treated as the response-geometric phase, whose local datum is an admissible ordered second-order response form E_x = g_x + omega_x. The symmetric sector carries quotient geometry, metric, curvature, canonical energy, gravitational response, tail phases, and quadratic visibility readout, while the antisymmetric sector carries oriented response, reversible kernels, chiral and parity-odd readouts, and projected scalar response. Ordinary Hessian geometry is recovered as the specialization omega_x = 0, whereas non-Hessian ordered response is retained through the antisymmetric sector. The paper also formulates logic, information, thermodynamics, and description as internal readable phases, and presents entropy production and information-theoretic reduction through the same kernel/non-kernel response pattern. External reference is excluded as an admissible operation: no exterior mathematical universe, metalanguage, observer, physical law, logical authority, informational primitive, or final completion is used. The final statement of the theory is universal visibility without external foundation: all exact readable presentations occur as compatible phases of U, with all admissible compatibility internal to Adm(U).