Quantitative Local Persistence Certificates for Symmetric Nash Equilibria in Common-Kernel Coordinates

This paper develops a quantitative, auditable framework for certifying local persistence of symmetric Nash equilibria in common-kernel bimatrix games. The central objective is to bridge classical qualitative equilibrium-stability theory with finite, computable robustness certificates for a declared exact equilibrium under explicit perturbation budgets. The analysis begins with the common-kernel coordinate map R_T = (J + 2T − Tᵀ)/3, C_T = R_Tᵀ, and derives the exact full-dimensional parallelotope of admissible symmetric payoff matrices, its facet geometry, inverse map, and sharp coordinate norm bounds. For a support-regular isolated symmetric equilibrium, the equilibrium equations are reduced to an explicit support matrix A_S(T), and the local symmetric Nash index is computed algebraically as sgn(det[−A_S(T)]). This converts a topological persistence condition into a directly checkable finite-dimensional certificate. A central solver-to-topology result is the aligned zero-sum saddle identity Q = B_Sᵀ D_SS B_S = 3A_S(T), which establishes an exact correspondence between the reduced saddle KKT system and the reduced Nash system. The associated tangent saddle determinant satisfies det(K_tan) = (det Q)², showing explicitly why a generic saddle determinant cannot recover the sign of the local symmetric Nash index. The paper therefore separates invertibility from orientation and identifies the precise additional information required for a valid local index certificate. The nonuniqueness of auxiliary zero-sum saddle points is treated without imposing an artificial global selector. The full saddle set is handled as a compact-valued upper-hemicontinuous correspondence, while active regularity and strict margins yield a unique quantitative local branch with an explicit perturbation radius. The framework also provides finite continuation radii, displacement estimates, boundary-margin certificates, and exact or interval-verifiable pass/fail tests based on support probabilities, inactive payoff gaps, inverse-matrix or singular-value data, and the common-kernel perturbation scale. The contribution is deliberately coordinate-specific rather than foundational: the paper does not propose a new abstract theory of essentiality, fixed-point index, or parametric stability. Its novelty lies in integrating these classical mechanisms into a closed, explicit, and verifiable certification architecture for common-kernel symmetric equilibria. The resulting framework is intended for ex-post robustness verification after model estimation, recalibration, numerical optimization, or finite-precision computation, while clearly separating regular isolated equilibria from positive-dimensional components and unaligned saddle configurations.