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

This paper develops a quantitative, mathematically rigorous, and computationally auditable framework for certifying the local persistence of symmetric Nash equilibria in common-kernel bimatrix games. It translates classical essentiality, fixed-point index, and parametric stability theory into explicit, verifiable robustness certificates for declared equilibria under payoff uncertainty, parameter perturbations, and finite computational error. The revised framework integrates strengthened continuation theorems, sharper perturbation bounds, exact support-based regularity tests, and constructive certificates for auxiliary zero-sum saddle nonuniqueness. The analysis employs the affine common-kernel representation R_T = (J + 2T − Tᵀ)/3, C_T = R_Tᵀ, with directional matrix D_T = 2T − Tᵀ = 3R_T − J. Building on the previously established affine inverse T = 2R + Rᵀ − J and parallelotope characterization of the admissible kernel cube K = [0,1]^(m×m), the paper derives sharp coordinate-norm bounds, exact boundary-distance formulas, and explicit conversions between kernel and symmetric-payoff perturbations. It proves that common-kernel-relative symmetric essentiality and ambient symmetric-strategy essentiality coincide for equilibrium points and specified isolated components throughout the closed cube, including boundary kernels. This equivalence holds without requiring equilibrium regularity. For a support-regular symmetric Nash equilibrium, the support equations reduce to an explicit matrix A_S(T) = BᵀR_TB. Nonsingularity of this reduced matrix, together with strict inactive payoff inequalities, establishes isolation in the entire strategy simplex. The local symmetric Nash index is computed algebraically as ind_N(T, x*) = sgn(det[−A_S(T)]). The resulting nonzero-index certificate guarantees qualitative local essentiality. A sufficient positive-index condition is also obtained from negative definiteness of the symmetric part of the reduced directional matrix. These results concern the symmetric-strategy index and distinguish it from the full product-space equilibrium index. The quantitative continuation theorem is extended to arbitrary ambient reference kernels, rather than being restricted to K. Explicit sufficient perturbation radii guarantee preservation of support, strict inactive payoff gaps, local isolation, and bounded strategy displacement. The bounds depend on the smallest active probability, the minimum inactive payoff gap, the reduced-matrix inverse norm, and the reference payoff scale. Direct payoff-space estimates eliminate the unnecessary factor-three loss associated with inverse-coordinate conversion, yielding stronger certified uncertainty tolerances. Strict pure equilibria are handled separately through a direct payoff-gap certificate, without invoking an inverse of a zero-dimensional reduced matrix. The auxiliary zero-sum connection is characterized precisely through support geometry. For a saddle pair (p, q) of D_T, the inclusion supp(q) ⊆ supp(p) guarantees that q is an exact symmetric Nash equilibrium; diagonal equality p = q is unnecessary. Crucially, strict support inclusion necessarily produces an unused best response and a zero inactive Nash gap. Consequently, the strict-gap support-regularity certificate cannot apply in this case, even if the reduced matrix on q's smaller support is nonsingular. An explicit counterexample demonstrates that persistence may fail. The revised certification workflow therefore distinguishes exact equilibrium verification from regularity and persistence certification, rather than incorrectly treating strict inclusion as sufficient for all three. Under equal support, the tangent identity Q = B_SᵀD_SS B_S = 3A_S(T) establishes the equivalence between active bordered-system regularity and reduced Nash invertibility. The two-block saddle determinant satisfies det(K_tan) = (det Q)², so it detects nonsingularity but does not retain the orientation information required for the signed symmetric Nash index. The complete auxiliary saddle correspondence is treated as a nonempty, compact-valued, upper-hemicontinuous set-valued mapping, without assuming a globally continuous selector. For active-regular equal-support saddle pairs with strict inactive inequalities for both players, improved probability-vector residual estimates yield explicit perturbation radii, support-preserving smooth continuation, and quantitative displacement bounds. These estimates remove the explicit multiplier involving the absolute saddle value, although the full bordered inverse norms may still depend on that value. A local-isolation argument, followed by convexity of the complete zero-sum saddle set, proves global uniqueness of the continued saddle pair throughout the certified neighborhood. Its column strategy is an exact support-regular symmetric Nash equilibrium and coincides with the independently certified Nash branch wherever both continuation certificates apply. A further contribution of the revised analysis is a constructive nonuniqueness theorem. After successful equal-support Nash regularity tests, a zero inactive column gap is shown to imply saddle nonuniqueness under the stated nonsingularity and strict inactive row-gap hypotheses. The new result explicitly constructs a nonconstant segment of saddle column strategies, with a positive admissible step bound and an exact rational witness for rational inputs. Thus the certification procedure can establish saddle nonuniqueness while retaining a valid Nash persistence certificate. Nash exactness, Nash regularity and persistence, and saddle uniqueness are reported as logically separate outcomes. The computational framework comprises two explicit certification algorithms supported by exact rational arithmetic and outward-rounded interval verification. Its checks include determinant signs, validated matrix inverses and residuals, active probability positivity, inactive payoff margins, solver-output reconstruction, and finite-error perturbation budgets. Worked examples cover gap-limited proper supports, equal-support saddles with p ≠ q, strict-inclusion failure of persistence, Nash regularity despite saddle nonuniqueness, exact boundary-mesh certification, and complete interval reconstruction from approximate numerical output. Additional computational audits and independently replayable solver records strengthen the reproducibility of the certification procedures without being substituted for the mathematical proofs. For isolated equilibrium components, the framework provides quantitative persistence guarantees conditional on a supplied isolating neighborhood, an independently certified nonzero component index, and a validated boundary covering. The finite-cover displacement estimate does not independently establish these prerequisites. General persistence for arbitrary degenerate or positive-dimensional components is not claimed. The contribution is a coordinate-explicit synthesis and quantitative refinement of established equilibrium-index, sensitivity, and perturbation principles, rather than a claim of foundational priority over classical stability theory. It provides finite, independently checkable certificates for regular isolated symmetric Nash equilibria and precisely delineates when auxiliary zero-sum solutions yield exact equilibria, persistent Nash branches, globally unique saddle branches, or constructive nonuniqueness witnesses. The framework supports ex-post robustness assessment in game-theoretic models after estimation, recalibration, numerical optimization, scenario modification, and finite-precision computation. All reported perturbation radii are sufficient guarantees; failure to meet a certified bound does not by itself establish instability or nonexistence of nearby equilibria.