Sharp Selector Bounds and a 1/5 Saddle Algorithm for Common-Kernel Bimatrix Games: Recognition, fixed-normalization structure, robustness, and certified compiler barriers

This paper develops a structured algorithmic theory for a class of symmetric bimatrix games generated by a common kernel. It combines sharp selector bounds, a polynomial-time approximation algorithm, exact recognition and kernel recovery, robustness certification, structural complexity results, and conditional barriers for the design of Nash-equilibrium hardness reductions. The study is organized into six methodological components and six corresponding theorem-level results. First, the paper analyzes a normalized full-subset selector game and proves a sharp finite-dimensional regret-to-uniformity bound. The constant in this bound is shown to be optimal for the declared payoff family by an explicit matching construction. This result provides an exact calibration benchmark for selector gadgets used in approximate-equilibrium reductions. Second, the paper introduces and studies a common-kernel class of symmetric bimatrix games. For every rational game in this class, a single zero-sum saddle-point computation is sufficient to construct a rational 1/5-approximate Nash equilibrium in polynomial time. The corresponding threshold policy is shown to be tight relative to the two branch estimates used in the proof. The 1/5 guarantee is a fixed-normalization result for the declared payoff representation and is not claimed to be invariant under arbitrary positive affine rescaling. Third, the common-kernel representation is made fully algorithmic. The paper gives a polynomial-time membership test for arbitrary rational input games, proves uniqueness of the recovered kernel, and shows that every normalized symmetric bimatrix game has a positive-affine embedding into the common-kernel class. Under this embedding, every unilateral deviation gain, and therefore approximation regret, is scaled by exactly one third. Fourth, the paper determines the structural position of the common-kernel class relative to several neighboring game families. It characterizes its common-payoff slice, shows that the class has no dimension-independent rank bound, separates it from fixed-rank symmetric games at the declared normalization, and transfers PPAD-hardness of computing an exact symmetric Nash equilibrium into the class. These results distinguish approximation tractability at constant error from exact-equilibrium complexity. Fifth, the exact theory is extended to arbitrary rational square games through an explicit linear-programming projection. The nearest common-kernel game and a corresponding witness kernel can be recovered in polynomial time. This yields a robust approximation certificate whose quality degrades continuously with the distance from the common-kernel class. As a result, the paper identifies an algorithmically checkable neighborhood in which the structured approximation method remains useful. Sixth, the paper develops conditional reduction-theoretic consequences. Under an explicit fine-grained source-hardness premise, compiler outputs that lie exactly in the common-kernel class, or sufficiently close to it, cannot support the targeted approximation-threshold bridge. A separate threshold-searchability theorem shows that a semantic high region cannot simultaneously be nonempty, uniformly searchable in subexponential time, and guaranteed to decode to a valid source solution. The practical contribution is computational and certification-oriented. The results provide directly implementable procedures for recognizing common-kernel games, recovering their unique kernels, computing certified approximate Nash equilibria, projecting nearby games onto the structured class, quantifying approximation guarantees, and auditing proposed fine-grained reductions by explicit linear-programming tests. The paper also isolates several design limitations that are useful beyond the common-kernel setting. In particular, diffuse semantic validity need not imply concentration on a heavy valid witness, compressed selector constructions cannot inherit sharp full-subset constants without separate analysis, and easily searchable semantic regions cannot themselves sustain the intended fine-grained hardness mechanism. The work is positioned as a structured-game algorithm and certification paper. It does not claim to solve the global deterministic approximation-threshold problem for bimatrix Nash equilibria, does not claim that 1/5 is optimal among all polynomial-time algorithms for the common-kernel class, and does not claim a global impossibility result for all genuinely two-sided compiler architectures. Overall, the paper contributes a unified framework combining sharp quantitative analysis, polynomial-time approximation, exact structural recognition, robust projection, equilibrium-complexity separation, and mathematically checkable barriers for future Nash-equilibrium reduction design.