Sharp Selector Bounds and a 1/5 Saddle Algorithm for Common-Kernel Bimatrix Games: Recognition, fixed-normalization structure, robustness, and consequences for the 1/3-threshold compiler program

This paper develops a structured algorithmic and certification theory for a class of symmetric bimatrix games generated by a common kernel. The methodology emerged from a broader research program aimed at the deterministic one-third approximation threshold for bimatrix Nash equilibria. A candidate reduction architecture was found to contain a tractable common-kernel region, turning part of the hardness-design program into an independently verifiable approximation and certification framework.The main result is a polynomial-time algorithm that computes a rational one-fifth approximate Nash equilibrium for every rational game in the common-kernel class using a single zero-sum saddle-point computation. The paper further strengthens this procedure through an exact one-dimensional post-processing step that searches the segment between the two saddle strategies and can produce strictly better instance-specific equilibria while preserving the same worst-case guarantee.The common-kernel representation is made fully algorithmic. The paper provides an exact membership test, proves uniqueness of kernel recovery, characterizes the class as an affine image of a compact convex polytope, and explains the structural origin of the factor-three decomposition in terms of symmetric and antisymmetric interaction components. It also shows how normalized symmetric bimatrix games embed positively affinely into the class, with exact control of how unilateral deviation gains and approximation regret rescale.A sharp selector theorem is established for the declared full-subset selector family, providing an exact calibration benchmark for uniformity gadgets used in approximate-equilibrium reductions. The paper also clarifies that this sharp constant is specific to the full-subset representation and cannot automatically be transferred to compressed or fixed-cardinality selector constructions.The theory is extended beyond the exact class through an explicit linear-programming projection for arbitrary rational square games. The nearest common-kernel representation, its witness kernel, and a dual optimality certificate are all computable in polynomial time. This yields a directly checkable robustness guarantee for games close to the common-kernel class and supports three explicit procedures: Recognize-and-Recover, Common-Kernel Saddle Solver, and Project-and-Solve.The reduction-theoretic consequences are deliberately local and conditional. Under an explicit fine-grained source-hardness premise, compiler outputs lying in the common-kernel class, or sufficiently close to it, cannot support the targeted threshold bridge. A separate searchability result shows that a semantic high region cannot simultaneously be nonempty, uniformly subexponentially searchable, and guaranteed to decode to a valid source solution. These results serve as design constraints for future Nash-equilibrium hardness constructions rather than as a global impossibility theorem.The paper also records two broader reduction-design lessons: diffuse semantic validity need not produce a heavy decodable witness, and easily searchable semantic regions cannot themselves sustain the intended fine-grained hardness mechanism. Reproducibility is supported by computational checks of the selector identities, the one-fifth approximation theorem on random rational instances, and the primal-dual projection certificates.Overall, the paper contributes a unified framework combining polynomial-time approximation, exact structural recognition, affine geometry, robust projection, primal-dual certification, selector calibration, and mathematically checkable barriers for future Nash-equilibrium reduction design. It is positioned as a structured-game algorithm and certification paper that arose from the broader one-third-threshold research program, without claiming to resolve the global deterministic approximation-threshold problem.