We develop a unified, layer-based framework for revealing and quantifying polynomial-time structure within classically NP-complete problems. Each problem is decomposed into constructor families (layers) that are constructible, canonicalizable, and countable under natural symmetry groups in polynomial time. We provide a comprehensive analysis for Sudoku-like exact-cover constraint satisfaction problems on k-squared by k-squared grids, establishing: (i) a separable (Kronecker-like) base layer with a closed-form class-count formula, validated computationally for k in the range 3 through 20; (ii) a broader linear/bi-affine layer with empirically tight class count; (iii) exact Polya enumeration forms for piecewise layers via multiplicative local type catalogs. We extend this framework to Boolean Satisfiability (cyclic/dihedral templates), Traveling Salesman Problem (Cayley/geometric templates), Integer Programming (Kronecker/unimodular templates), and Graph Coloring (lattice/group templates). With fixed layer ordering, orbit sums across layers recover total solution counts under position groups, providing a principled map of polynomial-time structure within NP-hard terrains without contradicting the NP-completeness of the general problems. This construction does not provide a polynomial-time procedure for solving arbitrary instances from partial givens; it is a template-generating method with clearly delineated scope.
Michael Rey (Thu,) studied this question.