The Representation-Escape Tournament: Which Hardness Falls to Which Mathematical Weapon?

Rather than first guessing at an ultimate invariant, this round pits five classic hardness families (Tseitin/XOR, Pigeonhole, Clique, general SAT, and the TSP/CUT/Stable-Set polytopes) one by one against six representation weapons (resolution, algebrization, monotone circuits, treewidth decomposition, knowledge compilation, and LP extended formulations), assigning each cell one of three verdicts: “escape,” “lower bound,” or “unknown.” The matrix immediately shows that no row forms a general lower bound across every column, and no single weapon escapes universally — Clique's strong lower bound against monotone circuits cannot be extrapolated to general circuits (the missing key primitive is exactly the forbidden negation), and the LP extended-complexity lower bound for the TSP/CUT polytopes constrains only linear representations. Proposes the representation-escape profile (REP), recasting the object of study from “hardness” into the binary relation “hardness relative to which representation.”