Bounded Ratios of Lorentzian Polynomials II: The Complete Quadratic Local-to-Global Classification
Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs $(n,d)$ these quadratic-slice ratios generate the full bounded-ratio cone for every $M$-convex support $S\subseteq\Delta_n^d$. For $d\geq 2$, this quadratic local-to-global principle holds universally if and only if \[ n\leq 3,\qquad d=2,\qquad\text{or}\qquad (n,d)=(4,3). \] In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in $n\geq 5$ variables and for polynomials of degree $d\geq 4$ in $n\geq 4$ variables. We identify the two minimal obstructions, at $(n,d)=(4,4)$ and $(n,d)=(5,3)$, and propagate them using cut-cone certificates and variable-lifting arguments. The quartic bounded ratio extends to every higher degree by differentiation, while perturbing a square-transportation direction yields an explicit degree-uniform family of separating functionals. As a conceptual byproduct, we show that transportation costs on finite median graphs give polar directions in arbitrary dimension and degree.