Inverse Cachazo-He-Yuan Integrand Construction from Factorization Channels with Exact Integer Constraint Propagation

In the Cachazo-He-Yuan (CHY) representation, many tree-level scattering amplitudes are written as integrals localized by the scattering equations, where the CHY integrand encodes the theory and must reproduce the physical factorization pattern of Mandelstam singularities. The inverse CHY problem therefore asks for a rational integrand whose physical poles match a prescribed set of factorization channels, a task that becomes rapidly combinatorial at higher multiplicity. We present an algorithmic approach that makes the subset structure of pole data explicit by assigning each particle subset A a generalized pole degree K(A), realized as a signed internal-edge count in a colored integrand graph. Additivity under integrand multiplication and an elementary face recursion on the subset lattice imply that all higher-channel K(A) are linear functions of the two-particle integers {K(si1)}, so the inverse step becomes a mixed-integer linear feasibility problem. The same subset lattice provides a fixed computation graph for exact constraint satisfaction: pole requirements and recursion identities are treated as hard constraints, updates are restricted to those that preserve feasibility, and whenever a proposed change would enter a fixed node its constraint residual is rerouted and redistributed through unfixed degrees of freedom so that all constraints remain satisfied throughout. After a factorial rescaling, every local propagation update is purely integral, keeping the entire procedure exact in integer arithmetic and avoiding a posteriori reconstruction from numerical evaluations. Since physical tree-level amplitudes must contain only simple factorization poles as dictated by unitarity and locality, we further organize generalized integrands by an n-regular grading under multiplication, where degree-zero factors act as M & ouml;bius-invariant insertions admitting a decomposition into products of four-point cross ratios, which guides efficient higher-order pole reduction. We illustrate the construction at six and eight points, including pole selection and higher-order pole reduction.