Exact CHY Integrand Construction Using Combinatorial Neural Networks and Discrete Optimization

TL;DR

This paper introduces a combinatorial neural network and discrete optimization approach to construct rational CHY integrands with prescribed pole constraints, reducing the inverse problem to a linear feasibility task.

Contribution

It presents a novel method combining combinatorial neural networks and discrete optimization for exact CHY integrand construction, avoiding training and numerical approximation.

Findings

Exact integrand construction demonstrated at six and eight points.

Pole hierarchy encoded via generalized pole degrees and linear relations.

Method efficiently handles higher-order poles and pole selection.

Abstract

Constructing a rational CHY integrand that realizes prescribed physical pole constraints is a discrete inverse problem whose combinatorial complexity grows with multiplicity. We encode the pole hierarchy through generalized pole degrees $K(A)$ (channels $s_A$), defined as signed internal-edge counts associated with particle subsets in a colored integrand graph. Additivity under integrand multiplication together with the elementary face recursion on the subset lattice expresses all higher-channel $K(A)$ as linear functions of the two-particle data $\{K(s_{ij})\}$ and reduces the inverse step to a mixed-integer linear feasibility problem. The subset lattice provides a fixed dependency graph for deterministic message passing with forward evaluation and backward residual propagation; this computation is parameter-free and involves no training. In factorial-rescaled variables $\widetilde K(A)=(|A|-2)!\,K(A)$, every local update is integral, so propagation is exact in the rescaled recursion variables and does not rely on numerical reconstruction. We further organize generalized integrand graphs by an $n$-regular grading under multiplication, where degree-zero (0-regular) factors act as M\"obius-invariant insertions that can be decomposed into four-point cross ratios. We illustrate the construction at six and eight points, including pick-pole selection and higher-order pole reduction.