This paper presents a new way to organize the algebra behind multi-loop Feynman integral reduction. The authors rewrite the usual integration-by-parts (IBP) identities as differential equations for sector-wise generating functions. In that language, many of the index shifts that appear in reductions become actions of differential operators, and the whole problem can be studied in a non-commutative algebra of those operators.

Concretely, the team packages all integrals in a chosen sector into a single generating function. Integration-by-parts identities then turn into differential equations for that generating function. Working at the level of operators instead of individual integrals leads to symbolic recurrence relations among operators. The paper develops an iterative algorithm that (1) generates candidate equations, (2) extracts symbolic reduction rules, (3) updates an active set of rules, and (4) tests whether those rules are complete on the lattice of integral indices.

At a high level the idea is to make the dependence on integral indices explicit and algebraic. Shifts of propagator powers or numerator degrees act like differential operators on the generating function. This shift makes it possible to produce and simplify operator identities, to derive “descendant” equations that follow from earlier rules, and to formulate completeness checks that do not rely on picking a manual seed set of integrals.

The authors illustrate the method on a range of examples. They work through the sunset topology, planar and non-planar massless double-box topologies, several representative subsectors, and a degenerate case in which the top sector contains no master integral. These examples show how symbolic reduction rules, descendant equations and completeness criteria can be brought together inside a single algebraic framework.