weingarten-toolkit: exact Weingarten calculus for networks of Haar-random SO(m) matrices

Abstract

Exact (symbolic, no sampling) Weingarten calculus for networks of Haar-random SO(m) matrices, via two structurally independent computation engines (a "cut and glue" symbolic surgery engine, and a "direct index sum" engine using union-find and bucket elimination). This is the reusable computational core extracted from a research project studying the Weingarten calculus underlying Nissim's SO(3) deconfinement mechanism and its generalizations (see the related technical notes below). It computes exact expectation values of products of traces of Haar-random SO(m) matrices whenever every edge in the network occurs exactly 2, 4, or m times -- the three cases with a known closed-form Weingarten identity. Implemented and tested for SO(m), m >= 3 (numerically exercised for m = 3, 5, 7); SU(N) is not implemented. Ships with a known-answer test suite (classical Haar moments, two real regression cases from development, and a cross-check that both engines agree). MIT licensed. Deliberately does not include the tube-specific geometry that originally used this engine -- this is the generic calculus layer, independent of any one application. See the included README.md for full documentation, scope, and a worked example.