A vanishing lemma for doubled plaquettes touching a loop edge, in Weingarten calculus for SO(m)

Abstract

We record a vanishing phenomenon for exact (non-Monte-Carlo) Weingarten calculus computations on SO(m), encountered while searching for non-local closing constructions for the residual 3⁻³⁴ discrepancy in reconstructing Nissim's SO(3) deconfinement tube. Call a plaquette a 4-cycle of directed edges, and say it is doubled in a trace network if it occurs twice, identically (in either relative orientation). We prove that if a doubled plaquette has an edge e₀ of loop type (total multiplicity exactly m in the network, so its expectation is governed by the order-m moment identity of Haar(SO(m))) and at least two of its other three edges are isolated (total multiplicity exactly 2, contributed only by the two copies of the plaquette), the value of the network is identically zero, for every m≥3 and both relative orientations (Theorem 3.2). We further report, as a computationally verified but not symbolically proven proposition, that the same vanishing persists far more broadly: across 108 exact rational-arithmetic evaluations covering every way of resolving a doubled plaquette's other edges (isolated or order-4, in any mixture, with 1, 2, or 3 of the plaquette's own edges themselves of loop type), for m∈{3,5,7} and both orientations, every case vanished (Proposition 4.1). As a control, the same construction with no loop edge at all gives 1 in all 36 cases checked, confirming the vanishing is tied specifically to the loop edge, not to doubling per se. Finally we report an explicit exception: a doubled plaquette all four of whose edges are loop edges (hence sharing no edge with the rest of the network) need not vanish -- we compute its value in closed form to be 1/27 for SO(3) (confirmed by three independent implementations) and 1/1000 for SO(5) (confirmed by two).