A vanishing theorem and computational obstructions for extending the SO(3) deconfinement mechanism of Nissim to SO(2N+1)

Nissim (arXiv:2605.16162) proved that SO(3) lattice Yang-Mills theory at strong coupling fails Wilson's confinement criterion, via an explicit "tube" cluster in the cluster expansion whose leading contribution is computed exactly using third-moment identities for Haar(SO(3)). A natural question, raised explicitly in Nissim's paper (Section 5), is whether the same mechanism extends to SO(2N+1) for N>1, which would be a step toward deconfinement results for general odd orthogonal gauge groups. We record two results toward this question. First, we give an independent derivation of the exact m-th moment formula for Haar(SO(m)) (a corollary of Lehrer-Zhang's invariant theory computation) in the form needed here. Second, we prove that Nissim's exact tube construction, transported verbatim to SO(2N+1), gives an identically zero contribution for every N>1 — not a numerically small one — because every loop edge in that construction only reaches total multiplicity 3, which is below the threshold m=2N+1 required for a nonvanishing odd moment. This yields an explicit necessary condition any candidate replacement construction must satisfy. We also report a negative computational finding obtained while testing the simplest such replacement, which we record to save a future attempt the same work: a "boosted-degree" variant of the tube for SO(7) that reaches the correct total multiplicity does not show a statistically significant nonzero signal up to N=1.5×10^7 Monte Carlo samples. The extension of the deconfinement mechanism to SO(2N+1), N>1, remains open. (A separate, unrelated discrepancy we encountered while reconstructing Nissim's own SO(3) tube combinatorially is reported in a companion note, https://doi.org/10.5281/zenodo.22778193.)