The Polar Side of Moonshine: Cusp Poles of Twined Traces and Counterexamples to a Conjecture of Dong and Mason

Let v₁₂ be the unique Monster-invariant Virasoro primary of weight twelve in the moonshine module, normalized by Z(v₁₂, τ) = Δ(τ). We compute its twined one-point functions for the classes 2A, 2B, 3A, 3B, 4A, 4C, 5A and 5B. For the four Fricke classes the traces have simple poles at the cusp 0, with exact positive eigenvalues on the one-dimensional twisted ground spaces. The four non-Fricke traces are holomorphic but have nonzero constant terms at that cusp. None of the eight is a cusp form, so all are counterexamples to a conjecture of Dong and Mason (Conjecture (B)). We determine every primary compression scalar through weight eight; the compressed zero mode generates the full Monster commutant of each of these primary spaces. The reconstruction uses character identities, Virasoro Ward identities and twisted-module bounds. Unused Fourier coefficients and the absence of weight-one states in the four Fricke-twisted modules provide exact checks. At level four we also compute the constants at the third cusp. Modular covariance then gives a pole at a suitable cusp for all 94 nonidentity classes with a power in 2A, 3A, 4A or 5A.