Reflection Positivity and the Mirror Sector: The suppression rate is the gap, and a volume-independent symmetric mirror gap in three spatial dimensions

The rate is the gap [proved]. The reflection-positive form of any mirror operator is a sum of e^(−τE) with non-negative weights. Reflection positivity therefore suppresses the mirror exactly at the rate of its gap. A canonical mirror field is never a null vector, because its particle and hole channels sum to one. One-body blindness [proved, checked]. The reflection that exchanges matter and mirror walls treats both walls alike at the one-body level, so any mechanism that suppresses the mirror alone is many-body. Measured in three spatial dimensions [computed]. A sign-free determinantal QMC computation of a reduced Spin(10) mirror model on L³ lattices (L = 4, 6, 8; β = 8; two flavours) finds a mirror gap independent of the volume: 0.667(10), 0.654(12), 0.677(5). There is no long-range order, no symmetry breaking, and the average sign is one. A momentum-resolved criterion, fixed before the largest volume was run, finds all six common momenta converged: the gap is open and symmetric. For the 16 the gap at L = 4 is 8.90(7). In this reduced model the result is the necessary condition for symmetric mass generation, now met in the dimension in which the question is posed. The decisive test, with an interaction that admits no invariant bilinear, is stated. Derivations, verification code and drafting were carried out in collaboration with Claude (Anthropic).