Sharp Negative-Association Thresholds and Ten Switching Events in a Single-Circuit Perturbation of a Wheel Forest Measure

We study a probability measure on the edge subsets of the five-vertex wheel: each of its 134 forests receives multiplicative edge weight, and the complete four-edge rim cycle receives an independent nonnegative mass lambda. When the rim activities are one and the common spoke activity is t>0, we prove that negative association of all disjoint increasing events holds precisely for 0<=lambda<=min{P(t),Q(t)}, with explicit rational functions P,Q. The active branch changes at one positive algebraic parameter. The proof combines elementary necessary inequalities with a reproducible certificate of 3,184,542 exact integer maximum-flow conditions. Separately, we give eight rational asymmetric activity coefficients for which the first pairwise-correlation boundary changes its active edge pair exactly ten times as t ranges over (0,infinity). The certificate covers all twelve candidate thresholds, 66 crossing equations, 49 pair-counted positive roots and 46 distinct locations. The wheel/whirl support, ordinary forest pairwise nonpositive correlation, and half-plane properties of homogeneous basis polynomials are established background. We claim the explicit certified statements given here, without asserting exhaustive novelty, optimality of ten switches, or a general theorem for all graphs.