Positive cubature on $S^2$:low-degree rigidity and uniform bounds

Let $N_t$ denote the least number of nodes in a positive cubature formula of degree $t$ on $S^2$. We prove that a formula of degree $2m+1$ cannot have exactly $(m+1)(m+2)+1$ nodes for any $m\ge 2$. Excluding equality in the Fisher bound then gives $$N_{2m+1}\ge (m+1)(m+2)+2$$ for $m\ge 3$, and known constructions yield the exact values $N_7=22$ and $N_9=32$. For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient $13\mathrm{/}8$, parity-dependent linear terms, and an $O(m^{2\mathrm{/}3})$ remainder. We determine the sharp constant $49\sqrt[3]{3}\mathrm{/}72$ for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP–Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.