Linear independence conjecture for higher-order blow-up scalar fields

Let $V$ be a set with one Poisson particle source for each element, and let a degree-$r$ arrival experiment receive particles in successive batches of $r$, silencing each source after it contributes at least one particle. An arrival sequence is a possible outcome, and its probability is a function of the source rates. Let $b\mathcal P_r\Lambda^0$ be the span of these probability functions. Higher-order blow-up basis conjecture. These probability functions are linearly independent and hence form a basis for $b\mathcal P_r\Lambda^0$. If true, this would provide the proposed higher-order scalar blow-up space with a canonical probability-function basis; the source gives no resolution.