Full-orbit and tie-randomized permutation p-values

In a setup as in Measurable finite-group permutation setups, the inclusive full-orbit upper-tail p-value at $x\in \mathcal{X}$ is $p_{\mathrm{o}\mathrm{r}\mathrm{b}}(x):=\frac{\mathrm{\#}\{g\in G:T(g\cdot x)\ge T(x)\}}{\mathrm{\mid }G\mathrm{\mid }}\mathrm{.}$ If $U$ is independent of the data and uniform on $[0,1]$, meaning $\mathbb{P}(U\le u)=u$ for every $u\in [0,1]$, the tie-randomized full-orbit p-value is $p_{\mathrm{t}\mathrm{i}\mathrm{e}}(x,U):=\frac{\mathrm{\#}\{g\in G:T(g\cdot x)>T(x)\}+U\mathrm{\#}\{g\in G:T(g\cdot x)=T(x)\}}{\mathrm{\mid }G\mathrm{\mid }}\mathrm{.}$ Independence is in the sense of Independent random elements. Each count is over group elements with multiplicity, so nonfree actions are handled without replacing $\mathrm{\mid }G\mathrm{\mid }$ by the number of distinct orbit points.

On the finite-space special case with invariant null laws, these formulas are the inclusive and tie-randomized ranks of Orbit upper-tail rank p-values. The null-law validity argument is supplied separately from this definition.