arXiv:2610.41255v1 Announce Type: new Abstract: For multinomial counts $X\sim\operatorname{Mult}(N;\mathbf p)$ on $n$ ordered bins and a reference law $\mathbf r$ we study the expectation of the reference-centered discrete total variation $d_{\mathbf r}(X)=\sum_{j=1}^{n-1}|(X_{j+1}-Nr_{j+1})-(X_j-Nr_j)|$, the natural extension of the comb-test statistic to a non-uniform null. For a non-uniform null the raw total variation of the counts carries a deterministic staircase of order $N$ that no centering of the statistic removes; the counts are therefore centered at the reference law, and under the null ($\mathbf p=\mathbf r$) the statistic is the matched $d_{\mathbf p}$. Exact finite-sample expectations are obtained by conditioning on the total mass in each adjacent pair, which reduces every edge to a translated binomial absolute-deviation problem. For fixed $n\ge3$ and a sampling law bounded away from the boundary of the simplex, the matched statistic admits a three-term asymptotic expansion in powers of $1/N$ whose coefficients are explicit continuous periodic functions of $N(p_{j+1}-p_j)$ (a generalized, non-Poincar\'e asymptotic expansion), obtained by a weighted local Edgeworth estimate and a corner Euler--Maclaurin summation; all half-integer orders vanish by parity. Each correction separates into a smooth Edgeworth part and an oscillatory lattice part, the latter an Edgeworth perturbation of one formal Euler--Maclaurin series of a Gaussian lattice sum. Under a sampling law $\mathbf p\neq\mathbf r$ the leading expectation is the folded-normal profile implied by the central limit theorem; under local alternatives we identify the $O(1)$ correction (a skewness term, plus a variance-gradient term for Pitman alternatives), and fixed alternatives give a law-of-large-numbers limit, the $\ell_1$ distance between the adjacent-gradient profiles of $\mathbf p$ and $\mathbf r$, on which threshold tests are consistent.