We study how two geometries of mass on a finite graph interact: the information geometry of the generalized entropy $F_q$ used for regularized transport, and the arithmetic of divisors governed by the Riemann--Roch theorem of Baker and Norine and by the Jacobian. Because the correction term of $F_q$ is affine, $F_q$-regularization at strength $γ$ is Tsallis regularization at temperature $γ/q$. Over the reals, chip-firing connects all configurations of equal degree, so the Jacobian enters transport only through integrality. We introduce the transport defect, the worst-case cost of steering $d$ chips towards a target by chip-firing, and prove $1/d\leδ_{\TV}(D)\le(d-r(D))/d=(g-1-r(K-D))/d\le\min(1,g/d)$ whenever $g\ge1$; vertex multiplicities give a lower bound and an equality criterion. A regularized topological gap $Γ_{q,γ}(D,E)$ tends to the chip-move defect as $γ\to0$, lies in an explicit window, has a high-temperature slope given by Tsallis coupling entropies, and admits a positive regime if and only if $g\ge1$. Dual potentials certify exactly the convex relaxation of a linear system; the remainder is an integrality gap. Zeros in optimal plans occur only for $q>1$, by a mechanism different from the sign change of $Γ$. Exact computations on $1784$ divisor classes find the rank bound $(d-r(D))/d$ attained in $1769$ of them, an observed frequency. Experiments on honeycomb and Erdős--Rényi networks with $10^2$ to $10^5$ nodes test the statements on regularized plans, not $Γ$; over $10^3\le n\le10^5$ their transport costs are consistent with growth like $n^{1/2}$ and $\log n$, respectively.

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