Let $K/\mathbf{Q}_p$ be a non-archimedean local field. For a semistable abelian variety $A/K$ with symmetric ample $L$, we prove a quantitative equivalence between tame monodromy on $H^1$ (vanishing Swan conductor; breaks in $\{0,1\}$) and uniform disc-wise regularity of the local Néron height: on every residue disc the function $\lambda_v(\,\cdot\,;L)$ is piecewise quadratic with globally-in-disc Lipschitz/curvature bounds. All constants depend only on the Weil–Deligne data of $H^1$ (slope polygon $\mathrm{Sl}(A)$ and toric rank $t(A)$). Consequences include: a uniform Diophantine density bound $\;N_{A,L}(D;T)\le C(1+T)^g$ on each residue disc; meromorphic continuation of the associated local height zeta with abscissa $s_0=g\log_q e$ and a simple pole; sharp lower bounds in the ordinary-good case; and a precise failure mode in the presence of wild inertia. For curves $C\hookrightarrow J$ we deduce analogous disc-wise bounds for $i^*L$, and for modular curves we obtain an explicit constraint for local Hecke orbits $\mathcal{O}_{\mathrm{loc}}(x;\ell^r)$ that is polynomial in $T$ and grows at most like $(1+\ell)^r$ in $r$. The arguments require no global rank hypotheses and no resolution data: both the finite definable partition and the coefficients of the quadratic pieces are controlled purely by $(g,\mathrm{Sl}(\cdot),t(\cdot))$. Worked examples (Tate curves and wild additive reduction) illustrate optimality.