We show that, for a broad class of gradient-based learning rules, scalar zero-sum games form a complete test family for convergence in unconstrained network zero-sum games. Specifically, suppose a learning rule achieves last-iterate or time-average convergence on every scalar game of magnitude $a$ whenever $η\leq f(a)$, where $f$ is nonincreasing. Then the same guarantee holds on an arbitrary network zero-sum game with stacked game operator $\mathbf{G}$ whenever $η\leq f(\lVert\mathbf{G}\rVert_2)$. More generally, finite-time rate guarantees transfer mode by mode: the network response is bounded by the largest response of its scalar canonical games, and under compatible linear initialization conventions the worst-case finite-time network response is exactly the largest scalar canonical rate. Thus, understanding the learning dynamics on scalar games is sufficient to characterize universal convergence and worst-case rates over the corresponding class of network games. The result follows from the classical planar decomposition of real skew-symmetric game operators. Related mode-wise reasoning has appeared implicitly in spectral analyses of particular learning algorithms and in prior work using these decompositions for game representation and visualization; here we isolate the resulting transfer principle for gradient-based learning. We discuss implications for simplifying convergence analysis, time-average guarantees, low-dimensional geometric arguments, accelerated-method design, and worst-case numerical testing.

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