Local large deviations for linear-region growth in random piecewise-linear networks

We study a random compositional model for the growth of affine regions in deep piecewise-linear networks. The model is generated by i.i.d. perturbations of the symmetric height-one tent map, and the main observable is the number (Nn) of affine pieces after (n) layers. We prove the existence of a submultiplicative pressure for (Nn), yielding exponential upper bounds for both tails of (n-1log Nn). The same argument applies to abstract submultiplicative complexity observables and gives higher-dimensional extensions for convex-polytopal affine-cover counts and worst-line affine-piece counts. Since the true branch count has no matching supermultiplicative inequality, lower bounds require a separate certified construction. We introduce a finite-state defect process that records branches whose future splitting can be guaranteed, and use bridge words to obtain constructive upper-tail lower bounds. In a uniformly favorable small-noise regime, this process is governed by a companion matrix whose Perron root tends to (2), implying eventual exclusion of lower tails below (log 2-xi).