A Witnessing Duality for ReLU Network Depth

A depth lower bound for ReLU networks asserts that every shallower network errs on some input; we ask when a weak theory can prove such a bound so that an error can be computed. For the exact computation of $\max(x_1,\ldots,x_n)$, recent work shows two hidden layers suffice with unrestricted rational weights. By contrast, Haase, Hertrich and Loho (ICLR 2023) proved that integer weights require depth logarithmic in $n$, and Averkov, Hojny and Merkert (ICLR 2025) extended the bound to nonbias weights whose denominators are powers of a fixed integer not divisible by a prime $q$. With integer weights the bound is sharp: the classical balanced construction has depth $\lceil\log_2 n\rceil$. For each fixed $q$ and depth cap $K$, we prove this known bound in $\mathbf{VTC}^0$, the bounded-arithmetic theory of uniform constant-depth threshold computation, together with a refuter that is itself such a computation and whose correctness the theory proves. Given a network with at most $K$ hidden layers, nonbias weights whose reduced denominators avoid $q$, arbitrary rational biases and $n>q^K$ inputs, the refuter returns a rational input of polynomial length on which the network errs. The proof makes the obstruction finite: an iterated intersection product of piecewise-linear functions is divisible by $q$ for the network but not for the maximum, so one of finitely many test points exposes the error. By witnessing, provable refutation corresponds to a provably correct refuter, and provable existence of a certified network to a provably correct constructor. On the construction side, we introduce certificates that label each maximum gadget by the inputs it aggregates; with them the theory verifies the balanced construction, which meets the refuter at the sharp threshold $n\le2^K$, and a rational family of polynomial description length and depth $O(\log n/\log\log n)$ built from known depth-two blocks.