Root Multiplicity Sets the Finite-Width Edge of Chaos in MLPs and Message-Passing Networks

How far a finite-width network's order/chaos crossing sits from its mean-field location is set by one integer, the multiplicity $r$ of the mean-field root: an $O(1/F)$ correction to the Lyapunov exponent moves the crossing by $O(F^{-1/r})$. In message-passing networks at zero bias the root is double. We prove this through an exact identity, $G-1={\kappa }_p(g-1{)}^2(1+o(1))$, whose constant is universal (${\kappa }_2=1/3$) across smooth saturating odd activations, so the finite-width crossing lies at distance of order $F^{-1/2}$, not $F^{-1}$, from both the linearized boundary and mean field, which coincide. For MLPs and regular graphs with uniform input the network reduces exactly to a two-state Markov chain, and we prove the critical-window law $\log g=\log g_{\mathrm{lin}}(F)+\sqrt{3/F}(1+o(1))$. Between $g_{\mathrm{lin}}(F)$ and the crossing the exponent is at least $-(1+o(1))/(2F)$, sharply; at $\log g-\log g_{\mathrm{lin}}(F)=x/F$ the chain converges to an explicit diffusion, and $2F{\chi }^{\star }\to -\min (1,2x)$ for every $x>0$. Pre-registered measurements on full networks confirm the window law: the displacement decays as $F^{-1/2}$ at zero bias (fitted exponent $-0.56$) and as $F^{-1}$ at nonzero bias ($-1.09$), and softsign is displaced more than $\tanh$ by the predicted factor $1.84$. Bipartite graphs follow a predicted second branch without the $1/F$ cost, which a registered field-level test confirms. We prove, and a registered test at held-out widths confirms, a $5/(2F)$ shift for ReLU, whose root is simple.