Sharp Small-Height Estimates for Subcritical Special Lagrangian Equations in Dimension Three

We prove an interior Lipschitz estimate with a sharp height threshold for continuous viscosity solutions of the three-dimensional special Lagrangian equation with subcritical phase $\mathrm{\mid }\mathrm{\Theta }\mathrm{\mid }<\pi \mathrm{/}2$. If $$os{c}_{B_R}u\le M{R}^2,0\le M<\sec \mathrm{\Theta },$$ then $Li{p}_{B_{\rho R}}u\le CR$ for some $\rho >0$ and $C>0$ depending only on $\mathrm{\mid }\mathrm{\Theta }\mathrm{\mid }$ and $M$. The threshold is optimal: at $M=\sec \mathrm{\Theta }$, there are entire real-analytic solutions whose Lipschitz seminorms are unbounded on every fixed interior ball. The proof uses a two-point maximum principle and algebraic identities specific to dimension three.