Explicit depth-three circuit lower bounds beyond the square-root exponent

Theorem 1.1: there is a language $L$ decidable by a deterministic Turing machine in time $C(n+1)^a$ whose $n$-bit membership functions satisfy $\log_2 S_3(f_n)/\sqrt n\to\infty$, counting all gates of OR-AND-OR circuits with no fan-in restriction. The language and machine are fixed before the constant $A$. It does not give a fixed exponent $n^{1/2+\varepsilon}$ (Hastad-Jukna-Pudlak's problem stays open), says nothing about AND-OR-AND circuits beyond what duality gives for the complement, and nothing about depth above three or Valiant's $2^{\omega(n/\log\log n)}$ threshold.

Theorem 1.1: there is a language $L$ decidable by a deterministic Turing machine in time $C(n+1)^a$ whose $n$-bit membership functions satisfy $\log_2 S_3(f_n)/\sqrt n\to\infty$, counting all gates of OR-AND-OR circuits with no fan-in restriction. The language and machine are fixed before the constant $A$. It does not give a fixed exponent $n^{1/2+\varepsilon}$ (Hastad-Jukna-Pudlak's problem stays open), says nothing about AND-OR-AND circuits beyond what duality gives for the complement, and nothing about depth above three or Valiant's $2^{\omega(n/\log\log n)}$ threshold.