We give a $3.5n-o(n)$ lower bound on the size of Boolean circuits over the $B_2$ basis computing an explicit Boolean function F in P. This improves the best previous lower bound of $3.1n-o(n)$ [Li-Yang22].

The proof of the lower bound uses the gate elimination technique, and builds heavily on ideas in previous works [Find-Golovnev-Hirsch-Kulikov23, Li-Yang22]. The main conceptual innovation is to consider substitutions on an underlying space of {\it independent parameters} specifying values of input variables, rather than on the input variables themselves. This enables us to handle certain bottleneck situations by {\it reparameterisation} rather than substitution. As a consequence, we are also able to obtain a simpler lower bound proof than in [Find-Golovnev-Hirsch-Kulikov23, Li-Yang22].