We reduce non-deterministic time T > 2^n to a 3SAT instance phi of size |phi| = T polylog T such that there is an explicit circuit C that on input an index i of log |phi| bits outputs the i-th clause, and each output bit of C depends on O(1) inputs bits. The previous best result was C in NC^1. Even in the simpler setting of |phi| = poly(T) the previous best result was C in AC^0. We also somewhat optimize the complexity of PCP reductions. As an application, we tighten Williams' connection between satisfiability (or derandomization) and circuit lower bounds. The original connection employed previous reductions as black boxes. If one instead employs the reductions above as black boxes then the connection is more direct.
Based on joint works with Hamid Jahanjou and Eric Miles, and with Eli Ben-Sasson.