The PCP-for-PPAD conjecture of Babichenko, Papadimitriou and Rubinstein (quasilinear form)

Theorem 1.1: there are fixed rational $0<\varepsilon<1/10$ and $0<\delta<1$, and deterministic polynomial-time algorithms $R,D$, such that $R$ maps an End-of-Line instance of length $N$ to a generalized circuit of total binary length at most $AN\lceil\log_2(N+2)\rceil^b$, every generalized circuit has an acceptable rational assignment of polynomially bounded length, and from any such assignment $\varepsilon$-satisfying all but a $\delta$ fraction of gates $D$ recovers an End-of-Line solution. This also implies the weaker polynomial-size PCP-for-PPAD statement. It does not by itself give new Nash lower bounds; those (Rubinstein, Golowich) still need an exponential-time hypothesis for PPAD.

Theorem 1.1: there are fixed rational $0<\varepsilon<1/10$ and $0<\delta<1$, and deterministic polynomial-time algorithms $R,D$, such that $R$ maps an End-of-Line instance of length $N$ to a generalized circuit of total binary length at most $AN\lceil\log_2(N+2)\rceil^b$, every generalized circuit has an acceptable rational assignment of polynomially bounded length, and from any such assignment $\varepsilon$-satisfying all but a $\delta$ fraction of gates $D$ recovers an End-of-Line solution. This also implies the weaker polynomial-size PCP-for-PPAD statement. It does not by itself give new Nash lower bounds; those (Rubinstein, Golowich) still need an exponential-time hypothesis for PPAD.