Using rigorous high-precision floating point arithmetic we compute very tight rigorous bounds on the auto-convolution constant \[ ν_2^2 = \inf_f \|f \ast f\|_2^2 = \inf_f \int_{-1}^1 (f \ast f)^2 \] where the infimum is taken over all unit mass functions $f \in L^1(-1/2,1/2)$. This quantity arises in additive combinatorics, particularly in the study of Sidon sets. Our bounds give the first 128 digits of $ν_2^2$, and so substantially improve previous bounds on this quantity due to White, Green, and Martin & O'Bryant.