Failure of Uniform Bounds for Fractional Carleson Operators at Exponent Zero

We prove that the L2 operator norm of the even-phase fractional Carleson maximal operator, with kernel exp(i lambda |t|^epsilon)/t and supremum over real lambda, grows at least as sqrt(log(1/epsilon)) as epsilon decreases to zero. Explicitly, the norm is at least sqrt(3N/(2048*pi)) for every N>=1 whenever 0<epsilon<=2^(-16-12N). This disproves the universal estimate in Guo's AIM Question 12 (equation (0.17), Carleson theorems and multilinear operators), also for compactly supported smooth inputs. One bounded, finite-valued modulation parameter aligns packets centered at exp(16^j); an explicit three-region kernel estimate transfers the logarithmic construction to positive exponents. The inherited logarithmic reduction and multiplier in the frozen UnsolvedMath report are credited. The odd-phase question near exponent one and optimal growth are not addressed. This six-page English preprint was developed with AI assistance and self-audited; it has not undergone independent peer review or formal verification. Supporting exact and high-precision checks supplement the written analytic proof. No absolute priority is certified.