A Zero-or-Infinity Modulus Dichotomy on the Real Line

Let E be a compact subset of [0,1], regarded as a subset of the real axis L in the plane. We prove that the 2-modulus of all rectifiable curves joining E to L minus E is infinite whenever E has positive conformal capacity. This gives an affirmative answer to A. A. Gonchar's question recorded as Hayman-Lingham Problem 7.58 (UnsolvedMath AMR-022-7058). When E has zero capacity, the modulus is zero, giving a zero-or-infinity dichotomy. The proof applies the classical condenser capacity-modulus identity to separated compact approximations, uses the H^1 to H^(1/2) trace theorem and critical rigidity of indicator functions, and retains the capacitary constraint on null-length sets by finite Mazur combinations and approximation away from the line. The four-page English manuscript supplies the complete analytic argument and supporting exact algebra checks. Classical tools are credited. AI-assisted, self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.