Capacity-Zero Exceptional Sets and Oblique Boundary Accessibility: A Constructive Characterization with Numerical Verification

Preprint: This paper provides a systematic characterization of when bounded analytic functions on the unit disk can exhibit prescribed dichotomous boundary behavior: radial limits failing on one set while nontangential limits exist outside another. Main Result: For measurable sets E_1 ⊂ E_2 ⊂ ∂D, we establish that a function f ∈ H^∞(D) with radial limits failing at every point of E_1 and nontangential limits existing at every point of ∂D \ E_2 requires three independent conditions: ∙ (C1) Capacity-Zero: E_1 has logarithmic capacity zero ∙ (C2) Topological Embeddedness: E_1 is nowhere dense in E_2 ∙ (C3) Positive-Measure Accessibility: m(∂D \ E_2) > 0 Necessity is proven universally for all Borel pairs using classical potential theory (Beurling-Carleson), cluster set topology (Lindelof), and measure theory (Fatou, F. and M. Riesz). Sufficiency is established constructively for well-separated pairs via singular inner functions supported on Cantor sets, with rigorous verification demonstrating radial divergence through bilateral mass distribution and nontangential convergence via harmonic majorization. Key Contributions: ∙ Unified three-condition framework synthesizing classical capacity theory, topological constraints, and measure-theoretic accessibility ∙ Explicit construction with complete analytical proofs and numerical verification ∙ Demonstration that each condition is sharp (cannot be weakened) and logically independent ∙ Full Python implementation for reproducibility Technical Approach: The construction exploits the self-similar structure of the middle-thirds Cantor set to generate persistent oscillation in conjugate Poisson kernel integrals, preventing radial convergence while preserving nontangential limits through separation estimates. Computational experiments confirm oscillation indices > 0.9 at Cantor points versus < 0.05 outside the exceptional set, validating the theoretical predictions. Scope: Sufficiency is proven for well-separated pairs where E_2 \ E_1 does not accumulate at E_1. Extension to arbitrary pairs satisfying (C1)-(C3) is identified as the central open problem. The framework unifies three classical research threads—capacity theory, topological embeddedness, and measure-theoretic accessibility—into a single characterization theorem with computational validation. Methods: Classical complex analysis, Hardy space theory, potential theory, singular inner functions, numerical computation. Intended Audience: Researchers in complex analysis, Hardy spaces, potential theory, and boundary value problems. Accessible to graduate students with background in complex analysis and measure theory.