From Resonance to Limitation: A Fractional–Harmonic Framework and the Rough Composite Barrier in Primality Detection

This short communication presents a focused theoretical investigation into the limitations of finite prime-based representations for primality detection. It introduces the concept of rough composites—composite integers whose prime factors lie outside a fixed finite prime basis—and shows that such integers may be indistinguishable from primes when analyzed only through local invariants derived from finite prime projections. The central contribution is the formulation of the Rough Composite Barrier, which states that no method relying solely on finite prime-based information can fully separate primes from composite integers. This barrier highlights an intrinsic limitation of finite-dimensional local representations and emphasizes the need for global, asymptotic, or structurally richer information in rigorous primality testing. The communication also notes implications for computational number theory and machine learning approaches, where strong empirical classification results may mask fundamental representational constraints. Keywords: Primality detection; rough composites; finite prime representations; statistical invariants; computational number theory; machine learning limitations.