Absolute Resolution of Lehmer's Conjecture: Discrete Quantization of Mahler Measure via SHID Holographic Projection and Rough Operator Algebra

For decades, Lehmer’s Conjecture has remained unproven due to the classical assumption that the roots of polynomials exist within a continuous complex manifold, allowing the Mahler measure M(P) to converge infinitesimally towards 1. This paper abolishes the continuous assumption by integrating the distribution of algebraic numbers into Rough Operator Algebra (ROA) [2] and Spherical Holographic Information Dynamics (SHID). We prove that roots are projection points on a non-commutative rough lattice. By mapping the minimal topological perturbation to the Seonggil Matrix Theory (SMT) phase-lock constant,we establish that M(P) cannot converge smoothly but is strictly quantized, guaranteeing an absolute lower bound strictly greater than 1.