Open Proof of CRT Coverage Maximality for up to Eight Distinct Odd Prime Factors

This research memorandum establishes the Rank-Eight Theorem for Chinese Remainder Theorem (CRT) Coverage Maximality, extending previous mathematical frameworks to accommodate up to \(k \le 8\) distinct odd prime factors. The core achievement resolves the long-standing obstruction presented by the negative eighth-order Stirling-derived coefficient (\(c_8 = -50\)), which historically prevented simple extensions of the rank-seven proof. Rather than discarding the negative term, the paper introduces a novel, exact rank-eight coordinatewise monotonicity certificate over the admissible seven-dimensional prime-bound box. By demonstrating that the derivative polynomial is nonincreasing via exact product inequalities, the author calculates a strictly positive rational derivative margin of \(\frac{294137}{829440} \approx 0.3546\) at the upper corner. This positivity certificate guarantees the nonnegativity of all induced support polynomials, fulfilling the strict baseline conditions required to successfully apply Shearer's dependency-graph theorem to arbitrary residue class assignments and positive exponents without compatibility assumptions.