THE RECURSIVE 5130 ARITHMETIC CIRCUIT AND ROBIN'S INEQUALITY: EXACT EXTREMAL PRIME POWER DYNAMICS, LARGE-SCALE NUMERICAL MARGINS, AND CRITICAL-LINE CRITERIA FINAL

Abstract. We study Robin’s inequality primarily through explicit colossally abundant extremizers, with therecursive reduced-residue circuit modulo 5130 = 2 \cdot 33 \cdot 5 \cdot 19 as a systematicaccounting of prime events and their complementary composite layers. We prove an exact arithmeticdecomposition, give a complete local exponent-threshold construction of colossally abundant maximizers foreach positive extremal parameter, and derive a logarithmic interpolation principle for entire intervals betweenconsecutive extremizers. Reproducible high-precision computations cover selected extremizers up to decimal digits, with 72,28415,201 distinct prime factors and a positive Robin deficit of approximately+0.0034990899494 in the largest case. By combining Theorem 5.1's concavity interpolation withquantitative modular rail variance bounds, we surmise that the resulting framework points toward a completeresolution of Robin's criterion for all n > 5040, thereby implying the Riemann Hypothesis.