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

We study Robin’s inequality primarily through explicit colossally abundant extremizers, with the recursive reduced-residue circuit modulo $5130 = 2 \cdot 3^3 \cdot 5 \cdot 19$ as a systematic accounting of prime events and their complementary composite layers. We prove an exact arithmetic decomposition, give a complete local exponent-threshold construction of colossally abundant maximizers for each positive extremal parameter, and derive a logarithmic interpolation principle for entire intervals between consecutive extremizers. Reproducible high-precision computations cover selected extremizers up to 72,284 decimal digits, with 15,201 distinct prime factors and a positive Robin deficit of approximately 0.0034990899494 in the largest case. A plotted sequence shows that the margins are positive in the sampled cases but diminish substantially at large scales. We give the precise implication from a universal endpoint inequality to the Riemann Hypothesis, and provide brief complementary Li–Weil and spectral observations.