A Finite Computer-Assisted Verification of Robin's Inequality via Colossally Abundant Profiles, with Exact Prime-Power Residual Dynamics

This English preprint presents an unconditional finite, computer-assisted verification of Robin's inequality, σ(n) < eγ n log log n, for every integer 5041 ≤ n ≤ 107.1×1022. The finite certificate combines: exhaustive directed-interval enumeration of 3,341,978 colossally abundant exponent profiles; an analytic prime-power reduction; a smoothed explicit formula; rigorous finite-height verification of the Riemann hypothesis; and an explicit absolute estimate for the unverified zero tail. Robin's convexity proposition between consecutive colossally abundant numbers transfers the certified endpoint result to all integers in the stated range. The certified support computation extends through 1.64967×1023. Separately, the paper derives exact prime-side residual, cell, kick, and signed-triangular identities and reports exploratory scans over all 5,762,859 prime-power events through 108. These diagnostics isolate a precise remaining uniform eventwise target but do not prove it. Scope: the conclusion is finite. The work neither proves Robin's inequality for all integers nor proves the Riemann hypothesis. Source code, archived interval reports, tests, research notes, and figure-reproduction files are available at github.com/robopol/Riemann-hypothesis.