We studied the Robin defect associated with the inequality σ(n)<e^γ nlogu2061logu2061n, express its Laplace transform through Ramanujan’s transformation for the divisor Lambert series and isolate the precise difference between smoothed positivity and coefficientwise positivity. The main results give us an exact Ramanujan-transformed identity for the Robin defect, an equivalent coefficientwise formulation of the Riemann Hypothesis showing why transform-level positivity cannot by itself prove the hypothesis and an extremal reduction to colossally abundant and highest abundant numbers. Numerical data for early extremal integers illustrate how the normalized defect behaves past the exceptional value 5040. Ramanujan’s identities provide powerful global control, but the Riemann Hypothesis requires pointwise positivity at the extremal divisor-rich integers.