A Decomposition of the Mertens Discrepancy into Oscillatory and Pretentious Components

We derive a decomposition of the signed Mertens discrepancy into two structurally distinct terms: an oscillatory component governed by the prime harmonic error and a smooth positive tail arising from higher-order prime contributions. The resulting formula separates fluctuation from geometric bias and suggests a modest pretentious interpretation of the lower-order structure of the Mertens product.