Paper 125B: Centered Divisor Inversion and the Exact Möbius Complement Critical-Power Control of the Short-Divisor Tree and the Remaining Signed Correlation

This paper derives an exact centered divisor representation of the error in weighted prime counts, separating a controlled short divisor contribution from an exact Mobius complement. The kernel formed by subtracting the summatory divisor background from the logarithmic factorial satisfies an elementary square root bound. A discrete sampling argument at square boundaries uses the classical truncated Voronoi formula while retaining resonant frequencies and endpoint floor terms. It bounds the short divisor tree energy uniformly by a constant times the scale parameter raised to any fixed power greater than two, multiplied by the cutoff raised to the power three halves. The constant may depend on the chosen power. Every cutoff growing as a fixed power of the logarithm therefore permits quadratic energy growth with an arbitrarily small fixed positive power loss. The corresponding family of energy bounds for the complementary Mobius increments, required to hold for every fixed positive power loss, is equivalent to the Riemann hypothesis through the calibration in Paper 125A and remains unproved. Exact summation by parts preserves cancellation at artificial band boundaries, while finite records illustrate the cost of discarding their cross terms. Repeated square root frequencies are combined into explicit arithmetic channels containing a factor equal to the reciprocal of the square of the Riemann zeta function. A direct variation argument places the rational zero frequency component of a specified Voronoi main term below the required energy scale. These component estimates do not control the complete complement or its combined truncation and floor remainders. A self contained finite audit, Windows runner, and inline figure accompany the derivations.