Paper 125C: Square-Root Frequencies and the Möbius Near-Resonance Large Sieve Exact Channel Aggregation, Diagonal Closure, Algebraic Defects, and the Remaining Off-Diagonal Theorem

This paper formulates the square root frequency estimate needed for the canonical oscillatory part of the exact Mobius complement in Paper 125B. Repeated ratios are combined before their coefficients are estimated, and the rational zero frequency family is treated separately. The actual block difference multiplier bounds the complete frequency diagonal uniformly by a constant times the square of the shell scale multiplied by the ninth power of one plus the logarithm of twice that scale. A finite local arc inequality separates the shell energy into an amplitude drift statistic and a signed oscillatory statistic, both retaining aliases introduced by block length. The natural denominator window length is proportional to the denominator scale raised to three halves, divided by the shell scale and the square root of the numerator scale. This separates sparse boxes from intervals requiring Mobius cancellation. The theorem of Robert and Sargos controls specified separated rectangular boxes at large block lengths. An integer resonance defect excludes a deterministic region of distinct active pairs, while a separate argument bounds the primitive oscillatory diagonal by a fixed power of the logarithm. Archived and independently recomputed finite statistics show the substantial cost of removing Mobius signs in the tested cases. The signed estimates for distinct frequencies remain open. Applying the frozen main term bounds to the exact complement also requires justified treatment of moving truncations, floor corrections, and assembly over the full tree. The paper supplies reductions and specified component bounds while leaving the Riemann hypothesis unproved. The figure, finite verification program, and Windows runner are self contained.