Paper 125A: Prime Corridors and an RH-Calibrated Multiscale Energy Exact Reductions, Controlled Terms, and the Global Cancellation Problem
Year: 2026
Type: article
Abstract: This paper establishes an exact framework across multiple scales in which the Riemann hypothesis is equivalent to a uniform bound on corridor energy by a constant times the square of the scale parameter multiplied by the fifth power of its logarithm. Weighted prime powers are grouped between consecutive squares, and their signed discrepancies are summed over every aligned block whose length is a power of two. A real variable sampling argument establishes the implication from the Riemann hypothesis to the energy bound, while deterministic control of initial cumulative sums establishes the reverse implication. Higher prime powers, centering corrections, logarithmic variation within corridors, diagonal prime terms, a specified range of nearby separations, and the high frequency part of the complete signed tail admit bounds within the required allowance. Exact identities for overlapping windows preserve the signs when bands of longer separations are combined. Explicit nonprime selections show why the stated size bounds, increasing finite sieving, residue contrasts, and subtraction of one terminal mean do not force the missing cancellation. A further calibration identifies the zero free half plane that would follow from a fixed improvement in the power of the scale parameter in the complete energy bound. The unresolved contribution is the low frequency, long range signed tail. The paper provides a reduction and an analysis of obstructions, while leaving the unconditional energy bound required for the Riemann hypothesis unproved. A self contained finite audit and its Windows runner accompany the derivations.
Author: Sarnowski, Michael
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225717308
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