The Dusart Operator: Discrete and Continuous Prime Guarantees with Asymptotic Obstruction (Preprint)

This paper contributes a function-theoretic reformulation of Dusart’s explicit short-interval bound, yielding reusable sufficient conditions for primes in monotone families of intervals and a threshold theory for standard growth scales. This version substantially strengthens and corrects the continuous side of the paper. The discrete prime guarantee is retained, while the earlier continuous discussion is replaced by a rigorous h-window theory centered on the sliding-interval quantity D̃ₕ (f) (x) = ( (f (x+h) − f (x) ) / ln f (x+h) ) · (1 − 1/ln f (x+h) ). In particular, the continuous prime-existence criterion is now formulated in a form directly supported by Dusart’s explicit bound, the integral comparison arguments are corrected, and the development is extended by new comparison, stability, and asymptotic classification results. The resulting framework gives explicit sufficient conditions for primes in intervals (f (x), f (x+h) ], together with threshold laws for standard growth families and a variable-window analysis at the critical boundary f (x) ≍ x log x. Statement on the present version. The present version revises the paper by replacing the earlier continuous formulation with a rigorous window-based theory derived directly from Dusart’s explicit interval estimate. The discrete theorem remains unchanged in substance. On the continuous side, the principal sufficient condition is now expressed through the sliding-interval quantity D̃ₕ (f) (x) = ( (f (x+h) − f (x) ) / ln f (x+h) ) · (1 − 1/ln f (x+h) ), which yields an explicit prime guarantee for (f (x), f (x+h) ] whenever D̃ₕ (f) (x) > 1 and f (x) lies beyond Dusart’s range threshold. The integral estimates have been reorganized accordingly, the monotonicity arguments have been made exact, and the examples have been restricted to admissible monotone families. The revised paper also adds several new results: an h-window distortion theorem linking lower bounds on the differential operator D (f) to prime guarantees, a comparison principle, stability under asymptotic perturbation, fixed-window threshold classifications for standard model families, and a variable-window boundary law at f (x) = x log x. The paper therefore now presents not only a corrected formulation, but a broader analytic framework with explicit asymptotic phase transitions. Key results to highlight Discrete Dusart prime guarantee. For an increasing sequence f (n), if ( (f (n+1) − f (n) ) / ln f (n+1) ) · (1 − 1/ln f (n+1) ) > 1 eventually, then every interval (f (n), f (n+1) ] contains a prime. Continuous h-window prime guarantee. For increasing C¹ functions f, if D̃ₕ (f) (x) > 1, then (f (x), f (x+h) ] contains a prime. Distortion theorem. Lower bounds on the pointwise differential quantity D (f) (x) = (f′ (x) / ln f (x) ) · (1 − 1/ln f (x) ) imply prime guarantees once the logarithmic distortion across x, x+h is controlled. Comparison principle. Prime guarantees transfer between monotone families when their increments and endpoints are ordered appropriately. Perturbative stability. If a model family is above threshold, sufficiently small asymptotic perturbations preserve the prime guarantee. Asymptotic threshold classification. The paper now classifies the behavior of D̃ₕ (f) for standard families such as x^α, x (log x) ^β, x log x (log log x) ^γ. Boundary law at x log x. For the critical family f (x) = x log x, D̃ₕ (f) (x) → h, so the threshold occurs exactly at h = 1: below threshold for 0 1, and borderline at h = 1. Variable-window transition. At the boundary scale x log x, the paper identifies the moving-window transition at size h (x) ≈ 1 + (log log x) / (log x), giving a sharper asymptotic description of when the explicit prime guarantee turns on.