Paper 4. From Cycle to Window

Osman, Mohamed

Paper 4. From Cycle to Window

What survives when an exact periodic law is evaluated on a short interval

Mohamed Osman · ORCID 0009-0004-5912-999X · independent researcher · 2026-10-16

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Abstract. An exact law on a sieve cycle is not an exact law on a window, and the loss is not uniform: soft weights transfer almost exactly, sharp indicators do not. The twin-to-product ratio is measured close to e^(2 gamma)/4, the value the Hardy-Littlewood conjecture predicts.

No progress toward the twin-prime conjecture is claimed. Priority is not claimed for any result. Every number printed is regenerated by a script in code/ of the repository. Licence: CC BY 4.0 (text), MIT (code).