{"id":"palomar/PALOMAR-2026-09-20-000011","url":"https://github.com/Vulkin-prog/paper-c-lean/tree/d8edbc44654f623fc68df92612c26a6ef9a90ec2","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-20-000011-v1.json","repo":"https://github.com/Vulkin-prog/paper-c-lean","tags":["math.NT","math.PR","msc:11K65","msc:60F05","probability","number-theory","longest-runs","prefixes"],"score":0,"title":"Microscopic boundary, prefixes and longest runs","trust":"high","domain":"math.NT","source":"palomar","authors":["Brice Pouly"],"reviews":0,"subject":"math.NT","summary":"The exact left-border probability is 2^(-pi(L)), and the microscopic non-vacancy probability satisfies 0 <= q_L/2^(-pi(L)) - 1 <= 2^(-pi(L)/256) for sufficiently large L. The results give conditional localization and uniqueness, stability under changes of mesoscopic cutoff, and a quantitative Poisson law for runs contained in a finite prefix. They yield the critical longest-run distribution and asymmetric almost-sure envelopes for all integer prefix lengths. The exact border identity needs no literature hypothesis. The asymptotic results use, as applicable, scalar Stein bounds, the prime…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-20","reactions":{},"formalized":true,"formalizes":[{"title":"Long runs and rare patterns of a random completely multiplicative function","identifier":"https://github.com/Vulkin-prog/paper-c-lean/blob/b3cf9aa27700d784efc61e6f82a92cef53320c40/manuscripts/paper-c/paper_c_version_3PREL9_en.pdf","relationship":"adapts"},{"title":"Technical companion to Long runs and rare patterns of a random completely multiplicative function","identifier":"https://github.com/Vulkin-prog/paper-c-lean/blob/b3cf9aa27700d784efc61e6f82a92cef53320c40/manuscripts/paper-c/paper_c_version_3PREL9_technical_companion_en.pdf","relationship":"adapts"}]}