{"id":"palomar/PALOMAR-2026-09-20-000012","url":"https://github.com/Vulkin-prog/paper-c-lean/tree/d8edbc44654f623fc68df92612c26a6ef9a90ec2","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-20-000012-v1.json","repo":"https://github.com/Vulkin-prog/paper-c-lean","tags":["math.NT","math.PR","msc:11K65","msc:60F05","probability","number-theory","poisson-laws","runs"],"score":0,"title":"Microscopic–bulk crossover and conditional affine laws","trust":"high","domain":"math.NT","source":"palomar","authors":["Brice Pouly"],"reviews":0,"subject":"math.NT","summary":"Conditioned on the presence of a run contained in a finite prefix, the first such run has a mixture law combining a microscopic border contribution with a uniform bulk location, geometric excess and fair sign. The results retain the marked record, describe both location scales in finite and infinite phase regimes, and determine the sign bias. Under compatible affine conditions on small primes, the border probability is exactly 2^(-d), where d is the row-space deficit. Affine localization and sign bias use the rank information budget without future neutrality; the full marked-clock limit also…","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"}]}