Critical Poisson fields for long constant stretches of a random completely multiplicative function

Let f be a Rademacher random completely multiplicative function. The article studies the left endpoints of long constant stretches. A refined weighted two-window relation estimate and conditioning on the small primes yield total-variation Poisson approximation in a dyadic block at the critical scale. The same lattice comparison controls positions, exact excesses, the threshold staircase, and a discrete Poisson–Gaussian bridge. On a finite prefix, the exact left-border event has probability q_L = 2-π(L)(1+O(2-π(L)/256)); combining this with mesoscopic negligibility and a relative bulk lattice-process approximation gives an unconditional microscopic–bulk crossover for the first contained exceedance. The technical companion, deposited under the same record, supplies detailed arithmetic, rank, threshold, and relative-error verifications. The pinned Lean development targets the earlier English manuscript v0.9 and does not certify the results added or modified in Version 2. Version 2 substantially expands and retitles the article. It adds the PNT-optimized dyadic formulation, the refined growing-moment two-window profile, the direct macroscopic profile, moving exact-mark fields, the exact microscopic boundary asymptotic, and the unconditional boundary–bulk crossover. The technical companion is deposited as part of the same research object. The Lean development remains pinned to manuscript v0.9.