A critical Poisson law in a dyadic block

English edition (reference version) and original French edition ("Loi de Poisson critique dans un bloc dyadique") of the paper. Abstract. Let f be an extended Rademacher random multiplicative function, i.e. a completely multiplicative function whose values at the primes are independent Rademacher variables. We count, in the dyadic block [N,2N), the starting points of maximal constant stretches of length at least L. Uniformly for |L−log₂N| ≤ C, we prove that this count is Poisson in total variation with parameter N·2^(−L), at rate O_C((log log N)^(−2)); the dominant term comes from the probabilistic conditioning, while the arithmetic core leaves an exp(−c·√(log N)/log log N) margin. The proof combines an exact affine formulation, an absolute homogeneous two-block estimate, conditioning on the small primes, and the Chen–Stein method. We also prove a uniform deterministic-mask version, vague convergence of the spatial start process to a Poisson point process, and convergence of the excess-length marked process to a Poisson point process with geometric marks. The gluing of [1,M] is proved for interior starts, and the longest-run law in the finite prefix (f(1),…,f(M)) follows by coupling. Files: - paper_C_complete_v08_en.pdf — English edition (reference), 71 pages, SHA-256 2f7c7b9fe3522059f0eb5fb7bf7871f0c3247e30aec534b1caa63abff5c8c927 - paper_C_complete_v08.pdf — French original, 72 pages, SHA-256 c53b66ad467b2637d764124c20b9d788f1489b91cd90f3d907bf1eb814a17bc5 The deductions of the main results are verified with the Lean 4 proof assistant, conditionally on seven explicitly registered published results; the formalization (DOI 10.5281/zenodo.21735482, GitHub Vulkin-prog/paper-c-lean, tag v0.47.0) identifies these exact PDFs by their SHA-256 fingerprints.