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

This companion provides the verification layer for the article’s arithmetic and discrete-field results. It fixes the parameter hierarchy and dictionary, gives the Runge calculation, identifies the obstruction to an invalid channel reduction, and records the Pell interface. It verifies the weighted two-window estimates through the bounded-ratio and macroscopic decompositions. Growing-order moments and direct L1 channel summation remove the 7/4 Cauchy–Schwarz losses outside the terminal sector, while isolating the terminal clustering problem. A valuation-matrix argument establishes qL = 2−π(L)(1 + O(2−π(L)/256)), with explicit product thresholds and the uniform Laishram–Shorey input. A mesoscopic sieve, relative bulk lattice-process Chen–Stein estimate, and stable product lift yield the unconditional first-contained-exceedance crossover. The companion also develops the moving exact-length field, strong Poisson staircase, resolved thinning and reverse-immigration kernels, and discrete Poisson–Gaussian bridge. The macroscopic transport of the complete moving exact-mark field remains open. Unqualified statement numbers refer to the article; appendix-letter labels refer to this companion.