Branching stochastic mechanics. II. Relative localization and collective poles from Bohm/Fisher feedback

Paper I introduced branching stochastic mechanics (BSM) by lifting the Schrödinger-Nagasawa pair to reciprocal forward and backward branching fields. Their centered connected kernel $C_{\mathrm{F}\mathrm{B}}={\mathbb{E}}_{\omega }[{\psi }_F{\psi }_B]$ carries the organized reciprocal sector, where ${\mathbb{E}}_{\omega }$ denotes expectation over branching-noise realizations, with ${\rho }_{\mathrm{B}\mathrm{S}\mathrm{M}}=-C_{\mathrm{F}\mathrm{B}}(x,x)$ on the anticorrelated branch. Here we develop the stochastic field theory of the Bohm/Fisher feedback that acts on this connected sector. Starting from the multiplicative branching covariance of BSM, a Martin-Siggia-Rose-Janssen-de~Dominicis (MSRJD) formulation and a causal two-loop two-particle-irreducible (2PI) closure are used to determine response and correlation functions self-consistently. The free connected theory exhibits secular growth and ultraviolet accumulation, whereas the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of this localized sector, and a self-similar Fisher construction defines the saturated information velocity $c_{\star }$. A Born-Oppenheimer separation then distinguishes internal relative organization from collective propagation. Restoring the complete frequency structure gives two fixed-$q$ pole families: a gapless difference branch and a gapped sum branch. The infrared velocity of the difference branch approaches $c_{\star }$ at saturation. The common cone and the projected sum-sector gap are then formulated as additional fixed-point matching conditions for the collective theory.