Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger--Nagasawa pair. Its centered forward--backward kernel
CFB=Eω[ψFψB] resolves an organized connected sector, with weight
ρBSM=−CFB(x,x) on the anticorrelated branch.
Here we investigate how this sector forms, localizes, and propagates under Bohm/Fisher feedback. We retain the branching covariance on the prescribed background as the bare noise kernel and truncate the nonlinear interaction to Bohm/Fisher drift vertices. A Martin--Siggia--Rose--Janssen--de Dominicis formulation and a causal two-loop two-particle-irreducible (2PI) closure determine response and correlation functions self-consistently. While the free connected theory exhibits secular growth and ultraviolet accumulation, the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of the localized sector, and a self-similar Fisher construction defines a saturated information velocity
c⋆.
A Born--Oppenheimer separation connects this internal organization to collective propagation. The instantaneous adiabatic kernel has two pole families at fixed internal momentum: a gapless difference branch and a gapped sum branch. Projecting both inverse response kernels onto the same localized internal profile defines their collective coefficients. If both propagation speeds match
c⋆ and the projected gap matches
mc⋆2/ℏ, the gapped branch takes the infrared Klein--Gordon form. These matching conditions define a candidate relativistic fixed point whose dynamical realization remains to be tested.
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.