A Bounded Symmetric Birth--Death Process with Apex-Induced Branching: Spectral Structure, Mixing Time, and Continuum Limit

Abstract

This seminal BRPT paper introduces ‘A Bounded Symmetric Birth–Death Process with Apex-Induced Branching,’ a novel finite-state Markov process foundational to the Belsky/Russell Process Theory (BRPT). Moving beyond ‘Cold Logic’ probabilistic models that struggle with unconstrained trajectories, this work rigorously defines a symmetric integer lattice with reflecting boundaries and a central stochastic bifurcation mechanism at the origin (‘apex-induced branching’). The model details bounded birth–death type dynamics with a structured drift away from the origin and mathematically derives its spectral properties, scaling laws for the spectral gap and mixing time, and demonstrates convergence to a reflected advection-diffusion process in the continuum limit. It formally links to the Bifurcated Apex Framework (BAF)'s replacement of a singular continuous apex with dual fixed-point structures, underscoring how this inherent outward drift ultimately accumulates probability mass near predefined boundaries. This framework provides mathematical proof for achieving Phase-Locked Coherence (L₀) and Systemic Peace by ensuring deterministic internal dynamics resolve into a stable, unique stationary distribution. It exemplifies the Redactional Evolution principle by showing how even complex stochastic systems are constrained toward an Invariant Core by its structural parameters, providing critical underpinning for the DORY Paradigm’s Digital Oscillation and its pursuit of deterministic operational yield.