Distributed majority consensus in synthetic microbial consortia with bursty gene expression

Engineering microbial consortia to perform robust population-level computations requires principled ways to achieve majority consensus in the presence of multiscale stochasticity. We develop a hybrid framework that couples time-varying intracellular gene-expression dynamics, including burst-like jump noise, to population-level birth, death, and collision reactions through the random time-change representation. Birth and death hazards are allowed to depend on evolving intracellular states through bounded hazard maps, while interspecific collisions are modeled by biologically interpretable SD or NSD encounter rules. The construction uses instantaneous hazard integrals, rather than frozen-rate SSA propensities, so that reaction times remain consistent with continuously evolving intracellular states. Under explicit bounded-per-capita hazard assumptions, we prove order-wise sufficient majority thresholds for two biologically motivated collision rules. With self-destructive collisions, where one cell of each species is removed per encounter, a new potential-supermartingale argument controls temporary excursions of the minority population above a given level and shows that only O(log n) non-collision events occur before consensus with high probability. Consequently, an Ω(log n) initial gap suffices for correctness with high probability. With non-self-destructive collisions, where only one side dies per encounter, the species-count difference after subtracting its predictable drift is a bounded-increment martingale at embedded gap-changing reaction times. Thus, if the number of such steps is O(nlog n) with high probability and the predictable drift is neutral or favorable to the initial majority, an Ω(nlogn) initial gap suffices. Heavy-tailed intracellular fluctuations alter event timing and consensus-time distributions, but not these order-wise sufficient gap scales, provided hazard maps are bounded. We also establish non-explosion of the hybrid process, clarify the distinction between the exact random-time-change model and numerical simulation, and treat representative-pool hazard approximations as simulation accelerations rather than assumptions used in the proofs. Expanded numerical diagnostics test the SD non-collision bound, the NSD step-count assumption, multi-size NSD normalized transitions, heavy-tail versus light-tail ablations, representative-pool accuracy, and thinning acceptance rates. The result is an assumption-transparent mathematical and computational framework for majority-like decision making in noisy synthetic microbial consortia.