Dispersion-Domain Detection for Mobile Molecular Communication Under Multiplicative Geometry Uncertainty

Mobile molecular communication (MC) links with counting receivers are sensitive to transmitter--receiver geometry especially when nodes are mobile. We study binary detection from within-symbol count observations with unknown finite-memory inter-symbol interference (ISI) and a block-constant multiplicative geometry gain. Under a mixed-Poisson view mobility and geometry uncertainty can randomize the latent received intensity and create extra-Poisson dispersion. We propose a profiled dispersion-domain statistic $T_k^{(\Delta)}$ formed after profiling the deterministic mean shape. The statistic subtracts the intrinsic Poisson component and normalizes by the squared profiled mean to target threshold stability under the stated multiplicative-gain model. Activity gating makes conditional and gate-integrated false-alarm probabilities explicit. We characterize $T_k^{(\Delta)}$ using a time-series central-limit-theorem (CLT)-motivated Gaussian working approximation with long-run-variance dependence correction yielding Gaussian-approximate receiver operating characteristic (ROC)/bit-error-rate (BER) formulas and separability design metrics. Simulations with symbol-dependent active-Brownian mobility and finite-memory ISI support the proposed mechanism show empirical threshold stability over the tested gain range and indicate usefulness when mean-domain differences are weak unreliable or intentionally suppressed.