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Trail interactions occur when past particle trajectories bias future motion, rendering the system out of thermodynamic equilibrium. While such systems are abundant in nature, their understanding is limited to the single-particle level or phenomenological mean-field theories. Here, we introduce a minimal model of many trail-interacting particles that extends this paradigm to the fluctuating collective level. Particles diffuse while depositing long-lasting repelling/attracting trails that act as a shared memory field, coupling their dynamics across time and space. Using stochastic density functional theory, we derive fluctuating hydrodynamic equations and analyze analytically and numerically the resulting behaviors. We show that memory, coupled with fluctuations, fundamentally reshapes collective dynamics; In the repulsive case, the particle density displays superdiffusive spreading characterized by transient clustering and ballistic motion; In the attractive case, the system condensates in finite time into frozen, localized states. Our results establish general principles for trail-interacting systems and reveal how persistent fields generate novel instabilities and self-organization.

 Sex differences in cognition are well documented, but their biological roots - especially network-level origins, remain elusive due to hormonal, environmental, and societal confounds. To isolate genetic effects, we used isogenic iPSC-derived neurons from a rare mosaic Klinefelter donor. Utilizing calcium imaging assays, we revealed a temporal divergence in maturation as XY networks show augmented connectivity patterns early on, while XX networks surpass them later on. Conversely, XY networks exhibit an increasing level of synchronization over time, while XX networks exhibit more connections. We demonstrate that such features alone accurately classify independent XX/XY networks, revealing a robust, generalizable signature.Simulating information flow revealed faster, broader spread in XY networks at later developmental stages, indicating differences in function. Modeling cognitive tasks, we found XY networks enable faster, more accurate focused problem-solving, while XX networks excel in parallel information processing. This suggests that chromosomal composition shapes cognition via inherent differences in network topology.To mechanistically unify the findings, we introduced a generative network model governed by a single parameter p (cooperativity), which controls how local synchrony is amplified into global connectivity. Varying p generated a family of networks spanning hypocooperative, optimal, and hypercooperative regimes, simultaneously moderating topology and link weights. Remarkably, empirical XX and XY networks map onto distinct regions of the cooperativity landscape, as XX networks cluster closer to an intermediate p-range, whereas XY networks exhibit higher effective cooperativity.Together, our results identify cooperativity as a unifying, quantitative biomarker linking chromosome composition to network topology and emergent cognitive function. This work reveals fundamental sex-based differences in cortical network organization and provides a principled framework for sex-aware neuroscience, with implications for personalized diagnostics and targeted interventions.