Finite-noise route polarization: coarse-graining bounds and compatibility gaps in nonequilibrium branch selection

Plain-language summary When a nonequilibrium system switches between long-lived states, the full transition trajectory may contain much more information than an observer actually records. In experiments and simulations, trajectories are often reduced to a finite route label—for example, one reaction channel versus another, an upper versus lower passage, or a small set of trajectory classes. This work asks a precise question: how much of the directional distinction between matched forward and reciprocal transition paths survives such route coarse-graining? The starting point is an exact relative-entropy decomposition. Information contained in the full path ensemble separates into an observable contribution carried by the route probabilities and a nonnegative hidden contribution describing directional information that remains unresolved inside each route class. This yields finite-noise bounds on route polarization and an exact criterion for when a route label retains all directional information. Two asymptotic regimes are then treated separately. At fixed noise and increasingly fine route resolution, the hidden information loss follows a high-resolution quantization law and suggests how route boundaries should be placed to preserve directional information efficiently. In the weak-noise regime, the hidden terms instead approach compatibility gaps that quantify whether the paths preferred in the two directions remain compatible within the same coarse route. For finite path families, these compatibility gaps become discrete rerouting events. Three paths are sufficient—and minimal—for a nondegenerate deterministic reversal between route-level and path-level directional polarization, and an explicit three-path construction shows that this sign reversal can persist at every positive noise level. The results do not constitute a universal branch-selection law. They apply conditionally to a specified matched forward–reciprocal transition-path experiment. The final part of the paper also shows that total forward–reciprocal likelihood asymmetry does not, by itself, determine a unique decomposition into physical entropy production and a residual dynamical contribution; that identification requires additional model-dependent structure. Why it matters Coarse-graining is unavoidable in experiments, simulations, transition-path sampling, reaction-network analysis, and data-driven descriptions of nonequilibrium systems. A reduced route label may appear to capture the dominant switching mechanism while still hiding enough within-route information to alter, or even reverse, the directional ordering inferred from the full trajectory ensemble. The framework developed here makes that information loss explicit. It identifies when a finite route observable is sufficient, when refinement must recover additional directional information, and when apparent route-level polarization can disagree with the underlying path-level structure. The same mathematical problem also appears more broadly whenever high-dimensional trajectories are compressed into finite representations. Modern machine-learning systems routinely map trajectories, state histories, or action sequences into latent states, clusters, skills, or discrete labels. In such settings, the present framework suggests a principled way to ask how much directional or discriminative information survives compression, when a reduced representation remains sufficient, and when coarse representation can reverse an ordering visible at the full-trajectory level. This connection is a potential application of the mathematical framework rather than a result directly tested on machine-learning models in the present paper. More generally, the distinction between statistical likelihood asymmetry and its thermodynamic interpretation is important. The forward–reciprocal experiment determines an invariant information-theoretic object, whereas its decomposition into physical entropy production and other dynamical contributions generally requires additional assumptions. Keeping these levels separate helps prevent coarse-grained irreversibility from being over-interpreted as a unique thermodynamic mechanism. Reproduction package The complete reproduction code and numerical data are available at Mendeley Data: doi: 10.17632/5h2vnpg4sn.1 The package contains Python source code, numerical outputs, figure-generation scripts, independent consistency checks, SHA-256 manifests, dependency and build-environment information, and LaTeX sources used to reproduce the numerical results, figures, manuscript, and Supplementary Material. Independent checks reproduce the principal algebraic identities and numerical results without relying on the corresponding production routines. Preprint; not peer reviewed. This upload contains the complete manuscript, including Supplementary Material.