Geometry and the Emergence of Molecular Order in Early Life: A General Phenomenological Framework for Disorder-Limited Surface Symmetry Breaking, Applied to Prebiotic Homochirality

<title>Abstract</title> <p>Many physical and chemical systems exhibit emergent macroscopic order despite substantial microscopic disorder; we investigate this general phenomenon through mineral-surface homochirality in prebiotic chemistry. We introduce a new, experimentally measurable order parameter, surface uniformity (U_χ ∈ [0,1]), capturing the geometric regularity of chiral adsorption sites and inferable from AFM-derived terrace-width and step-edge statistics. Using a two-level Boltzmann discrimination model with spatially disordered surface sites, we test whether a given mean chiral discrimination energy survives realistic disorder to produce useful net enantiomeric excess (ee). The qualitative response to disorder is gradual rather than threshold-like, and this conclusion is robust across multiple σ_χ(U_χ) functional forms, three distinct disorder models, and a sensitivity sweep over the free parameters. A Monte Carlo propagation of parameter uncertainty shows the median outcome favors gradual disorder tolerance, though pessimistic parameter combinations fail to produce useful amplification except near-perfect uniformity. We propose patterned chiral nanochannel arrays as a fabricable geometry for isolating this geometric contribution experimentally, and state five falsifiable predictions. This work presents a general, testable framework for investigating whether geometric order is an independent control variable in disorder-limited symmetry breaking, rather than a quantitative prediction for any specific mineral system.</p>