Harmonic Mode-Locking and Discrete Toolmark Emergence in the Gallo-Roman Rolling Engine: Empirical Simulations and Mathematical Proof

This technical note serves as an empirical and computational companion to Version 2.0 of the primary framework (The Gallo-Roman Dodecahedron as an Aeroacoustic and Kinetic Rolling Engine). While the primary treatise models structural metallurgy, wear mitigation, and viscoelastic arrestor mechanics within a beeswax-silica sand composite bed, this addendum delivers quantitative evidence of emergent kinematic toolmarks. Classical archaeology has traditionally categorized Roman running guilloches and geometric medallions (including Solomon's knots, triquetras, and stellate envelopes) as skéuomorphs—purely optical imitations of woven hemp cordage, twined rope, or leather braiding laid out manually via compass drafting and snapped chalk lines. Using an open-access numerical simulation engine implemented in Python, we demonstrate that these canonical decorative motifs emerge deterministically as stationary standing waves and harmonic mode-locks produced by a rolling 12-sided nodular body. Key deliverables included in this companion note: Kinematic Equations of Motion: Parameterization of Cartesian translation (linear trough) and polar confinement (circular bowl) incorporating stochastic quartz grain asperities and operator drift. Volumetric Footprint Generation: Implementation of closed 2D polygonal footprints (PolyCollection) modeling tapered lenticular shear scars with parabolic depth profiles. Harmonic Mode-Locking Proof: Analytical derivation showing that rational frequency ratios (k = R-r/r = p/q) inside an isotropic circular vessel break symmetry to generate 3-fold (triquetra), 4-fold (Solomon's square knot), 5-fold (pentagonal torus), and 8-fold (stellate rosette) standing envelopes without manual substrate trimming. Forensic Metrology Protocols: Non-destructive 3D optical profilometry, focus-variation microscopy, and synthetic arrestor bed replication procedures to test the rolling engine hypothesis against physical bronze artifacts in museum collections.