Geometric Dressing in 1D Spin Chains: Classical vs. Triple-Harmonic Viviani Manifolds Under Extreme Thermal Stress

We present large-scale GPU-accelerated classical Langevin simulations of a 1D cyclicHeisenberg spin chain (N = 500 000 sites) dressed with site-dependent transverse fieldsaligned to the local normals of two Viviani curve variants: the classical parametrizationand a novel triple-harmonic deformation. At strong exchange coupling (J = 200) and weaktransverse field (h ≈ 1/137), we perform a temperature sweep from 400 to 1200 (effectivethermal energy units) with 20 independent repeats per point. Component-wise absoluteprojections ⟨|sα nα |⟩ (α = x, y, z) reveal a striking, temperature-independent asymmetryin the triple-harmonic case: y-component enhancement (> 2×), z-component suppression(∼42%), and x-component reduction (∼18%), yielding ∼1.62% better overall suppressioncompared to classical. This intrinsic directional “breathing” under extreme thermal stresssuggests the triple-harmonic manifold provides adaptive frustration channeling — poten-tially encoding emergent force-like hierarchies (strong = z compression/relaxation, weak =y buffering, electromagnetic = x propagation). The results support the hypothesis thathigher-harmonic geometric dressing induces qualitatively richer dynamical resilience, withimplications for fault-tolerant topological matter and theoretical unification of interactions.