A 32-Tooth One-Dimensional Simulation of the Bolton Analysis on Orthodontic Arches and Its Geometric Correlation with Hyper G-Matrices

Abstract: This study recasts the classical Bolton Analysis within a matrix-theoretic framework and simulates occlusal balance in a complete adult dentition (32 teeth) in a one-dimensional, small-deformation regime. Each dental arch is represented by a 16 × 16 symmetric positive definite M-matrix whose diagonal entries encode normalized tooth support and whose off-diagonal entries encode elastic coupling between neighbouring teeth; under ideal conditions this matrix is a Hyper G-matrix. Ideal occlusal compatibility is expressed by D₁AD₂ = B⁻ᵀ, where D₁, D₂ are positive diagonal scaling matrices representing interproximal reduction, bonding, or arch expansion. We prove exact singular-value reciprocity for this duality relation (Lemma 2). The classical Bolton ratio corresponds to a trace-level projection of the raw tooth-width matrices, while the spectral Bolton index ρ(A)/ρ(B) provides a diagnostic refinement. A full 16×16 numerical simulation, including off-diagonal crowding blocks, is performed, and Monte Carlo perturbation analysis with effect-size and convergence diagnostics quantifies the compatibility between the spectral index and the classical Bolton ratio. A retrospective validation on a cohort of 60 adult patients (Angle Class I–III, 20 per class) is reported, with the anterior compartment index mapped to Angle classification via separate ROC analyses for the total and anterior indices. The results support the proposed matrix model and clarify its scope, which is limited to a one-dimensional, small-deformation, statically-loaded linear spring representation of the periodontal ligament.