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Finsler Metric Clustering in Weighted Projective Spaces

This paper establishes a foundational framework for geometric learning in weighted projective spaces $\mathbb{P}_{\mathbb{q}}$ by introducing a hierarchical clustering algorithm governed by Finsler geometry. We define a scaling-invariant Finsler metric $d_F([z], [w])$-and its rational analogue $d_{F,\mathbb{Q}}([z], [w])$-derived from an optimization-based Finsler norm that effectively quotients out the weighted scaling action. Unlike previous approaches that characterized these spaces via non-metric dissimilarity measures, we rigorously prove that our construction satisfies the triangle inequality, providing a true metric framework that ensures the stability of hierarchical clustering via the Gromov-Hausdorff distance. We demonstrate that this metric approach preserves the intrinsic scaling symmetries and weighted topology of $\mathbb{P}_{\mathbb{q}}$ without the topological distortions inherent in Euclidean approximations. The algorithm's efficacy is explored in the context of arithmetic geometry (clustering moduli spaces of genus two curves), arithmetic dynamics, and quantum state-space analysis, where the weights $\mathbb{q}$ represent anisotropic physical constraints and noise profiles. This work establishes a robust theoretical foundation for the development of graded neural networks and other machine learning techniques for graded algebraic varieties.