Metric Geometry of Recursive Interval Algebras over Ω-1: Ω-Weighted Inner Product and Recursive Structure
Abstract
We construct Recursive Interval Geometry (RIG) as a strictly inductive algebra over \( R=\mathbb{Z}[\Omega^{-1}] \), generated by the binary structural projections \( \textrm{pos} \) and \( \textrm{len} \). On the resulting filtered configuration space \( \mathcal I \) we introduce an \( \Omega \)-weighted inner product and its induced norm \( \|\cdot\|_\Omega \), yielding an anisotropic pre-Hilbert geometry with an explicit orthogonal decomposition across recursive scales and an ultrametric-like, depth-dominant metric profile. We establish the associated Cauchy--Schwarz inequality and develop Gevrey-type structural estimates that quantify the decay of higher recursive layers. We further describe measure-theoretic and topological properties of the arithmetic substrate underlying the recursion. Finally, we define a canonical recursive tensor product driven by the principal component of the first argument, and prove its basic algebraic and metric properties, including depth additivity and strict \( \Omega \)-norm factorization. These results provide a compact, parameter-minimal mathematical foundation: beyond the discrete resolution parameter \( \Omega \), no additional free parameters are introduced, making the framework suitable for subsequent physical constructions.
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