The Pythagorean angle lattice and the omni-metallic framework: arithmetic bridges and spectral relations

Abstract

This paper establishes a synthesis between the Pythagorean angle lattice and the Omni-Metallic family, a four-parameter generalisation of the classical metallic means defined as the unique positive real root of a prescribed polynomial. The arithmetic kernel of the Omni-Metallic side is the counting function that records the number of Universal Metallic representations of a positive integer. The Mellin transform of this kernel is evaluated in closed form as a combination of the Riemann zeta function at shifted arguments, an identity here called the Bridge Identity, and its finite part at the double pole yields an alternative expression for the circle constant. The Borel transform of the same kernel is evaluated at unity and yields an alternative expression for the base of the natural logarithm. A regularised infinite product over the kernel is computed by zeta regularisation and is expressed in terms of the Glaisher and Kinkelin constant. The Generalized Integer Value Theorem is proved for all admissible parameters, characterising exactly when a member of the Omni-Metallic family is a positive integer. A Quarter-Angle Bridge is proved, giving in exact and unconditional form the value of the lattice under quarter-angle projection for every primitive Gaussian integer, and identifying precisely when that value is a classical metallic mean of integer index. The Crown Identity connecting the Universal Metallic family to primitive Pythagorean triples is established. For the cubic projection an exact irreducibility criterion is obtained: the associated cubic is reducible over the rationals precisely when the generating Gaussian integer is itself a cube, and explicit reducible instances are exhibited. The integer values attained by the lattice are determined completely: they occur only at even projection indices, and the associated multiplicities are computed. A spectral decomposition over odd projection indices is stated as a conjecture.

1. Introduction

The classical metallic means, introduced by de Spinadel, are the positive roots of x² − nx − 1 = 0, n ∈ ℕ, with explicit form M_n = (n + √(n² + 4)) / 2. These numbers include the Golden Ratio (n = 1), the Silver Ratio (n = 2), and their higher analogues [1–3].

This page shows an excerpt of the introduction. The full text, including the theorems, proofs and examples, is available in the PDF.

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Cite this article

Rajput, C. (2026). The Pythagorean angle lattice and the omni-metallic framework: arithmetic bridges and spectral relations. Asia Mathematika, 10(2), 83–98. https://doi.org/10.5281/zenodo.23122625

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