Hierarchical Harmonic Weighting and an Exact Zeta-Value Decomposition: A finite construction whose infinite limit is ζ(2) + 2ζ(3)

Abstract

This work began with a series of thought experiments about the nature of light, its propagation, and the way light may interact with recursively structured geometric space. While exploring these ideas, a simple hierarchical pattern emerged from the changing scales of the construction. This geometric intuition was then translated into a precise mathematical framework involving harmonic sums and scale-dependent weights. The resulting construction leads to an exact relationship with special values of the Riemann zeta function. Once the construction is formally defined, the derivation is entirely mathematical and does not depend on any physical assumption about light. The purpose of the work is therefore not to propose a new law of optics, but to show how an idea originating from reflections on the nature of light can lead to a well-defined mathematical structure and an exact analytical result.