Introduction to Spectral Number Theory

In this introductory presentation on Spectral Number Theory (SNT), we propose a previously unrecognized vector space intrinsic to the natural numbers, extending smoothly to the rationals and reals. Unlike conventional vector spaces that use numbers to represent abstract objects, the spectral vector space is made of numbers. Each number has a unique position—a spectral fingerprint—determined by its prime decomposition, forming the coordinate geometry of arithmetic itself. We argue that this is the natural basis from which both number theory and quantum arithmetic phenomena emerge.This is a complimentary addition to the daily Xpost lectures, https://x.com/LoiseauZeta presented 31 May - 6 June 2025 in the presentation of "Chapter 1" on Spectral Number Theory.