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Main Author:	Loiseau, Robert
Format:	Recurso digital
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Published:	Zenodo 2025
Online Access:	https://doi.org/10.5281/zenodo.16937789
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This document is Chapter 1 of the developing book Spectral Number Theory, presenting the Spectral Basis of the Integers. Earlier versions of this upload were titled The Spectral Basis Theorem (SBT). Upon further reflection, I recognized that while the underlying construction is rigorous and important, it is not a new theorem in the sense of establishing a result beyond the Fundamental Theorem of Arithmetic (FTA). Rather, it is a reinterpretation: integers can be represented as finite-support prime-exponent vectors, with multiplication corresponding to coordinate addition. Instead of minimizing this idea to a short 2-page “curiosity,” this chapter develops it fully as the foundational framework of Spectral Number Theory. It introduces: <ul> <li> The spectral basis representation of integers. </li> <li> Extension to $0$ and $\mathbb{Z}$ via the sign operator. </li> <li> Rational and irrational extensions (roots, transcendentals). </li> <li> Spectral approximations and the real line. </li> <li> Interpretive connections to analysis, signal processing, quantum mechanics, and cryptography. </li> <li>Corrected Figure 1.3 as \vec{a}(4)=(2,0,0,..) (not \vec{a}(4)=(0,2,0,..))</li> </ul> This reframing avoids “over-reaching” claims, but shows that what initially seemed a small notational curiosity — treating prime exponents as coordinates — actually opens a constructive and rigorous path toward a new branch of number theory. Chapter 1 thus serves as the pedagogical and structural foundation for Spectral Number Theory, with later chapters (to appear separately) addressing prime-logarithmic vector spaces, operators, invariants, and connections to analytic number theory.

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