Spectral Expansion of Transcendental Approximations (corrected)

This entry is Day 6 xpost lecture series and sets foundational concepts crucial to Spectral Number Theory (SNT). In particular we introduce: Transcendental numbers such as e and π that cannot be expressed as finite rational numbers, yet their decimal expansions can be approximated to arbitrary precision by rational fractions. A canonical example is the Maclaurin series for e:(corrected: prime 314159 is the 27131th prime from "the 70th prime," version 2)(corrected: spectral vector for 3.141592, Table 2, corrected from 6_3 to -6_3, version 3) This is the continuation of the groundwork for the broader SNT framework, which treats integers as waveforms, primes as basis modes, and arithmetic as resonance. This document is intended as both a mathematical reference and an accessible companion to the daily Xpost lectures, https://x.com/LoiseauZeta