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| Main Author: | |
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| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19254692 |
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Table of Contents:
- <p>Fourier analysis is derived entirely from the Tree of Continua C and the three primitives<br>— same, different, opposite. No postulate of Fourier analysis is assumed.<br>The Discrete Fourier Transform (DFT) at depth d is exact finite arithmetic in the<br>cyclotomic field Q(ωN ) where N = (k + 1)d : Q(ωN ) ⊂ Per(C).</p> <p>Every quantity is a periodic orbit. The DFT is a unitary operator on Vd = Q(ωN )N —<br>the Fourier basis diagonalises every observable that commutes with the shift (translation-<br>invariant observable).<br>The continuous Fourier transform is the IPG reading at [∞] of the compatible family<br>of DFTs. Parseval’s theorem is the statement that the DFT preserves the counting inner<br>product — unitarity of the DFT at finite depth. The convolution theorem is the product<br>structure of C: multiplication in frequency space corresponds to convolution in position<br>space because the Fourier basis diagonalises translation operators.<br>The Fourier transform is the unitary operator that rotates between the position basis<br>and the momentum basis of the Hilbert space — the same Hilbert space derived from the<br>counting measure on cylinder sets. The canonical commutation relation [X, P ] = iħI is a<br>statement about how the position and momentum labelings of cylinder sets interact under<br>the Fourier transform. The uncertainty principle is the statement that a function and its<br>Fourier transform cannot both be concentrated on small cylinder sets simultaneously —<br>a geometric fact about the counting inner product.<br>The connection to the Riemann hypothesis: the Riemann zeta function ζ(s) is a<br>Dirichlet series — a Fourier-type transform of the sequence {1, 1/2s , 1/3s , . . .} of periodic<br>orbits. Its zeros are the depths at which the transform fails to be a morphism. The<br>explicit formula connecting primes to zeros is a Fourier inversion formula — the same<br>inversion formula derived here.<br>Three primitives. One Fourier analysis. One connection to the primes.</p>