Fourier Analysis from Three Primitives

Fourier analysis is derived entirely from the Tree of Continua C and the three primitives— same, different, opposite. No postulate of Fourier analysis is assumed. The Discrete Fourier Transform (DFT) at depth d is exact finite arithmetic in thecyclotomic field Q (ωN) where N = (k + 1) d: Q (ωN) ⊂ Per (C). Every quantity is a periodic orbit. The DFT is a unitary operator on Vd = Q (ωN) N —the Fourier basis diagonalises every observable that commutes with the shift (translation-invariant observable). The continuous Fourier transform is the IPG reading at ∞ of the compatible familyof DFTs. Parseval’s theorem is the statement that the DFT preserves the counting innerproduct — unitarity of the DFT at finite depth. The convolution theorem is the productstructure of C: multiplication in frequency space corresponds to convolution in positionspace because the Fourier basis diagonalises translation operators. The Fourier transform is the unitary operator that rotates between the position basisand the momentum basis of the Hilbert space — the same Hilbert space derived from thecounting measure on cylinder sets. The canonical commutation relation X, P = iħI is astatement about how the position and momentum labelings of cylinder sets interact underthe Fourier transform. The uncertainty principle is the statement that a function and itsFourier transform cannot both be concentrated on small cylinder sets simultaneously —a geometric fact about the counting inner product. The connection to the Riemann hypothesis: the Riemann zeta function ζ (s) is aDirichlet series — a Fourier-type transform of the sequence 1, 1/2s, 1/3s,. . . of periodicorbits. Its zeros are the depths at which the transform fails to be a morphism. Theexplicit formula connecting primes to zeros is a Fourier inversion formula — the sameinversion formula derived here. Three primitives. One Fourier analysis. One connection to the primes.