Bergach, M. A.

Spectral methods for DNA sequence analysis---treating genomic data as a discrete signal and computing its Fourier transform---were proposed over three decades ago but remained impractical for whole-genome analysis due to computational cost. We present a quaternion Fourier transform framework that encodes DNA as a quaternion-valued signal q[n] [isin] {1, i, j, k} mapping to the four nucleotides {A, T, G, C}, and prove that the full quaternion spectrum is computable from exactly two standard complex FFTs: Q(k) = Z_1(k) + Z_2(N-k) {middle dot} j, where Z_1 = FFT(u_A + i {middle dot} u_T) and Z_2 = FFT(u_G + i {middle dot} u_C). We establish that the resulting spectral fingerprint F(k) = (|Z_1(k)|^2, |Z_2(k)|^2) is invariant under both cyclic shift and reverse complement---the two fundamental symmetries of double-stranded DNA. Building on this theoretical foundation, we develop three computational tools: (i)~a 4x4 Hermitian cross-spectral matrix with inter-channel coherence analysis, (ii)~a genome spectrogram via sliding-window short-time Fourier transform, and (iii)~an alignment-free spectral variant detection algorithm with O(N log N) complexity. Applying Welch's cross-spectral coherence analysis to E.~coli K-12, we discover that the DNA helical repeat (~11~bp) is invisible to the standard power spectrum but clearly detected through the cross-spectral matrix condition number ({kappa} = 6.5), demonstrating that multi-channel analysis reveals structural periodicities that single-channel methods miss. Phase spectrum analysis recovers the characteristic nucleotide ordering within codons (A [-&gt;] T [-&gt;] G [-&gt;] C), while three distinct frequency regimes of inter-nucleotide coupling emerge: complementary-dominated (long-range), purine/pyrimidine-dominated (structural), and codon-position-dominated (coding). Cross-species validation on 18 genomes spanning all three domains of life---Bacteria~(5), Archaea~(3), and Eukarya~(10)---with GC content from 19.6% (P. falciparum) to 69.5% (T. thermophilus) confirms the universality of these findings. The helical repeat is detected via cross-spectral coherence in 18/18 organisms (100%). All 10 eukaryotes show A-T dominance at the helical repeat---a spectral signature of nucleosome wrapping absent from prokaryotes. Non-complementary pairs (A-C, T-G) dominate the coding frequency in 17/18 organisms. Validation on human chromosome 21 (46.7 Mb, processed in 5.0 s on Apple M1) reveals eukaryote-specific spectral signatures---nucleosome positioning at 10.67 bp, nucleosome spacing at 170.7 bp, and Alu repeat dominance at 341 bp---absent from prokaryotic spectra. A proof-of-concept spectral variant detection experiment achieves 100% read-matching accuracy (100/100 reads) and statistically significant discrimination of SNPs from sequencing errors (t = 14.80, p < 0.001, Cohen's d = 1.64), scaling to d = 8.96 at 30x coverage. The full human genome can be spectrally analyzed in approximately 3--4 seconds on an M1 GPU and under 1 second on M4 Max, enabling interactive spectral genomics on commodity hardware.