The Primitive Spectral Transform: A Unified Framework for Spectral Analysis

We present the Primitive Spectral Transform (PST), a unified framework for constructing spectral transforms from the geometry of the sampling domain. The central insight is that every spectral method—Fourier, Laplace, Mellin, wavelet, graph Fourier, and their discrete, irregular, and tensor variants—is a configuration of four fundamental choices: the lattice (where samples live), the operator (how neighbors interact), the amplitude (how signals are weighted), and the polarity (how boundaries and phase are handled). What this paper provides: · A unified framework for all spectral methods. Given any discrete sampling set, the PST constructs a self-adjoint operator from the local geometry—a Jacobi matrix from sample spacings, a graph Laplacian from connectivity, a cumulative sum operator from prime exponents. The eigenvectors of this operator form an orthonormal basis adapted to the sampling structure. The eigenvalues define the natural frequency spectrum. Projecting data onto this basis yields the spectral transform. The forward transform produces spectral coefficients; the inverse transform reconstructs the signal exactly.· Reduction to classical transforms. For uniform sampling with appropriate boundary conditions, the PST reduces to the standard discrete Fourier transform (DFT). For the half-line with Dirichlet conditions, the PST reduces to the Laplace transform. For the multiplicative domain, the PST reduces to the Mellin transform. For graph vertices, the PST reduces to the graph Fourier transform. Every classical spectral method is a special case of the PST.· Generalization to irregular sampling. For arbitrarily spaced sampling points, the PST constructs a Jacobi matrix from the local spacings. The resulting eigenbasis is orthogonal and adapted to the sampling density: rapid oscillation in dense regions, slow oscillation in sparse regions. Compression and denoising follow naturally by retaining the largest spectral coefficients. No interpolation is required.· The four primitives as a periodic table. The lattice (Order primitive) can be uniform, irregular, warped, prime exponent, graph vertex, or continuous. The operator (Acceleration primitive) can be the discrete Laplacian, cumulative sum, graph Laplacian, fractional Laplacian, or Berry-Keating dilation. The amplitude primitive includes uniform weighting, energy weighting, threshold gating, and von Mangoldt weighting (for prime lattices). The polarity primitive includes Dirichlet, Neumann, periodic, polarity-aligned, and Möbius-weighted boundary conditions. The paper presents a complete periodic table of 35+ transforms, each specified by its choice of lattice, operator, amplitude, and polarity.· The Plank threshold for mode selection. Two modes with amplitudes A_i, A_j are mutually resolvable if A_i A_j > \sigma^2/(N |\omega_i - \omega_j|). This provides a principled criterion for determining which spectral coefficients are physically meaningful—a direct application of the Threshold Condition from the canvas model.· Tensor transforms on prime lattices. When the lattice is the set of prime exponents, the PST generates the Tensor Laplace Transform (TLT), Tensor Fourier Transform (TFT), Tensor Mellin Transform (TMT), and the Tensor Adele Class (TAC) operator—a self-adjoint operator whose spectral determinant is the completed Riemann zeta function \xi(s). These are the spectral methods underlying the theory of L-functions, constructed rigorously from the PST framework.· Algorithmic implementation. A step-by-step algorithm is provided for constructing the PST from any lattice configuration, operator choice, amplitude weighting, and polarity condition. A reference implementation is available in the supplementary materials. For uniform lattices, the operator is circulant and the FFT applies in O(N \log N). For arbitrary irregular lattices, the first K modes can be computed via Lanczos iteration in O(NK). Why this matters: The Fourier transform was invented to solve the heat equation. The Laplace transform for transients. The Mellin transform for multiplicative domains. Wavelets for multiscale analysis. Graph Fourier for networks. Each was discovered independently, each with its own formalism. The PST reveals they are all the same engine—a self-adjoint operator diagonalized, its eigenfunctions forming a basis, data projected onto that basis. The differences are only the four primitive choices. The PST is derived from the Canvas Model (Emergence I), a unified framework in which all physics and mathematics emerge from eight primitives governed by three equations. The mathematical foundation is formalized in Canvas Temporal Mathematics (Emergence 31). The complete classification of all transforms is in the Periodic Tables monograph (Emergence 34). The operator-theoretic foundation for the tensor transforms is in the Tensor Toolkit (Emergence 33). This paper is the practical application. It requires no familiarity with the theoretical papers. It presents the PST as a self-contained computational framework for spectral analysis on arbitrary domains. Keywords: Primitive Spectral Transform, PST, spectral analysis, Fourier transform, Laplace transform, Mellin transform, wavelet, graph Fourier, irregular sampling, Jacobi operator, Plank threshold, tensor transforms, TAC operator, Riemann zeta function, Canvas Model, periodic table of transforms