The Complete Periodic Table of Mathematical Concepts: A Unified Classification of Lattices, Operators, Transforms, Hilbert Spaces, and Mathematics Under the Canvas Model

For centuries, mathematics and physics developed side by side, sharing tools but never fully unified. The Fourier transform was invented to solve the heat equation. Hilbert spaces were formalized for quantum mechanics. L-functions were studied in number theory. Each field built its own machinery, its own language, its own transforms. The canvas model reveals that all of these are the same thing. Three equations govern everything. Eight primitives configure everything. Every integral transform, every Hilbert space, every spectral theorem, every L-function—they are not separate inventions. They are the same framework applied to different lattices with different property settings. What this monograph contains: · A complete classification of 26 lattice types (order primitive configurations), from continuous real lines to prime adele lattices to graph vertices· A classification of 19 operator types (acceleration primitive configurations), from standard Laplacians to Dirac operators to graph Laplacians· A classification of 12 amplitude configurations (from uniform weights to von Mangoldt weighting to information density)· A classification of 9 polarity configurations (from Dirichlet to Neumann to periodic to Möbius-weighted)· The complete periodic table of spectral transforms (48 transforms total), including classical continuous transforms (Fourier, Laplace, Mellin, Hilbert, wavelet, etc.), classical discrete transforms (DFT, DCT, Z-transform), irregular lattice transforms, amplitude-gated transforms, polarity-enhanced transforms, tensor transforms (Tensor Laplace, Tensor Fourier, Tensor Mellin, TAC Spectral), graph and network transforms, and 11 predicted but not yet discovered transforms· A classification of 31 Hilbert spaces, from continuous $L^2$ spaces to prime lattice spaces to Hardy spaces to graph spaces to Sobolev spaces· Systematic mappings of every major branch of mathematics—including Logic and Foundations, Algebra, Number Theory, Algebraic Geometry, Topology, Differential Geometry, Analysis, Functional Analysis, Complex Analysis, Probability and Statistics, Combinatorics and Graph Theory, Dynamical Systems, Optimization and Control, and Mathematical Physics—to the canvas model's eight primitives and three equations Why this matters: The universe does not distinguish between physics and mathematics. Neither should we. Every mathematical structure—every space, every operator, every transform, every algebraic object—is a configuration of a single unified wave equation with four dynamic primitives, filtered by a threshold condition with four property primitives, and organized by an eigenvalue equation. This is not a metaphor. It is a classification theorem. For each known mathematical concept, we specify the exact values of the eight primitives that generate it. The periodic table of transforms is the periodic table of physics is the periodic table of mathematics. All are configurations of eight primitives governed by three equations. This monograph is the culmination of the Emergence series, building on the Riemann Hypothesis proof, the primitive ontology, the unification of physics, and the extension to psychology and cosmology. It demonstrates that the canvas model provides a complete foundation for all of mathematics. Keywords: canvas model, periodic table of mathematics, spectral transforms, Hilbert spaces, lattices, operators, classification, unification, mathematical physics, number theory, functional analysis