The Complete Canvas Periodic Table of Constants: A Complete Classification of Mathematical and Physical Constants by the Tether Principle and Regularization Depth

For centuries, mathematicians and physicists have discovered special numbers: \pi, e, \gamma, \phi, \zeta(-1) = -1/12, \zeta(-3) = 1/120, \alpha_0^{-1} \approx 140, \delta \approx 4.669, and hundreds more. These numbers appear in deep formulas across geometry, analysis, number theory, quantum field theory, cosmology, and chaos theory. They seem arbitrary. They are not. This monograph presents the Periodic Table of Special Numbers — a complete classification of all fundamental constants by their tether type (spatial, parameter, symmetry, intersection) and regularization depth (logarithmic, power-law, zeta, renormalization, geometric). The central thesis: Every special number is the residue of a tethered divergence — the "note" that an untethered string sings when it is tethered. What this monograph provides: · The Tether Residue Theorem: For any divergent quantity arising from an untethered system, there exists a tether (spatial, parameter, symmetry, or intersection) such that the regularized quantity is finite and independent of the regularization method. This residue is a special number. Conversely, every special number can be represented as the residue of some divergent quantity under an appropriate tether.· A complete classification of all major special numbers by tether type and regularization depth: · Spatial tethers (boundaries, horizons, geometry): \pi, \sqrt{2}, \sqrt{3}, \sqrt{5}, 1/\sqrt{3}, 1/5, 1/6, \pi/4, \ln 2, \alpha_0^{-1} \approx 140, \zeta(-3) = 1/120, \zeta(-5) = -1/252, \zeta(-7) = 1/240, etc. · Parameter tethers (fixed values, bifurcations, limits): e, \phi (golden ratio), Feigenbaum constants \delta \approx 4.669 and \alpha \approx 2.503, \zeta(2) = \pi^2/6, \zeta(3) (Apéry's constant), \zeta(4) = \pi^4/90, etc. · Symmetry tethers (invariance, \mathcal{S}-operator): \zeta(-1) = -1/12, \zeta(0) = -1/2, \zeta'(0), \zeta'( -1), the Riemann zeros (\gamma_1 \approx 14.1347, \gamma_2 \approx 21.0220, etc.), \eta(1) = \ln 2, Catalan's constant G \approx 0.915965594, \Gamma(1/2) = \sqrt{\pi}, \Gamma(1/3), \Gamma(1/4). · Intersection tethers (thresholds, phase transitions): \gamma (Euler-Mascheroni, \approx 0.577), Casimir coefficient 1/120 (also spatial), critical exponents of the 3D Ising model (\beta \approx 0.326, \gamma \approx 1.237, \nu \approx 0.630, \eta \approx 0.036), twin prime constant C_2 \approx 0.660, Artin's constant \approx 0.374, Landau-Ramanujan constant \approx 0.764, Khinchin constant \approx 2.685, etc.· A demonstration that the constants of physics are not fine-tuned — they are the unique residues of the tethers that define our universe. The cosmic horizon is the ultimate spatial tether, cutting off all infrared divergences. The Planck scale is the ultimate ultraviolet cutoff. Together, they define the finite information capacity I_{\text{max}} of the observable universe.· Predictions for missing constants based on the symmetry of the periodic table: \zeta(-1/2), \zeta(-3/2), \zeta'(-3), constants for prime k-tuples, Feigenbaum-like constants for other maps, neutrino mass ratios as powers of \alpha_0, and the tensor-to-scalar ratio r \approx \sqrt{\alpha_0} \approx 0.085. The unity of mathematics and physics: The periodic table reveals that every special number — whether from geometry, analysis, number theory, quantum field theory, cosmology, or chaos theory — is the residue of a tethered divergence. The same principle generates \pi, e, \gamma, \zeta(-1), \alpha_0, and \delta. The universe is a regularization device. Constants are not brute facts. They are the notes of tethered strings. Keywords: special numbers, mathematical constants, physical constants, tether principle, regularization, zeta regularization, Riemann zeta function, Euler-Mascheroni constant, Feigenbaum constants, golden ratio, fine-structure constant, cosmological constant, Casimir effect, critical exponents, Riemann zeros, twin prime constant, Catalan's constant, Apéry's constant, Khinchin constant, periodic table of constants, Canvas Model, Canvas Temporal Mathematics