Prime Wave Bridge: The Four Primitives that Unify Physics and Number Theory

Note to Readers: This paper synthesizes results from several papers in the Canvas Model series. The author is seeking arXiv endorsement to make the work more widely accessible. If you are a registered arXiv user and have found this work valuable, an endorsement would be greatly appreciated. Please contact: [e-mail] Physics and number theory are traditionally separate disciplines. Physics describes the behavior of matter and energy through dynamical equations. Number theory studies the properties of integers and the distribution of primes. No established framework connects them. This paper shows that both emerge from the same four dynamic primitives—Order, Amplitude, Acceleration, and Polarity—governed by the same organizing angle \pi/2. The connection is not analogical. It is structural. What this paper provides: · A unified framework for both domains. The four primitives generate the Unified Wave Equation (physics) and the Primitive Spectral Transform framework (number theory). The same four choices—Lattice (Order), Operator (Acceleration), Amplitude weighting, and Polarity conditions—produce the Fourier, Laplace, and Mellin transforms on one side and the Tensor Adele Class (TAC) operator whose spectral determinant is the completed Riemann zeta function \xi(s) on the other.· The \pi/2 organizing angle. The primitives map onto the four quadrants of the unit circle. The angle \pi/2 is the boundary between the first two quadrants, separating Order from Amplitude. In physics, this ratio produces the waveform asymmetry T_{\text{rise}}/T_{\text{fall}} = \pi/2, verified in 3+1D numerical simulation. In number theory, the same angle organizes the PST periodic table of 44+ spectral transforms.· The Primitive Spectral Transform as the bridge. Entry T19 in the PST periodic table is the TAC operator—a self-adjoint operator on a restricted tensor product of \ell^2 spaces over the primes whose regularized spectral determinant is \xi(s). The eigenvalues of the TAC operator are in bijection with the non-trivial zeros of \zeta(s). The Steering dynamics drive them toward the critical line \operatorname{Re}(s) = 1/2.· The six primitive pairings and their realizations in both domains: Order × Acceleration (wave dynamics / Berry-Keating), Order × Amplitude (linear propagation / von Mangoldt potential), Order × Polarity (time asymmetry / Dirichlet boundary), Amplitude × Acceleration (mass generation / real symmetric Jacobi), Amplitude × Polarity (threshold crossing / tensor product over primes), and Acceleration × Polarity (nonlinear oscillation / free operator reference).· The four tether types (spatial, parameter, symmetry, intersection) as the four ways constraints produce discrete structure, appearing in both physics (particle in a box, fine-structure constant, Noether's theorem, spacetime voxel threshold) and number theory (Dirichlet boundary, fixed r in logistic map, functional equation of \zeta(s), Möbius zeros). Why this matters: The bridge is not analogical. It is structural. The same four primitives that generate the waveform asymmetry in physics generate the TAC operator in number theory. The Riemann Hypothesis and the \pi/2 waveform asymmetry are consequences of the same underlying principles. If the asymmetry is confirmed experimentally, it validates not only the physics but also the mathematical structure. The same experiment that tests a wave equation also tests a conjecture about the Riemann zeta function. Keywords: four primitives, Order, Amplitude, Acceleration, Polarity, Unified Wave Equation, Primitive Spectral Transform, waveform asymmetry, \pi/2, Tensor Adele Class, Riemann zeta function, Riemann Hypothesis, L-functions, periodic table of transforms, tether types