Unified Arithmetic Dynamics Explorer

Visualizing the H = xp Hamiltonian and arithmetic geodesics

Unified Arithmetic Dynamics
Berry-Keating (Spectral)
H = xp Zeros of ζ(s) are eigenvalues of this Hamiltonian.
x(t)=x₀eᵗ The "Flow" that thins the primes over time.
Grand Action (Operational) Link
S = ∫L dt Primes are Geodesics (paths of Least Action).
∇V(n) Arithmetical functions create "Potential Friction."
Synthesis: While Berry-Keating identifies the vibration (Primes as harmonics), the Grand Action Principle identifies the medium. In this view, Primes are the "least-action" stable paths where the buoyancy of the Prime Number Theorem $\pi(x)$ perfectly balances the gravitational "drag" of arithmetical density $\Omega(n)$.