A Tessarine Reformulation of the Feynman Propagator: Rapidity Cutoff, Vacuum Condensate, and the Planck Frequency as a Geometric Mean

We propose a reformulation of the Feynman propagator in terms of the tessarine algebra, in which the vacuum is represented as a stationary condensate of hyperbolic rapidities rather than as a transient ensemble of virtual particles. The construction begins by expressing the momentum-space measure d³p in rapidity coordinates and identifying the atemporal, isotropic section of the tessarine wave function Ψ(0, β) = 2 cosh β as the natural vacuum amplitude. The resulting integral reproduces the standard vacuum energy density ρᵥ while revealing a quartic exponential divergence ∼ ∫ e⁴ᵝ dβ. We introduce a physically motivated rapidity cutoff yₘₐₓ ≈ 1.93, constrained by the observed upper bound on cosmological recession velocity and the lightest neutrino mass scale, and show that this cutoff may offer a new perspective on the vacuum energy discrepancy. We further observe that the Planck frequency arises as the geometric mean of the Schwarzschild frequency and the Margolus–Levitin computational bound, a relation that follows directly from the conformal inversion r ↦ 1/r underlying the tessarine framework. This inversion is interpreted as a geometric barrier analogous to confinement in gauge theories. Throughout, we distinguish carefully between identities inherited from established physics and the interpretive extensions proposed here.