The Odd Hyperbolic Partner and the Cornell Potential: Split-Complex Vacuum Trichotomy, Born-Rule Spatiality, and the Three Luminal Causalities

In previous works we parametrized the vacuum zero-point energy, the Schwarzschild interior metric, the charged-lepton mass spectrum, and the regularized Coulomb field using even hyperbolic functions of rapidity (cosh β, sech β, and sech² β). In the present note we examine the role of the odd hyperbolic partner, sinh β, and show that it naturally parametrizes the phenomenological Cornell potential of quantum chromodynamics (QCD), V꜀ₒᵣₙₑₗₗ(r) = -A/r + Br. Factoring the potential at its crossover radius R꜀ = √(A/B) with characteristic energy scale V₀ = √(AB) = σR꜀, and setting r/R꜀ = eᵝ, yields the exact identity V꜀ₒᵣₙₑₗₗ(β) = 2V₀ sinh β, whose rapidity derivative recovers the vacuum zero-point structure, dV꜀ₒᵣₙₑₗₗ/dβ = V₀Ψᵥ(β) with Ψᵥ(β) = 2 cosh β. Within the split-complex (j² = +1) subalgebra of the tessarines, the confining potential corresponds to the perplex imaginary component Im_j(eʲᵝ) = sinh β that cancels out in the symmetric vacuum sum Re_j(eʲᵝ) = cosh β. This establishes an algebraic trichotomy over the unitary pair (eʲᵝ, e⁻ʲᵝ): symmetric sum (vacuum energy and mass), antisymmetric difference (chromodynamic confinement), and Born-rule conjugate product (emergent spatial metrics and inverse-square fields). Moreover, expanding eʲᵝ = eᵝ e₊ + e⁻ᵝ e₋ in the canonical idempotent zero-divisor basis e_± = ½(1 ± j) shows that conjugate tessarine states occupying opposite null-cone branches satisfy an identically vanishing effective metric separation dₑ = |z₁z₂| = 0 at arbitrary spatial scale r = R₀eᵝ, providing a scale-free algebraic realization of bipartite and multipartite quantum entanglement. Finally, we show how function parity distinguishes horizon permeability across the three luminal forces, and how the weak interaction acts outside this spatial triad as a generalized tessarine Wick projection (ij · j = i) that relaxes hyperbolic contraction by shedding discrete Riemann-sheet windings and decompressing the Gaussian localization core. Mathematical identities are strictly separated from physical conjectures.