The Hyperbolic Vacuum: A Zero-Parameter Exact Solution of Einstein's Field Equations and the Geometric Origin of Apparent Dark Energy

The spacetime interval ds² = c²dt² − dr² defines a hyperbolic geometry that is an exact vacuum solution of Einstein's field equations, with k = −1, ρ = 0, Λ = 0, and a (t) = ct. The geometry is Milne's; what is new is the Hamiltonian-constraint selection that singles it out as the unique expanding member of the vacuum family, a measured perturbation channel that Milne's empty universe cannot carry, and its confrontation with modern data. We advance this vacuum not as an asymptotic limit but as the expansion history of the universe after recombination. Everything is scoped to z < 100; the earlier universe is not modelled and is not claimed. With zero adjustable parameters the geometry reproduces the Type Ia supernova luminosity distances (Pantheon+, χ²/dof = 0. 910 over 1580 supernovae), the 1998 dimming attributed to acceleration, and the Hubble constant from the age of the universe alone (H₀ = 1/t₀ = 71. 9 ± 2. 6 km/s/Mpc). Fitting flat ΛCDM to distances generated by this geometry returns Ω_Λ = 0. 52–0. 75, spanning the observed value, with no dark energy in the input. Two direct measurements are reported: the effective-curvature channel Ωᵦam = −0. 03 ± 0. 14 (≤ 0. 12 at 68%), and a homogeneous matter component bounded at Ωₘ ≤ 0. 038 (95%) — below the known baryon density, consistent with matter residing in walls rather than as a homogeneous component. The framework predicts redshift drift identically zero, an algebraic identity of H (z) = H₀ (1+z), falsifiable within ELT-ANDES design sensitivity. Standing open problems are stated with named tests: the DESI DR2 Alcock-Paczynski shape mismatch (χ² = 135. 3/6 under ΛCDM-fiducial extraction), a Pantheon+ residual of Δχ² = 49. 6, and the CMB acoustic angle, which no member of the family with a standard sound horizon reproduces and which is posted as an explicit constraint on any early-universe completion. A decision table commits in advance to which claims each scheduled measurement would kill. This is submitted as a solicitation: the decisive tests require collaboration-scale pipelines the author does not hold, and are specified precisely so that groups holding them can perform them.