The Eigenvalues of Reality

The Eigenvalues of Reality: Jacobian Analysis of the Self-Referential Friedmann System The axiom σ = 1/(1+σ) generates a coupled dynamical system whose fixed point is a four-sector energy partition: φ⁻¹ + φ⁻³ + φ⁻⁵ + φ⁻⁶ = 1. This paper derives the full 3×3 Jacobian of the system, catalogues its eigenvalues (all real, negative, and in ℚ(√5)), and shows that the observed dark energy fraction ΩΛ = 0.6847 equals the attractor value φ⁻¹ plus a geometric frustration correction ε = 1/[Δ(Δ−2)] = 1/15, derived in six forced steps from the axiom's discriminant Δ = 5. The formula matches the Planck 2018 central value to four significant figures with zero free parameters. The frustration arises because the regular pentagon (forced by Δ = 5) cannot tile flat three-space. Three pentagons at a vertex leave a 36° deficit; normalised by the pentagon's total interior angle, this gives ε = 1/15. The Gauss–Bonnet theorem generalises the correction to curved space: ε(x) = 3(1−x)/[5(9+x)], where x measures the fraction from flat space to dodecahedral closure. At flat (x = 0), ε = 1/15. At closure (x = 1), ε = 0. The derivative at flat, combined with the slowest eigenvalue λ₁ = −(3√5−5)/2, predicts a dark energy equation of state w₀ ≈ −0.97, consistent with DESI DR1. One function unifies three predictions: the current dark energy fraction, the equation of state, and the endpoint of spatial curvature evolution. Five kill conditions are stated. The framework is falsified if: w₀ = −1 exactly at 1% precision; w₀ < −1 (phantom); w₀ falls outside −0.98 to −0.95; the matter fraction deviates from 0.3153 ± 0.008; or the frustration derivation is shown to be circular. DESI DR2, Euclid, and Rubin/LSST will test these predictions within 2–3 years. Part of the Pentagon Physics programme. Thirty-six companion papers derive all fundamental constants from the same axiom.