The Complete Lagrangian of Inverted Hypersphere Cosmology: A Single Action from Real Projective Four-Space

All predictions of the Inverted Hypersphere Cosmology series follow from a single action principle on real projective four-space. This paper assembles that action explicitly, proves that every term is forced by the topology, derives all equations of motion, and recovers the full set of zero-parameter predictions as their solutions.

The complete action has four components: the Einstein-Hilbert term with cosmological constant derived from the ultraviolet-infrared Casimir seesaw; the conformally coupled cohesion field with quartic potential forced by the antipodal identification; the SO(10) gauge sector derived from the isometry doubling of real projective four-space; and the matter sector accommodating exactly three Standard Model generations as three cycles of the 24-cell polytope. Nothing is added by hand.

From the Euler-Lagrange equations, with no further assumptions, the following are derived: dark energy density 0.6882, sound horizon 153.2 Mpc, expansion-rate step at redshift 0.754, coherence factor 6cos(pi/23), weak mixing angle 0.23176, inverse fine structure constant 137.036, the charged lepton mass formula, proton-to-electron mass ratio 1836, and grand unification energy 1.005 times 10 to the 15 GeV. All 17 predictions are solutions of the same Euler-Lagrange equations. None is a free parameter. The theory is falsifiable, internally consistent, and complete. Verification: 28 independent tests pass across three tools: Cadabra2, SymPy, and NumPy/SciPy. All validation scripts are included as supplementary material.