The Last Kervaire Manifold as a Physical Origin: From Gödel Incompleteness Oscillation to Riemann Zeros Spectrum

This mathematical foundation expands physical principles to a unique last 126-dimensional smooth closed manifold with Kervaire invariant one, serving as the underlying space for a self-projective topological system subject to the following arithmetic condition. The complex structure modulus is locked at the Heegner point $\tau_{163}$, yielding arithmetic invariants that determine all physical parameters. From five axioms we derive the standard model and general relativity as projections, and obtain a spectral density for the chaotic background that is the unique fixed point of a Fourier self-projection equation, relying on the Riemann hypothesis, making it testable. Testable predictions include a universal power-law tail in the CMB small-scale power spectrum, periodic modulations in the expansion history, and discrete features in the stochastic gravitational wave background. All predictions are parameter-free and accessible to current or near-future observations.