Realizability and the Origin of Causality

This work derives spacetime causality as a logical necessity rather than a physical postulate.
By imposing a minimal consistency requirement—that variational laws admit attainable extrema—we show that positive-definite (Euclidean) geometries are logically incompatible with nontrivial dynamics, leading to unavoidable variational collapse.

The analysis establishes the Incompatibility Lemma: in any positive-definite metric, the stationarity condition $\delta \mathcal{S}=0$ implies only trivial displacements. Realizability therefore requires the existence of nontrivial null directions, uniquely enforcing an indefinite (Lorentzian) metric structure and its associated causal boundary (the light cone).

Beyond spacetime physics, numerical investigations suggest that analogous realizability constraints arise in arithmetic dynamics. Experiments involving oscillatory modes associated with the nontrivial zeros of the Riemann zeta function indicate that the critical line $\mathrm{Re}(s)=\tfrac{1}{2}$ functions as a unique stability boundary: oscillations dissipate below it, collapse into trivial smooth behavior above it, and persist only at the boundary itself.

From this perspective, causality is not an added assumption but a structural mechanism preventing the collapse of variational laws. The appearance of causal cones in physics and critical oscillatory spectra in arithmetic may reflect a common realizability principle governing the existence of nontrivial structure across domains.