Realizability and the Origin of Causality

We show that spacetime causality—light cones and an invariant maximal speed—arises as a logical necessity from the requirement that a fundamental variational law admit an attained global minimum (realizability). Under positive-definite (Euclidean) geometry, the minimal action principle admits only an unattained infimum, rendering nontrivial dynamics unrealizable. Realizability enforces the existence of zero-cost (null) propagation modes, which exclude Euclidean geometry and force an indefinite (or degenerate) metric signature. Special Relativity emerges as the minimal geodesic dynamics compatible with this constraint, rather than as a foundational postulate.

**Table 1: Classification of Spacetime Realizability**

| Parametric Condition | Geometric Type | Physical Consequence  |  Perception |

| \(ab>0\) | Euclidean (Positive-Definite) | The global minimum (\(\min=0\)) is attained only by the trivial process; restricted to nontrivial processes the infimum is \(0\) but unattained. Hence nontrivial dynamics are unrealizable (realizability fails). | The “Collapse Limit” (Unrealizable) |
| \(ab<0\) | Lorentzian (Indefinite) | Null directions exist (\(\exists\,\delta x\neq 0\) with \(ds_{\mathrm{info}}^{2}=0\)). Define \(c_{\max}:=\sqrt{-a/b}>0\); the null cone (\(dr/dt=c_{\max}\)) provides zero-cost propagation modes, making a nontrivial attained global minimum (realizable dynamics) possible. | Our familiar Causal World |
| \(ab=0\) | Galilean (Degenerate) | The null cone “opens up” (\(c_{\max}\to\infty\), equivalently \(b\to 0\)). Proper time \(d\tau\) decouples from spatial displacement, becoming an absolute time increment. | Newton/Galileo’s Ideal_