Realizability and the Origin of Causality

Zeno's paradox implicitly assumes that spatial displacements carry irreducible positive
cost at every scale. We negate this: if spatial displacement sequences exist whose cost-to-
magnitude ratio vanishes (Assumption R), while pure temporal displacements retain positive
cost (Assumption T), then in (1+1) dimensions any quadratic leading-order expansion of the
cost function is forced to be nondegenerate and indefinite. After normalisation,
Q = dt^2 − c_{max}^{-2} dr^2.
Lorentzian signature therefore emerges from the asymmetry between temporal and spatial
displacement costs rather than from any geometric postulate. In (1+3) dimensions, under a
strengthened three-directional zero-threshold condition together with nondegeneracy of Q,
the only possible signature is Sig(Q) = (1, 3).
The present derivation differs from the EPS axiomatisation [1] in its starting point: no
particles, light signals, or kinematic objects are presupposed. This provides a more primitive
geometric foundation for the thermodynamic derivation of Einstein’s equations [2].