Realizability and the Origin of Causality

Zeno's paradox implicitly assumes that spatial displacements carry irreducible positivecost at every scale. We negate this: if spatial displacement sequences exist whose cost-to-magnitude ratio vanishes (Assumption R), while pure temporal displacements retain positivecost (Assumption T), then in (1+1) dimensions any quadratic leading-order expansion of thecost 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 spatialdisplacement costs rather than from any geometric postulate. In (1+3) dimensions, under astrengthened 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: noparticles, light signals, or kinematic objects are presupposed. This provides a more primitivegeometric foundation for the thermodynamic derivation of Einstein’s equations [2].