TDN — Numerical Falsification Tests: Two Central Numerical Assumptions Are Not Supported, and a New Scaling Law Emerges

Independent numerical follow-up study to the Time Dilation Network (TDN) framework (Version 1, DOI 10.5281/zenodo.20625572). This is a standalone study with its own methodology and results, not an erratum or revised version of the original paper. We test, by large-scale simulation of random ordered directed acyclic graphs up to N=2000, two central numerical claims associated with the current TDN formulation: (1) the existence of a universal scaling exponent in L_max ~ N rho^beta, and (2) the existence of a second-order phase transition at rho_c ~ 1/N marking the emergence of time. Results: (a) Strong data collapse establishes the scaling form L_max = N f(rho) with finite-size corrections below the percent level. (b) However, f(rho) is not a single power law; effective exponents drift from ~0.63 to ~1.0, so no universal exponent exists. (c) Susceptibility, derivative, and direct-threshold tests show no second-order transition at rho_c; the behavior is a smooth crossover. (d) Critically, the scaling law is ensemble-dependent: causal-set sprinkling gives L_max ~ N^(1/d), so the exponent encodes spacetime dimension and the independent-edge ensemble corresponds to effective dimension d=1. Results are reproduced with three independent algorithms. These findings falsify two specific numerical assumptions but not the core TDN definition of time as longest-path length, which lies outside the scope of these tests. They show that no stable quantitative prediction can be extracted before the physically relevant generating ensemble is specified (the generator problem).