Phi‑Emergence Theory

We present a complete axiomatic framework for the emergence of spacetime and gravity from discrete, stochastic dynamics on graph sequences. The theory, called Φ-Emergence Theory (ΦET), is built on six fundamental axioms (A1–A6) that specify: (i) a discrete substrate with scaling relations, (ii) Gromov-Hausdorff con-vergence to continuous geometry, (iii) heat kernel convergence, and (iv) dimensional constants calibrated to cosmological observables. From these axioms, we derive the emergence of continuous spatial geometry, effective gravitational potential, Newto-nian dynamics, and post-Newtonian corrections. The theory is formulated using a dual-track methodology that separates structural derivations (Track A, parametric) from empirical calibration (Track B, numerical), preventing circular reasoning. We prove dimensional consistency, linear stability of the Newtonian regime, universality of effective couplings, and provide explicit falsifiability criteria. No-Go Theorems establish that certain quantities (dimensions, density, action) cannot emerge with-out additional structure, justifying our axiomatic choices. The manuscript includes a fail-fast calibration test demonstrating feasibility before full theory development.