This paper presents the Z-Canvas Theory, a unified theoretical framework that reconciles discrete topology with quantum and cosmological phenomenology by abolishing the classical postulates of microscopic unitarity and spacetime continuity. We postulate that the observable universe operates as a thermodynamically open, stochastic causal graph (Layer 0), driven by the continuous injection of information (Layer +1) and stabilized by a dissipative sink bounded by the Bekenstein limit (Layer-1). The overall dynamics of spacetime are analytically dictated by the Master Equation of Z. We demonstrate that this formalism guarantees global stability, recovers the macroscopic Bianchi identities, and resolves fundamental crises without invoking dark entities. On a cosmological scale, cosmic acceleration emerges naturally as stochastic inertial feedback, and dark matter is mathematically deduced as galactic topological hysteresis. On a quantum scale, the model provides mass with an exact geometric origin and explains the chiral anomaly as a necessary thermal leakage. To establish its strict Popperian demarcation, the paper provides an empirical refutation matrix with falsifiable predictions for the next generation of detectors: a specific topological noise floor in gravitational-wave interferometers, a Compton wavelength-dependent variance in the measurement of the proton charge radius, and an anomalous transition towards a thermal power law in the decoherence of macro-quantum systems. In summary, the framework redefines physical reality not as a passive geometric arena, but as a continuous topological computation in rigorous pursuit of thermodynamic equilibrium.