Paul Dirac's Large Numbers Hypothesis (1937) identified a recurring ratio of 10^40 between fundamental physical constants and cosmological scales, most notably the ratio of electromagnetic to gravitational force between elementary particles. Historically dismissed as a numerical coincidence due to the lack of a causative physical mechanism, standard models provide no geometric or thermodynamic engine to explain this correlation. This paper demonstrates that Dirac's ratio is the strict mathematical consequence of a volumetric scaling architecture. By applying the General Theory of Correspondence (GTOC), we establish a universal mass-energy floor of 10^-31 kg and a corresponding volumetric matrix limit. Correcting for the historically miscalibrated Planck ruler, this zero-latency computational architecture is explicitly scaled to the 10^122 baseline. Within this rigidly bounded matrix, we prove that Dirac's 10^40 ratio is not an arbitrary constant, but the exact 1-dimensional linear extraction of the GTOC 3-dimensional spatial matrix constraint (10^120). Dirac did not discover an isolated cosmological anomaly; he calculated the 1-dimensional linear shadow of the volumetric architecture that governs all physical propagation.