PRIM II: The Recursive Architecture of the Physical World

PRIM II develops the structural layer that follows the arithmetic construction of PRIM I. Its central proposal is that the physical world is not organized by a different foundational mechanism at every scale. Instead, it is modeled as a finite observable tower of recursively generated levels in which the same primitive PRIM grammar reappears. The objects carried by those levels need not be physically identical; what recurs is the architecture by which a structure is retained, enlarged, connected to a neighboring level, and transported through a boundary. The primitive unit is a typed PRIM cell with a carrier, admissible relation family, invariant and provenance data, and ordered state and bridge counters. Genuine level formation is NOLOSS: the inherited Old sector remains reconstructible while a nonzero Fresh sector adds relation capacity unavailable on the lower level. Counters across levels form an interwoven provenance network rather than independent clocks. In the PRIM realization, a complete rooted binary response carrier with \(p\) directions has \(2^p-1\) non-root states. Admissible Mersenne values are used as structural opening points, where Fresh relation space becomes available and the adjacent boundary process is activated. Opening, occupation, criticality, completed transition, and physical elapsed time remain distinct. The observable universe is treated as another PRIM cell within the same architecture. A reconstructed scaling comparison gives a continuous relative structural depth \(r_{\mathcal U}\approx41.84\), corresponding under the stated convention to 41 completed adjacent transitions, 42 realized level objects including the reference object, and nearest-shell label 42. Chemistry, Cooper-pair structure, and black-hole causal dynamics are included as current embedding tests of the same structural grammar, not as proofs of universality. The manuscript also contains an explicit disclosure of AI-assisted research, verification, and manuscript preparation.