The Conservation at the Boundary Theorem: How the Finite Edge of Preonic Space Completes the Causal Lifecycle of the Universe

In Quantum-Geometry Dynamics (QGD), space is constituted by a finite discrete lattice of preons(−) — the fundamental spatial units that are space itself. The lattice has a definite finite extent: it ends at the last preon(−). When a preon(+) executing a directed leap reaches a boundary-adjacent preon(−) with a momentum vector pointing outward, there is no adjacent preon(−) in that direction. This paper derives what happens from Axiom 2 (directed leap) and conservation of momentum alone, without additional postulate. The Conservation at the Boundary Theorem establishes that conservation of momentum at the boundary of the finite preonic lattice requires each preon(+) to execute its directed leap to the corresponding preon(−) on the opposite face of the lattice. Any other outcome — stalling, reflection, or forced parallelisation of the momentum vector — would require momentum non-conservation across a causal transition, which the axioms prohibit. The theorem implies that the preonic lattice has the topology of a three-torus: finite, discrete, and closed. This topological conclusion is derived, not imposed. The theorem generates different consequences for different types of composite structures. A composite object in QGD does not cross the boundary as a unit. Only individual constituent preons(+) cross, each conserving its own individual momentum vector. In any composite whose constituent preons(+) have non-parallel momentum vectors — every known composite except photons — boundary crossing distributes the crossed preons(+) across different regions of the opposite face according to their distinct momentum directions. The crossed preons(+) scatter. They cease to satisfy the binding conditions relative to each other and relative to the constituent preons(+) remaining on the original side. The composite structure dissolves into free preons(+), which propagate independently across the lattice below the electromagnetic detection threshold. These constitute the free preonic dark matter background. Photons are the unique exception. A photon in QGD is a composite whose constituent preons(+) all have parallel momentum vectors. They cross the boundary sequentially — not simultaneously, because they are spatially staggered along the propagation direction — but because all momenta are parallel, all constituent preons(+) cross to the same direction on the opposite face. The transverse arrangement of constituent preons(+) is preserved because parallel momenta carry each one to the transversely corresponding position. The longitudinal staggering is preserved because sequential crossing reproduces the original spacing. P-gravity maintains the binding configuration throughout the crossing. The photon reconstitutes intact on the far side of the finite lattice. This difference between photons and all other composites generates a specific observational prediction: photons from distant galaxies that have crossed the boundary appear as mirror images at antipodal sky positions with reversed chirality. A spiral galaxy winding clockwise as seen in the original appears counterclockwise in the reconstituted image. The image appears at the antipodal sky position, and its redshift differs from the original by an amount proportional to the additional path length traversed. This prediction is falsifiable with existing wide-field galaxy survey data and is independent of the QGD empirical grounding programme — it requires only the topology result and morphological galaxy survey coverage. Together, the Conservation at the Boundary Theorem, the dissolution mechanism, and the photon exception complete the derivation of the four-stage causal lifecycle of the preonic universe from the axioms: (1) the isotropic initial state as a uniform fog of free preons(+); (2) condensation of composite structures through the p-gravity cascade; (3) material expansion as large-scale separations exceed d_Λ and n-gravity dominates; (4) dissolution of structures at the boundary as constituent preons(+) cross individually, scatter, and reconstitute the free preonic background. The lifecycle is continuous rather than catastrophic — all four stages operate simultaneously in different regions of the universe — and closes causally: the dissolved free preons(+) are the isotropic initial state. The entire lifecycle of the universe — from its first structure to its final dissolution and reset — is a theorem derivable from three axioms and two constants. No external cause, no additional postulate, and no fine-tuning is required at any stage. This paper supersedes the description of Stage 4 given in the visual companion presentation (The Preonic Universe: A Deterministic Lifecycle from Origin to Edge). That presentation described the boundary mechanism as forced parallelisation of momentum vectors — a description that would require momentum non-conservation. The Conservation at the Boundary Theorem gives the correct account: wrap-around through conservation, not reflection or parallelisation.