Deterministic SU(n) Projection onto Phi5 Space Reveals a Closed Membrane Structure, an SU(2)–SU(3) Existence Band, and a Finite Closure Horizon

We construct a deterministic projection of Haar-distributed SU (n) matrices (n = 1–5) into a fixed five-constant space defined by: φ, √2, √3, ln 5, π. The projection produces normalized simplex coordinates, revealing that the system does not occupy the full state space but collapses onto an intrinsic 4-dimensional manifold embedded in a 5-dimensional simplex. This manifold behaves as a closed membrane structure with directional axes corresponding to anchoring (φ), correction (√2), expansion (√3), dissipation (ln 5), and closure (π). A unique mapping between SU (n) sectors and constants is identified and ranks first among all permutations tested, indicating a non-random structural correspondence: SU1→φ, SU2→√2, SU3→√3, SU4→ln 5, SU5→π. The system exhibits intrinsic threshold behavior, converging toward independently defined values: 0. 61, 0. 66, 0. 78, 0. 87, 0. 946, and 0. 99952. A coupling observable between SU (2) and SU (3) defines a finite stability interval: 0. 61 ≤ κ ≤ 0. 87, which we identify as an existence band where stable configurations occur. Outside this band, the system becomes either over-constrained (symmetry-dominated) or unstable (expansion-dominated). A phase transition is observed through the balance between expansion (√3) and dissipation (ln 5), with convergence toward equality near structural saturation. This defines a transition from expansion to a crystallization regime where additional structure no longer increases coherence. A closure coordinate associated with π converges toward a limiting value: s_π → 0. 99952, which defines a horizon beyond which coherent configurations are not maintained. This establishes a finite closure boundary for the system. The global structure is interpreted as a closed toroidal membrane supporting constrained trajectories. The observable system corresponds to trajectories confined within this membrane and restricted to the SU (2) –SU (3) stability band. The results demonstrate that SU (n) systems, when projected onto a fixed constant basis, exhibit: intrinsic dimensional reduction non-random structural mapping bounded stability regions phase transitions governed by internal balance a finite closure horizon This suggests the existence of a constrained geometric layer underlying SU (n) behavior.