LFIS–13: Multi-Ray Closure and Triangle Admissibility

LFIS–13 formalizes multi-ray closure and triangle admissibility in Light Frame Cadence Geometry. The volume distinguishes single-ray closure from joint multi-ray closure, showing that individual representability along separate ray-relations does not guarantee simultaneous representability across a network. The central result is that the triangle is the minimal configuration in which loop consistency can impose a genuinely nontrivial representational constraint. In tree-like or single-ray settings, loop obstruction is absent; in a three-ray closed loop, mutually compatible edge relations can still fail to admit a single jointly admissible representational state. This establishes triangle inadmissibility as the first structural source of selective representability. LFIS–13 defines routing as the selective retention of representable closure when full joint closure is inadmissible. The volume is explicitly constraint-level: no transport mechanism, global time parameter, energetic depletion, or dynamical evolution is introduced. It supplies the formal multi-ray closure foundation used by the later LFIS volumes on routing, completion, representational rest, and foundational constraint completeness.