LFIS–13: Light Frame Infrastructure Series — Multi-Ray Closure and Triangle Admissibility: Volume 13

LFIS–13 records the minimal multi-ray constraint governing joint representability in Light Frame Cadence Theory. This volume establishes the distinction between single-ray closure and multi-ray closure and identifies the three-ray loop (the triangle) as the minimal configuration in which joint representational admissibility can fail, even when all pairwise relations remain admissible. The paper formalizes triangle inadmissibility as a structural condition arising from coupled closure constraints under finite representability. It shows that failure of joint closure is not dynamical and does not result from interaction, transport, or evolution. Instead, it reflects the absence of a jointly admissible representational solution once multiple closure obligations must be satisfied simultaneously. Within this framework, routing is identified as the canonical representational response when full joint closure is not admissible. Routing does not repair closure, transmit information, or resolve contradiction; it preserves coherence by reallocating representational obligation under constraint. All results in LFIS–13 are stated at the level of structural constraints. No dynamics, transport mechanisms, field equations, probabilistic rules, or global time assumptions are introduced. “Inadmissibility” denotes only the absence of a jointly admissible representational configuration. It does not imply depletion of cadence, loss of coherence, physical inconsistency, or breakdown of relation. LFIS–13 serves as infrastructure for later treatments of routing, representational completion, and boundary formation within the Light Frame Infrastructure Series.