From Closure to Content: Matter Representation Selection in the Quantized Dimensional Ledger

This paper develops the next stage of the Quantized Dimensional Ledger (QDL) program by asking whether structural admissibility constrains matter content as well as operators and interaction architecture. In the QDL framework, physical constructions are assigned integer ledger charges and judged by closure rather than by dimensional homogeneity alone. Earlier stages of the program argued that admissibility can filter operators, organize closure classes, and constrain lawful interaction architecture. The present paper asks the next necessary question: once closure-compatible interaction organization is fixed, which matter representations can lawfully inhabit it? The paper introduces a minimal formal setting in which a matter representation is treated not merely as a group-theoretic label, but as a package of field assignment, kinetic structure, gauge-coupling behavior, Yukawa compatibility, and stability under admissibility-preserving completion. Within that setting, an admissible matter-sector theorem is established: a matter representation is structurally viable only if its kinetic, gauge, and interaction terms jointly close into the admissible action sector. A representation-selection corollary then states that representations failing to support such a closure-compatible interaction network are excluded from the lawful matter inventory. A Standard Model-style chiral matter embedding is presented as a positive worked example, and a nearby malformed alternative is used to illustrate exclusion by closure failure. The paper does not claim a unique derivation of the Standard Model fermion spectrum, family replication, or anomaly cancellation from closure alone. Its claim is narrower and more defensible: once closure-compatible interaction architecture is fixed, matter content is no longer a free inventory but becomes a structural admissibility problem. This paper therefore serves as the bridge in the broader QDL Unified Admissibility Theory program from lawful interaction organization to lawful unified content.