The Quantized Dimensional Ledger: A Lattice Structure for Dimensional Closure in Physical Theories

This paper introduces the Quantized Dimensional Ledger (QDL), a lattice-based framework for representing the dimensional structure of physical quantities and defining a structural admissibility condition stronger than conventional dimensional homogeneity. Dimensional quantities are represented as exponent vectors in a five-dimensional integer lattice generated by three geometric scaling directions and two frequency-like dynamical generators. Within this lattice a distinguished primitive vector corresponding to the dimensional structure of physical action defines the Quantized Dimensional Cell (QDC). The QDC generates a closure subgroup that provides the basis for a new structural condition called dimensional closure. Dimensional constructions can therefore be analyzed using lattice methods. Every dimensional vector admits a canonical decomposition into a closure-aligned component and a transverse obstruction component. This decomposition defines a closure index measuring the number of dimensional cells contained in a construction and an obstruction vector characterizing deviation from closure alignment. The lattice formulation yields several structural consequences. Dimensional closure is invariant under unimodular representation transformations that preserve the closure generator. Dimensional constructions can be classified by obstruction type. In effective field theory the closure generator induces a modular grading of operator monomials, leading to closure sectors whose conservation follows from the dimensional neutrality of interaction vertices. In metrology the ledger representation provides a dimensional ontology for physical constants and natural dimensional cells for measurement units. The paper provides a rigorous formulation of dimensional closure within an integer lattice framework and shows that dimensional admissibility can be defined as a structural property of dimensional quantities rather than merely a post-hoc consistency condition.