Integer Lattice Structure of Dimensional Quantities: The Algebraic Structure of the Quantized Dimensional Ledger

Dimensional analysis is traditionally enforced through dimensional homogeneity, requiringthat physical expressions share identical dimensions. While necessary, this condition is permissive:many constructions satisfy homogeneity while differing in their underlying dimensionalstructure. This paper develops an integer-lattice formulation of dimensional quantities in whichadmissibility is defined by subgroup closure after quotienting by declared transform equivalences.Physical quantities are represented as exponent vectors in a free abelian lattice with fivegenerators corresponding to directional length and frequency structure. A distinguished rankonesubgroup generated by the Quantized Dimensional Cell defines dimensional closure. Theresulting admissibility predicate states that a construction is QDL-admissible exactly when itsdeclared-equivalence ledger class lies in this subgroup. The lattice admits a canonical decompositioninto closure and transverse components, yielding projection operators, a closure index, andan obstruction map that measures deviation from admissibility. Two structural results follow.First, dimensional homogeneity does not imply dimensional closure except in degenerate cases.Second, the closure condition collapses to ordinary dimensional homogeneity when the quotientlattice has rank one. A worked example involving scalar effective-field-theory operators illustrateshow the closure index and obstruction classify dimensional constructions beyond ordinarydimensional analysis. The closure index and obstruction provide a compact workflow for auditingwhether homogeneous operator constructions remain structurally aligned under declaredrepresentational equivalences, thereby turning dimensional closure into an explicit diagnostictool rather than a purely formal restatement. These results establish dimensional closure as asubgroup constraint on dimensional representations and clarify conditions under which dimensionalreasoning provides additional structural constraint in theoretical physics.