The Quantized Dimensional Ledger: A Structural Admissibility Framework for Physical Theories

The Quantized Dimensional Ledger (QDL) is proposed as a structural admissibility framework for dimensional reasoning in theoretical physics. While dimensional analysis is widely used as a consistency check, it is typically applied informally and rarely treated as an explicit methodological constraint on theory construction. This work develops a formal framework in which dimensional structure functions as a pre-empirical admissibility criterion.

In the QDL framework, physical quantities are represented as integer exponent vectors within a five-component length–frequency lattice. Admissibility is defined through a closure predicate requiring that constructed expressions lie within a rank-one subgroup generated by a distinguished dimensional monomial referred to as the Quantized Dimensional Cell (QDC). This closure condition is strictly stronger than conventional dimensional homogeneity.

The framework introduces three methodological elements:

• a structured lattice representation of dimensional quantities
• an explicit closure-based admissibility rule
• a discipline of declared equivalence transforms governing representational freedom

An important internal diagnostic is the collapse condition: if the declared equivalence relations reduce the lattice quotient to rank one, dimensional closure becomes equivalent to ordinary dimensional homogeneity and therefore ceases to provide additional constraint.

A worked example drawn from scalar effective field theory demonstrates that the closure predicate can distinguish between constructions that are dimensionally homogeneous yet differ in ledger closure status. The example is illustrative rather than prescriptive; the goal is to demonstrate non-vacuity of the admissibility condition under explicit representational rules.

The QDL framework operates at a representational layer that is orthogonal to physical symmetries and dynamical laws. It does not propose new dynamics or empirical predictions. Instead, it provides a formal method for identifying when dimensional reasoning contributes genuine structural constraint and when it collapses into methodological vacuity.

If dimensional closure proves non-vacuous across relevant model classes, QDL functions as a pre-empirical admissibility principle for theory construction. If not, the framework contains explicit defeat conditions under which it should be restricted or rejected.