A Closure-Constrained Variational Principle for Typed Dimensional Field Theories: Class-II Operator Selection, Hyperbolicity, and the Minimal Quantized Dimensional Cell Continuum Action

This paper formulates a local variational framework in which a typed dimensional ledger acts as an additional operator-selection layer on top of ordinary effective-field-theory requirements. Its principal application is the Quantized Dimensional Cell, with primitive dimensional signature L3F2 and F = T^-1. The typed ledger is represented by the integer lattice Z5, with three permutation-equivalent length slots and two permutation-equivalent frequency slots. Under additivity, permutation symmetry, primitiveness, and neutrality on the primitive QDC vector (1,1,1;1,1), the unique closure functional, up to sign, is C_QDC = 2S_L - 3S_F. When two compact recurrence phases are introduced, this functional selects the primitive Fourier lock vector (2,-3). The minimal continuum field manifold is R_{\geq 0} × S2 × T2, consisting of a nonnegative closure-amplitude field, an S2 recurrence-orientation field, and two compact recurrence phases. Locality, reality, covariance, compact periodicity, bounded static energy, derivative truncation, vacuum existence, linear hyperbolicity, microscopic matchability, and explicit closure grading define the admitted action class. The resulting normal-form action contains amplitude, orientation, and compact-phase kinetic terms; bounded radial and orientation potentials; the primitive phase-lock interaction 1 - cos(2 theta_L - 3 theta_F); and a quartic recurrence-curvature term of Faddeev-Skyrme type. A concrete Class-II operator comparison shows that ordinary effective-field-theory requirements admit both the primitive 2:3 QDC lock and a conventionally admissible 1:1 comparator, whereas the QDC Fourier module Z(2,-3) admits the former and excludes the latter from the primitive QDC action. The two theories consequently possess different residual currents, vacuum windings, locked-mode masses, wall widths, and wall tensions. For the QDC lock, the phase vacuum is a connected circle with primitive internal winding (3,2). One exact residual global phase current survives the lock, while the orthogonal phase combination becomes massive. For amplitude exponent s = 2, the complete amplitude-phase equations admit an exact embedded sine-Gordon wall with finite one-dimensional energy and finite planar tension. The paper also derives coefficient conditions for positive quadratic energy and vacuum hyperbolicity, establishes the relevant Derrick virial balance, and gives a microscopic weighted body-centered-cubic matching framework for continuum stiffnesses, inertias, propagation speeds, and the ultraviolet scale. The paper does not claim a unique fundamental action, a derived Standard Model gauge group, a stable trefoil soliton, general relativity, or a completed quantum theory. It supplies a finite, calculable, falsifiable closure-constrained action class whose adequacy can be tested through nonlinear principal-symbol analysis, renormalization stability, microscopic coefficient extraction, numerical soliton studies, and comparison with conventionally admissible competitor theories. This record forms part of the QDL/QDC Completion Series and addresses the action-principle and continuum-dynamics gate identified by the QDC Completion Theorem.