This record contains the main preprint and its accompanying Technical Supplement. The study examines whether a weighted two-shell body-centered-cubic contact frame motivated by the Quantized Dimensional Cell can generate an autonomous massless spin-2 gravitational sector. Its principal result is an exact obstruction within a precisely specified microscopic model. When all sixteen coframe components receive positive componentwise inertia and the same weighted-BCC bond operator, the velocity Hessian is positive and full rank. The Legendre map is therefore invertible, no Hessian-generated primary constraints arise, and all sixteen frame components propagate. Fourth-order contact isotropy and a shared relativistic-looking characteristic cone are consequently insufficient to produce a constrained two-degree-of-freedom massless spin-2 theory. A second construction is developed under an additional integrable-frame null-energy hypothesis that is not derived from QDL closure. This hypothesis replaces ordinary componentwise frame-gradient energy with an antisymmetrized coframe derivative or torsion. The resulting discrete torsion-loop construction has the parity-even quadratic teleparallel or new-general-relativity class as its leading continuum limit. Weighted-BCC isotropy fixes the derivative geometry but does not determine the three independent torsion-contraction coefficients. The Einstein-equivalent branch therefore additionally requires the TEGR coefficient ratio, local Lorentz redundancy, absence of algebraic metric strain, suppression of preferred-frame operators, positive tensor normalization, matter-compatible constraints, and infrared closure of the discrete constraint algebra. For the specific tensor and matter lattice operators tested, the sectors share a leading cone but have different first dispersive corrections, producing the operator-pair residual DeltaTm^ (4) = -1/8. This value is not claimed as a universal QDC constant. The Technical Supplement provides the complete contact-moment, lattice-symbol, Hamiltonian, centered-derivative, sixth-order anisotropy, tensor-variation, transverse-traceless projector, degree-of-freedom, TEGR-diagnostic, preferred-frame, time-delay, and continuum-versus-discrete constraint-algebra derivations. The record establishes a rigorous obstruction to the minimal componentwise QDC frame and a finite conditional route to an Einstein-equivalent torsion branch. It does not claim an unconditional microscopic derivation of general relativity, Newton’s constant, the exact discrete gravitational constraint algebra, quantum gravity, or the cosmological constant.
James D. Bourassa (Fri,) studied this question.