Common-Characteristic Geometry and Conditional Einstein-Limit Recovery in a Closure-Constrained QDC Contact Model: Weighted-BCC Isotropy, Densitized-Metric Reconstruction, Universal-Coupling Gates, and Weak-Field Gravity

This paper develops a six-gate route from a microscopic weighted body-centered-cubic contact model to a conditional long-wavelength Einstein limit. The model is motivated by the Quantized Dimensional Cell construction of the Quantized Dimensional Ledger, but its principal mathematical results can be evaluated independently of that wider interpretation.

The microscopic system is a two-shell weighted body-centered-cubic contact complex. Each site possesses eight body-diagonal contacts with stiffness K_X and six axial contacts with stiffness K_X/2 for each physical perturbation sector X. Direct calculation shows that the weighted second and fourth contact moments are isotropic. The leading propagation geometry is therefore isotropic, and the first dispersive correction is also isotropic.

For a slowly deformed contact background, the continuum action supplies a sector-specific contravariant tensor density f_X^{mu nu}. The paper derives the exact lattice dispersion relation and its infrared expansion,

omega^2 = c_*^2 k^2 [1 - a^2 k^2/24 + O(a^4 k^4)],

where a is the microscopic contact spacing.

The first theorem establishes the common-characteristic criterion. Two isotropic sectors X and Y share one leading propagation cone if and only if

K_X/I_X = K_Y/I_Y.

The corresponding mismatch obstruction is

Delta_XY = K_X I_Y/(K_Y I_X) - 1.

The paper does not claim that QDC closure alone forces Delta_XY = 0. It identifies a sufficient single-contact normalization mechanism in which temporal inertia and spatial stiffness carry the same positive sector factor, causing that factor to cancel from the characteristic cone.

When all sector-specific tensor densities factorize through one common tensor density g^{mu nu}, the paper reconstructs a unique full Lorentzian metric directly from the densitized principal tensor:

g^{mu nu} = gfrak^{mu nu}/sqrt[-det(gfrak)].

No independent conformal density is inserted after the characteristic cone has been identified.

The paper then separates metric recovery from gravitational recovery. It formulates six logically distinct gates: common characteristic geometry, densitized-metric reconstruction, universal matter coupling, autonomous metric dynamics, the Newtonian limit, and the controlled Einstein limit.

Universal coupling is established under explicit single-frame factorization assumptions. Einstein-Hilbert dynamics is treated conditionally: if the slow contact-frame sector develops a local, metric-only, diffeomorphism-redundant branch with positive two-derivative tensor dynamics and no unsuppressed extra modes, then the leading geometric action is Einstein-Hilbert plus a cosmological term.

On that successful branch, the paper derives the Poisson equation, an attractive inverse-square force, composition-independent test-body motion, two transverse tensor polarizations, and the low-curvature conditions under which higher-derivative, preferred-frame, and lattice corrections decouple.

The resulting Einstein-limit equation is

G_{mu nu} + Lambda_eff g_{mu nu}
= 8 pi G_eff T_{mu nu}

• O(a^2 R^2, a^2 nabla^2 R).

The paper provides a conditional recovery architecture rather than an unconditional derivation of general relativity. The autonomous geometric sector, emergence of diffeomorphism redundancy, removal of antisymmetric and nonmetric modes, positivity of the Einstein-Hilbert coefficient, microscopic value of Newton’s constant, smallness of the cosmological constant, and construction of a graviton Hilbert space remain open gates.

This record forms part of the QDL/QDC Completion Series and supplies the effective-geometry and conditional-gravity-recovery stage following the closure-constrained variational principle.