Quantum Gravity: The Missing Link Between Quantum Theory and Einstein Gravity | G Model

This ten-page audit follows the structural chain four-null geometry to simple bivectors to holonomy curvature to the Einstein action to a quantum phase. It identifies a credible candidate bridge, not a completed or independently verified quantum-gravity theory.

A complete bridge must pass five gates: discrete variables must reconstruct Lorentzian geometry; discrete curvature must reproduce Einstein-Hilbert gravity; constraints must remove unphysical gauge components; a symplectic bracket and quantum amplitude must be defined; and a new observable must differ from general relativity or quantum field theory without fitting.

The native geometry uses k_a = L_c(1,n_a), k_a squared = 0, and n_a dot n_b = -1/3 for a not equal to b. The six bivectors X_ab = k_a wedge k_b form the natural antisymmetric two-form sector. The non-degenerate four-null Gram matrix reconstructs the Minkowski metric, and the signed boundary relation B_partial C = 0 is kept distinct from the unsigned endpoint-incidence operator K_0 = N transpose N - 2I_6.

The Einstein-Hilbert phase is S_EH/hbar = [c cubed/(16 pi G hbar)] integral R square-root(-g) d4x. Within the retained source-modulus and continuum-coupling branch, the MMM Theorem G_G K_M = c to the fourth/(4 pi) rewrites this as S_EH/hbar = [K_M/(4 c hbar)] integral R square-root(-g) d4x. This links the weak M-field source modulus and spacetime curvature through one stiffness scale.

CF141 provides group-valued Bianchi closure, CF142 recovers additive curvature near the identity, and CF146 contracts dual bivectors with holonomy curvature. Matching the parity-even CF147 microscopic phase to Einstein-Hilbert gravity gives zeta_eff = c cubed L_c squared/(16 pi G_G hbar) = 5 pi/48 = 1/(4 C_Sigma), reproducing CF148 without numerical fitting inside the retained branch. The corresponding one-cell phase is S_cell/hbar = (5 pi/72)L_c squared R_disc.

The local bivectors satisfy diagonal simplicity X_ab wedge X_ab = 0, cross-simplicity X_ab wedge X_ac = 0, non-degenerate four-volume det(k_0,k_1,k_2,k_3) = -16 L_c to the fourth/(3 square-root 3), and equal volume-simplicity magnitudes. This places them in a constrained gravitational sector rather than an arbitrary topological two-form sector. The beta_I term analogous to a Holst or Immirzi contribution remains open.

For the continuum graviton sector, ten symmetric metric components minus four gauge freedoms and four constraints leave two physical tensor modes. The branch-space mapping, gauge identity, and D^a E_ab[Q] = 0 conservation identity pass at the stated level. The two-mode count is a conditional continuum pass; finite-cell positive norm and ghost freedom have not yet been proved.

The candidate amplitude is Z = integral DW DX exp(i S_CF147[W,X]/hbar). A canonical theory would require a derived holonomy-flux bracket of the schematic form {X_f^IJ,W_f} proportional to J^IJ W_f, becoming [X-hat_f^IJ,W-hat_f] = i hbar J^IJ W-hat_f. CF147 does not yet provide the required discrete time split, boundary term, momentum, or Poisson bracket. Gauss/Lorentz and local simplicity are supported, while Hamiltonian evolution and inter-cell gluing remain open.

The precise missing link is a consistent neighboring-cell rule. Candidate shared-face matching is X_f^(x) = -Ad[W_xy]X_f^(y), and loop holonomy W_xy W_yz W_zx = exp[K_f] defines discrete curvature. The decisive finite-scale test requires rank(H_reduced) = 2 per wave vector with non-negative eigenvalues after gauge reduction. Extra positive modes would imply extra gravitational fields; negative modes would indicate ghosts; no local modes would leave a topological theory.

Final assessment: four-null Lorentz geometry and six bivector channels are exact; the coefficient zeta_eff = 5 pi/48 is exact within the retained bridge; the quantum cell phase is derived; local simplicity passes exactly; and the continuum Einstein sector has a structural pass. Inter-cell gluing, canonical brackets, finite-cell first-class constraint closure, ghost freedom, and a new non-GR prediction remain unresolved. The next calculation is a two-cell variation deriving shared-face matching, the boundary symplectic term, and the canonical holonomy-flux bracket.

Working-scale boundary: T_c = 2.186570237920 x 10^-43 seconds is accepted in the source report as a working value obtained using measured Newton G. The structural action-matching tests do not use numerical fitting, but the absolute scale must not be presented as an independent G Model prediction.

Authorship and verification boundary: G Model conceptual framework, research direction, interpretation, and final scientific responsibility belong to Gurbuz Dolu. AI assistance supported retrieval, symbolic and algebraic re-testing, status reconciliation, and document preparation. This internal audit is not peer review, experimental validation, or independent replication.