Abstract
A relation-first program does not obtain spacetime, Lorentzian reality, transport, or gravity merely by naming complementary roles. This paper asks what finite and conditional gravitational structure follows after one chooses a rank-two paired-chiral carrier and keeps every additional selector explicit. The construction begins with a four-complex-dimensional relation module and the big cell of . On a Hermitian real slice, line incidence is equivalent to a determinant null condition, its polarized determinant supplies a Lorentzian conformal metric, and faithful exchange of the chiral factors acts as spatial parity once orientation and time-orientation data are selected. Frame variation alone cannot produce a coframe; an affine vector sector is representation-theoretically necessary. Local paired transports then separate curvature holonomy from coframe nonclosure, while complete bivector simplicity, rank-four incidence-star gluing, and holonomy recurrence give a conditional route to common-coframe data. Under locality, polynomiality, curvature-linearity, nondegeneracy, direct simplicity, faithful complement parity, and scalar matter couplings, the propagating first-order direction reduces to Palatini rather than a constant Holst deformation. A principal-log finite functional reproduces Regge hinge terms only on an admitted local logarithm branch, so lifted angles, Magnus/Baker–Campbell–Hausdorff path data, and an action groupoid are required to retain winding information. Exact rooted simplicial and periodic calculations then exhibit a finite Palatini–Regge architecture, a six-dimensional physical metric quotient with two null polarizations on the lattice characteristic cone, and an independent-connection Schur response equal to the Regge Hessian on every frozen Bloch fiber. Weak-curvature and finite response studies support matrix-valued two-plane transport but do not select a universal local helix phase or a universal amplitude law. The result is a parent-side structural architecture, not completed quantum gravity, arbitrary-mesh invariance, continuum convergence, scale selection, or empirical validation.
The bridge paper
TCG-F1 is where Complementarity-First meets Twistor Configuration Geometry. It belongs to Release I by membership and provenance, and it is the parent architecture for both discrete-gravity descendants — CUD-G1 and CUD-G2.
Its most recognizable endpoint: on one frozen periodic lattice, the physical metric quotient has dimension six away from the characteristic cone; on the nonzero cone the relevant rank falls by two, leaving exactly two physical null polarizations — the polarization count of classical gravitational radiation. In the same finite construction, eliminating the independent connection produces a response agreeing coefficient by coefficient with the corresponding Regge Hessian on every frozen momentum-space fiber.
These are not quantum gravitons, an arbitrary-mesh theorem, a continuum limit, or a derivation of nonlinear Einstein dynamics. They are a demonstration that a carefully selected relational-incidence architecture can reach gravitational structure with an unexpectedly familiar signature.
Download paper (Zenodo) — 21 pages. CC-BY-4.0.