Complementarity-First TCG: Paired Incidence, Transport, and the Finite/Local Gravity Architecture

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 Gr(2,4). 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.