Modal Triplet Theory and Loop Quantum Gravity: A Conditional Holst/Canonical Embedding

This paper determines exactly how far the present Modal Triplet Theory (MTT) gravity program reaches into loop quantum gravity (LQG). It proves an exact conditional composition theorem. If the selected four-dimensional MTT action contains a normalized Holst term and if a global three-plus-one splitting, time gauge, nondegenerate tetrad, and required boundary conditions are supplied, then the standard canonical reduction yields the real Ashtekar–Barbero pair. In the convention used here, the coefficient ratio fixes the Immirzi parameter as its negative inverse. Current MTT results conditionally recover the two-derivative Einstein/TEGR tensor structure, but do not yet emit this parity-odd coefficient; therefore they do not yet predict the Immirzi parameter. The LOST representation, geometric spectra, constraint quantization, and EPRL/FK amplitudes are separately inherited only after their standard hypotheses and constructions are added. This establishes a precise LQG interface and a finite exit contract without treating imported LQG machinery as an MTT derivation.