Loop Quantum Gravity as a Possible Shadow of Coherent Fixed-Point Geometry: An Interpretive Dictionary and Its Missing Bridges

This paper develops an interpretive comparison between loop quantum gravity (LQG) and the coherent fixed-point sector of Modal Triplet Theory (MTT). The comparison is useful, but it is not a derivation of LQG from projection. Fixed-point stability, finite survivor sectors, compact reuse channels, and noninvertible projection suggest why graph-based states and discrete labels can be effective descriptions of coherent geometry. They do not by themselves construct the Ashtekar–Barbero phase space, the holonomy–flux algebra, the Ashtekar–Lewandowski state, area and volume operators, quantum constraints, or spin-foam amplitudes. Those objects require the explicit canonical and representation data stated in the companion technical paper. In particular, the Barbero–Immirzi parameter is selected by MTT only if the same four-dimensional source that fixes the parity-even gravitational action also emits a normalized parity-odd Holst coefficient. That coefficient is currently open. The result is a disciplined shadow dictionary: it identifies a plausible overlap regime, explains what fixed-point geometry contributes, and gives concrete conditions under which the analogy would become a mathematical embedding.