TAET-R V3: Structural Closure of a Candidate Relational Effective Field Theory - Gauge - Redundant Common Time, Gapped Relative Clock Modes, and a Controlled GR+QFT Infrared Limit.

TAET-R V3 presents the current effective-field-theory formulation of the Relational Space-Time Architecture (TAET-R) program. This version consolidates and revises the exploratory V1 and the more cautious phenomenological V2 into a structurally closed candidate EFT, while explicitly separating derived results, phase conditions, matching assumptions, and open ultraviolet problems. The effective theory is formulated using a Palatini/Einstein-Cartan gravitational sector together with a local relational clock fiber. The common clock direction is treated as an exact gauge redundancy, while the physical temporal degrees of freedom are the \(N_c-1\) relative clock modes. In the synchronized phase, these relative modes acquire a positive mass gap, yielding a finite correlation length and allowing the low-energy theory to reduce in a controlled way to general relativity plus ordinary quantum field theory: TAET-R_eff → GR + QFT for energies \(E \ll m_{\rm gap}\). The construction is tested against degree-of-freedom counting, constraint closure, ghost and tachyon absence in the minimal branch, locality, effective causality, radiative consistency, spectral positivity, and the infrared decoupling of the relative clock sector. A reciprocal local clock-link realization naturally yields a Laplacian Hessian with one gauge zero mode and \(N_c-1\) massive relative modes when the link curvature is positive. V3 does not claim a completed fundamental or ultraviolet theory. The microscopic origin of the clock-locking sign and scale, the value of \(N_c\), the Standard Model gauge structure, fermionic statistics, the three-family structure, and the full UV completion remain open matching or fundamentalization problems. The theory is therefore presented as a candidate relational effective field theory with an open UV completion, rather than as a completed theory of fundamental physics. The paper also identifies falsifiable consequences of concrete realizations, including positivity relations for low-energy amplitudes and the unavoidable gravitational spectral contribution of propagating massive clock modes above their pair-production threshold. Version note: V3 supersedes the exploratory V1 and the phenomenological V2 as the current effective-theory formulation of TAET-R.