3+1D Representation Completion from the Second Invariant and Lattice Admissibility in Finite Reversible Closure - Paper 14

3+1D Representation Completion from the Second Invariant and Lattice Admissibility in Finite Reversible Closure - Paper 14 ABSTRACT Paper 13b derived a first-order kinetic operator in 2+1D from discrete transport algebra and a linear infrared dispersion branch. Paper 14 completes the construction in 3+1 dimensions by incorporating the Second Invariant; the minimal orientation-bearing reversible update per primitive tick. We show that isotropic scalar closure in three spatial dimensions requires three mutually anticommuting spatial generators and an additional independent mass operator. No such algebra exists in two complex dimensions. Therefore representation enlargement is structurally required. Under locality and translational invariance, lattice admissibility enforces sector doubling. The minimal faithful 3+1D representation is four-dimensional. This enlargement emerges from algebraic closure constraints rather than from postulated continuum spinors. INTRODUCTION The Finite Reversible Closure (FRC) programme develops a strictly local, finite-dimensional substrate in which physical structure emerges from admissible reversible update. Paper 9 established U(1) recurrence universality.Paper 10 constructed a gauge-invariant composite excitation.Paper 11 derived Z2 parity from composite structure.Paper 12 operationalised involutive holonomy.Paper 13a showed that non-commuting transport algebra forces a two-component composite sector.Paper 13b demonstrated that a linear infrared dispersion branch forces a first-order effective generator. Paper 14 addresses the next structural requirement;- Given a two-component transport sector and a first-order kinetic operator, what is the minimal representation compatible with isotropic scalar closure in three spatial dimensions? We incorporate explicitly the Second Invariant; There exists a minimal non-trivial reversible orientation-bearing update per primitive tick that cannot be gauged away because it is relational. At the composite level this invariant appears as an involutive operator P satisfying P squared equals identity. We show that;- Isotropic scalar closure in three spatial dimensions forces anticommutation relations among spatial generators; Mass completion requires an additional operator anticommutes with all spatial generators and No two-dimensional complex Hermitian representation satisfies these relations. Therefore the minimal faithful representation in 3+1D is four-dimensional.