Emergent Two-Component Structure from Composite Parity Transport Algebra - Paper 13a

Emergent Two-Component Structure from Composite Parity Transport Algebra - Paper 13a Abstract Paper 12 established that composite excitations carrying an involutive plaquette defect Wp = -1 exhibit a measurable Z2 holonomy under closed transport. Paper 13a derives the minimal local propagation algebra compatible with that holonomy. We show that locality, finite reversible closure and involutive transport constraints force non-commuting spatial generators whose minimal faithful representation is two-dimensional. No spinors, Clifford algebras or continuum Lorentz symmetry are assumed. Thus the two-component structure of composite excitations emerges purely from lattice transport constraints and parity structure. Continuum coarse-graining and the resulting first-order kinetic operator are derived in Paper 13b. 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 charge-flux composite. Paper 11 showed that composite structure carries a projective Z2 parity. Paper 12 made that parity operational by deriving a measurable minus-one holonomy from involutive plaquette defects. Paper 13a addresses the next structural question;- Given operational Z2 holonomy, what is the minimal local transport algebra consistent with that holonomy? We derive;- A finite-step transport operator in each spatial direction A parity operator P satisfying P squared equals 1 and A commutator relation between spatial generators of the form Ki, Kj proportional to P. We then determine the minimal Hilbert-space dimension supporting;- Hermitian transport generators; A non-scalar parity operator and Non-commuting spatial generators. The result is that the minimal faithful representation is two-dimensional. This shows that composite excitations must carry two local components. Scalar propagation is incompatible with involutive holonomy. Continuum linearisation and the derivation of the minimal first-order kinetic operator are treated in Paper 13b.