Description (ASCII-safe, long form) This paper derives the minimal algebraic structure forced by involutive Z2 holonomy in the Finite Reversible Closure framework. Composite Parity Sector From Paper 12, a composite excitation carries a binary projective label sigma belonging to Z2. A minimal closed loop linking an involutive plaquette defect once yields a Wilson-loop value equal to -1. On the composite sector this defines a parity operator P acting as: P acting on state sigma equals (-1) to the power sigma times the same state. P squared equals 1. Finite-Step Transport Operators Let Ti(epsilon) denote coarse-grained transport by displacement epsilon in spatial direction i. Locality and finite reversible closure imply an expansion: Ti(epsilon) equals identity minus i times epsilon times Ki plus higher-order terms. Here Ki are Hermitian generators. Holonomy Constraint on Generators From Paper 12, a minimal square loop linking the defect once yields -1 holonomy. This induces the transport constraint: Ti Tj Ti inverse Tj inverse equals exponential of i times epsilon squared times kappa times P. Using standard operator expansion, this implies the commutator relation: [Ki, Kj] equals kappa times P. Thus spatial generators must fail to commute in proportion to the parity operator. Minimal Faithful Representation We determine the smallest Hilbert dimension supporting; Hermitian Ki, P with P squared equal to 1, Non-zero commutator [Ki, Kj] proportional to P. In one dimension, P would be scalar and the commutator would vanish. Therefore dimension one is impossible. In two dimensions, an explicit matrix representation exists with P diagonal and Ki off-diagonal. The commutator then reproduces the required proportionality. Therefore the minimal faithful representation is two-dimensional. Emergent Two-Component Structure The local excitation field must take the form; Psi(x) equals (psi0(x), psi1(x)). Parity acts diagonally, while transport generators mix components. Thus scalar propagation is impossible. The minimal excitation structure is two-component. Programme Flow;- Paper 12 - Operational Z2 holonomy from involutive defectPaper 13a - Non-commuting transport algebra forces two componentsPaper 13b - Continuum coarse-graining forces first-order kinetic operatorPaper 14 - Representation completion and relativistic covariance Paper 13a completes the algebraic stage. Paper 13b performs the continuum coarse-graining stage.

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.