Continuum Coarse-Graining and the Minimal First-Order Kinetic Operator in Finite Reversible Closure - Paper 13b

Continuum Coarse-Graining and the Minimal First-Order Kinetic Operator in Finite Reversible Closure - Paper 13b ABSTRACT Paper 13a derived a two-component composite sector and a non-commuting transport algebra induced by involutive Z2 holonomy. Paper 13b performs the controlled continuum coarse-graining step. Using only finite reversible closure, the existence of a linear infrared dispersion branch E(p) approximately equal to c times absolute value of p (established in Paper 3) and the composite parity sector derived in Paper 13a, we show that the minimal admissible long-wavelength generator must be first-order in spatial derivatives. If the leading operator were second-order, the dispersion would be quadratic near p = 0, contradicting the linear branch. Therefore first-order structure is forced by the IR behaviour. A manifestly unitary split-step construction demonstrates the result explicitly. Full representation completion and relativistic covariance in 3+1D are treated in Paper 14. 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 projective Z2 parity from composite structure.Paper 12 operationalised that parity as measurable minus-one holonomy.Paper 13a showed that involutive holonomy forces non-commuting transport generators and that the minimal faithful representation is two-dimensional. Paper 13b now addresses the next structural question;- Given a two-component composite sector and a linear infrared dispersion branch, what is the minimal admissible effective long-wavelength generator? We assume only;- Finite reversible closure and local unitary tick evolution; Existence of a linear infrared dispersion branch E(p) approximately equal to c times absolute value of p; An emergent causal cone with invariant speed c and The two-component transport algebra derived in Paper 13a. We show that the minimal Hermitian generator consistent with these inputs must be first-order in spatial derivatives. This is not an assumption of Dirac structure. It is a structural necessity following from linear IR dispersion and non-commuting transport generators.