lin-0069

4.8 Interactions and brackets

Suppose a realized model generates admissible vector fields \(X\) and \(Y\). Their Lie bracket \([X,Y]\) measures the antisymmetric discrepancy between the two local orders of flow. For a declared distribution \(\mathcal A\subseteq TM\), a nonzero class of \([X,Y]\) in \(TM/\mathcal A\) can indicate failure of that distribution to close. This is central to infinitesimal causality, BRIDGE/SKFM, ALLORA, and LASKO.

Yet a tangent category does not choose one universal interaction statistic. A connection \(\nabla \), when declared, also produces

\[ A_\nabla (X,Y) = \frac12(\nabla _XY-\nabla _YX), \qquad S_\nabla (X,Y) = \frac12(\nabla _XY+\nabla _YX). \]

For a torsion-free connection, \(2A_\nabla (X,Y)=[X,Y]\); the symmetric term depends on the connection and can carry different information.

Definition 4.7 Interaction signature

An interaction signature \(\Omega _D\) is a declared family of operations on the admissible tangent distribution of \(D\). It may include brackets, connection-dependent derivatives, curvature, canonical-flip comparisons, or higher jets.

Design principle

Choose an interaction signature because the application gives it meaning. Antisymmetry, curvature, and symmetric acceleration are different probes; none is a universal replacement for the tangent factorization obstruction.