lin-0223

18.6 Restricted repairs and integrability

Let \(\mathcal A\subset TM\) be the edits actually exposed by the architecture and set

\[ \mathcal D_{\mathrm{SID}} = \mathcal A|_Z\cap TZ. \]

Theorem 18.6 Admissible integrability

If \(\mathcal D_{\mathrm{SID}}\) has constant rank and is closed under Lie brackets, then each compatible family lies on a maximal local integral manifold whose tangent directions are precisely the available gluing-preserving repairs.

Proof ▼

This is the Frobenius theorem applied to the restricted repair distribution.

Bracket closure is informative only for a restricted repair language. If \(\mathcal A=TM\), then \(\mathcal D_{\mathrm{SID}}=TZ\) and closure is automatic. The theorem is local integrability, not convergence and not a universal argument for bracket regularization.