PDT-MC-1: Mathematical Closure Supplement

This supplement provides the mathematical foundations for the PTG/PDT programme. It formalises the closure axioms, constructs the closure-compatible metric, proves the existence and uniqueness of solutions to the differential transport system, establishes the minimal winding and loading results, and derives the return algebras of the triplet, doublet, and singlet sectors. The central result is the Return Algebra Theorem, which shows that the triplet sector has return algebra End($C^{3}$) of dimension 9, yielding the down-sector invariant $T_{3}$ = 9 used in PTG-5 and PDT-II. PDT-MC-1 provides the mathematical backbone for the physical interpretation developed in PDT-I, PDT-II, and PDT-III.