Marked Heegaard Boundary Modules: Filling, Joint Completion, and Integral Lattices
A marked handlebody state must retain its filling relations and coefficient control before complementary capping or root specialization. We construct boundary modules and regulated integral lattices that keep the disk and cyclotomic residues of one finite-stage representative together. Standard handle-attachment and recoupling results supply the geometric inputs; the comparison constructs endpoint lattices, separating transport, raw and saturated quotient towers, and exact cap domains. These objects supply the companion rootwise theory, while the two completed topologies remain distinct and…
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