math.GT — Geometric Topology
Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing
We construct rootwise and formal Rees boundary data from the marked modules and lattices of the companion boundary-module construction, keeping their distinct coefficient and completion domains. Using a Newton interpolation basis, we determine the Smith invariants and the integral image of the map induced by the Hopf pairing. Factorial/Rees covariance and compatible finite-stage representatives produce a strict rootwise/Rees module with its theta endpoint. At an admissible odd prime, no continuous homomorphism extends algebraic root specialization to the entire completed Rees source; further obstructions exclude unrestricted Newton padding and a state-natural lift of a contracted scalar identity. Established perturbative, Habiro, and stated-skein/Morita constructions give closed comparisons after the declared cap or sewing map, with their boundary state objects kept separate.