\documentclass[11pt,a4paper,onecolumn]{amsart}

\usepackage[margin=1.02in,includeheadfoot]{geometry}
\input{preamble}
\usepackage{xr-hyper}
% Read the existing auxiliary dependency, but link to the sibling delivered PDF.
\externaldocument[P4A-][nocite]{../../build/P4A/main}[P4A.pdf]

\hypersetup{
  pdftitle={Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing},
  pdfauthor={Haoyu Sun},
  pdfsubject={Integral root fibers, factorial Rees covariance, caps, and closed comparisons},
  pdfkeywords={root of unity, Hopf pairing, Rees completion, Habiro cap, Morita sewing}
}

\title[Integral root fibers and closed factorizations]{Integral root fibers and closed factorizations:
Pairing maps, Rees covariance, caps, and sewing}
\author{Hao-Yu Sun}
\address{Weinberg Institute, The University of Texas at Austin, 2515 Speedway, C1600, Austin, Texas 78712, USA}
\address{SpaceXAI, USA}
\email{hkdavidsun@utexas.edu}
\date{5 September 2026}
\input{ams-metadata}

\begin{document}
\begin{abstract}
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.
\end{abstract}

\maketitle

\tableofcontents

\input{frontdoor}

\input{sections/02_state_separation_and_product_chambers}
\input{sections/03_finite_defect_and_typed_targets}
\input{sections/04_smith_newton_and_saturation}
\input{sections/05_factorial_rees_and_joint_bridge}
\input{sections/06_root_rees_theta_closure}
\input{sections/07_hopf_riesz_caps_and_single_s}
\input{sections/10_lmo_taylor_operational_boundary}
\input{public_closed_inputs}

\input{closing}

\input{references}

\end{document}