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Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing

Primarily AI-generated textHuman understanding: some partsmath.GT — Geometric Topologymath.QA — Quantum Algebra

Contributed by Hao-Yu Sun ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

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.

Provenance statement

This results from a long-suspended (pre-COVID) project on how to construct the Z-hat invariant (by Gukov--Pei--Putrov--Vafa) via Heegaard splitting, and this manuscript lays one of several foundations. GPT-5.6, GPT-6, and GPT-6.1 were used. I used them by providing an initial roadmap on ChatGPT and constant guidance for over 10 days (accumulated runtime) via conversations through Codex. Wolfram MCP was used for symbolic computations.

Tools used

OpenAI
ChatGPT
OpenAI
Codex
Wolfram
Mathematica

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Version history

  1. v1Submitted by Hao-Yu SunInitial depositCurrentOct 18, 2026