From 3-dimensional skein theory to functions near <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math>

Motivated by the Quantum Modularity Conjecture and its arithmetic aspects related to the Habiro ring of a number field, we define a map from the Kauffman bracket skein module of an integer homology 3-sphere to the Habiro ring, and use Witten’s conjecture (now a theorem) to show that the image is an effectively computable module of finite rank that can be used to phrase the quantum modularity conjecture.