Homotopy-Theoretic Structures of Integrability, Thermal Time, and Coverings via Extension Classes

We introduce the Univalent Manifold Infinity Network (UMIN), a homotopy-theoretic framework for studying integrability and thermal time in open quantum systems. The theory rests on a single central assumption: the existence of a TremblingCore, a minimal non-trivial object with non-trivial self-extension Ext¹ (*, *) ≠ 0. We establish three main results: (1) Yang-Baxter-type structures arise as higher coherence of extension composition in a weak braided monoidal ∞-categorical setting; (2) KMS-type thermal time arises from Sasaki adjunction defects; (3) A Covering Classification Theorem (Theorem 5. 1) with complete five-step proof, lifting the classical π₁ (A) -set classification to Ext¹-data rigidified by a normalization principle. Core constructions are mechanically verified in Cubical Agda (--safe --cubical --guardedness) with no postulates, with the exception of the clutching higher inductive type in Step 1 of Theorem 5. 1 (in progress). GitHub: https: //github. com/Psypher33/UMIN