Pre-Monoidal Categories with Controlled Coherence Defects

This preprint introduces the block-cyclic category, a concrete and fully computable pre-monoidal category in which the associator carries a localized coherence defect determined by a single integer-valued function \delta : \mathbb{N} \to \mathbb{Z}.

Objects are finite cyclic sets, morphisms are rotations, and the tensor product is defined by block concatenation. Naturality and the triangle axioms uniquely determine the form of the associator, which acts nontrivially only on the rightmost tensor block.

We derive a necessary arithmetic condition under which the pentagon identity can hold and analyze explicit choices of the defect function, including the canonical non-additive deformation \delta(c)=c+1. This yields a minimal and transparent model of controlled pentagon failure, interpretable as a localized categorical curvature. The construction further induces a \mathrm{U}(1)-valued 3-cochain, connecting the categorical coherence obstruction to geometric notions of holonomy in abelian gauge theory.

This preprint omits a technical appendix containing some explicit endomorphism maps in the block-cyclic category.

The full appendix is included in the version submitted for peer review to the Journal of Mathematical Physics, and some of the explicit endomorphism maps are available upon request.