Cyclotensor Calculus I: Phase-Harmonic Duality, Lifted Cohomology, and Holonomy on Tensor Networks

We introduce a conservative mathematical framework for tensor networks whose index spaces, contractions, and global sectors carry explicit cyclic structure. The basic cyclic leg is the Hilbert space ℋ_cyc = L²(S¹), equivalently ℓ²(ℤ) under Fourier transformation. A cyclotensor is an oriented tensor over such legs; phase-space contraction is Haar integration, while harmonic-space contraction is ordinary summation over integer modes. On the finite-support core, we prove their equivalence, associativity, a finite-support trace identity, and the equivalence between U(1)-equivariance and an integer selection rule on harmonic labels. We then enrich a tensor-network graph by homotopy classes of phase paths on edges. Their integer lift data are defined up to coboundary and therefore determine classes in H¹(G; ℤ); a spanning-tree normal form gives one integer per fundamental cycle. A discrete U(1) connection is added separately, yielding group-valued holonomy. We prove a decomposition that keeps integer winding and connection holonomy distinct, and we show that both compose naturally under cycle addition. Amplitude-phase states form a cone over S¹; the phase fibre collapses at zero amplitude. Consequently, winding is invariant under continuous nonzero deformations and can change only when continuity, boundary compatibility, or nonvanishing fails. Finally, we define static cyclotensor networks, cyclic tensor processes as loops in regular tensor-state spaces, and the corresponding graph and state-space holonomies. The construction is a mathematical module of Cyclomathematics. Quantum registers, physical phase slips, dissipative stabilization, and FDS interpretation are deferred to Applied Cyclomathematics