Zeckendorf-k-bonacci Tensor Representations and Infinite Dimensional Hilbert Spaces

We establish a rigorous mathematical foundation for Zeckendorf-k-bonacci tensor (ZkT) representations and their embeddings in infinite-dimensional Hilbert spaces. By extending classical Zeckendorf decompositions to k-bonacci sequences with tensor structure, we construct a new class of mathematical objects that serve as the foundation for computational ontology frameworks. Key findings include: (1) ZkT tensors form well-defined mathematical structures with k× dimensions under binary, column-complementary, and no-k constraints; (2) these tensors naturally embed in non-separable Hilbert spaces for all k2, with separability only for finite k=1 systems; (3) the spectral properties of associated evolution operators exhibit kbonacci growth rates with characteristic roots approaching 2 as k→; (4) ZkT structures provide complete orthogonal bases for infinite-dimensional quantum systems; (5) information-theoretic entropy rates scale as log2(rk) where rk is the k-bonacci characteristic root. We prove convergence theorems for ZkT series expansions and establish the mathematical completeness of this framework for representing complex recursive systems. These results provide the essential mathematical infrastructure for observer-based computational theories and quantum tensor applications.