Formal treatment of implicit rerouting

Establish a formal characterization of when local tensor architectures and bond-dimension constraints rule out implicit rerouting of conditional dependencies between non-adjacent variables in fully connected tensor networks.

Background

The recovery proof assumes that a local tensor cannot internally reconstruct a conditional dependence between two non-adjacent variables from its incident bonds. The paper acknowledges that saturated bonds and sufficiently expressive local tensors may nevertheless encode such dependencies without activating the corresponding direct bond.

A formal result establishing sufficient architectural or bond-dimension conditions that exclude this phenomenon would make the rerouting-cost argument and the exact moral-graph recovery theorem rigorous without relying on the stated heuristic assumption.

References

We do not formally rule out implicit rerouting.

— Tensor Network Moral Graph Recovery of Discrete Probability Distributions  (2609.09258 - Olivas et al., 8 Sep 2026) in Remark 2.4, Section 5.2; Remark 2.5, Section 6.1

A rigorous landscape analysis is left open.

— Tensor Network Moral Graph Recovery of Discrete Probability Distributions  (2609.09258 - Olivas et al., 8 Sep 2026) in Section 8, paragraph “Non-convexity”

We do note that modeling the inner distribution of allowed variables may require transformers of depth larger than the lower bounds imposed by \Cref{thm:depth}. How the depth requirements of the inner distribution interact in this case with the bounds in \Cref{thm:depth} is left for future work.

— Constraint-Aware Training  (2610.02909 - Kim, 2 Oct 2026) in Section 3, footnote following Theorem 2 (Depth required to internalize an analysis)