Integrability of triangle-based non-bond kernel relations

Prove a general integrability theorem for the triangle-based non-bond kernel relations of weighted graph-state tensor networks, determining whether these first-order null directions extend to finite constant-ray curves.

Background

The paper distinguishes infinitesimal non-bond kernel relations from finite state-preserving transformations. Relations whose centers are pairwise nonadjacent integrate through explicit multiplicative constructions, while adjacent-center relations can encounter second-order obstructions.

Numerical continuation suggests integrability for tested triangle-based motifs, but the paper does not establish this in general. A theorem would clarify which triangle-based first-order redundancies belong to the tangent cone of the physical fiber and how they interact with adjacent-center relations.

References

Triangle-based relations have adjacent centers, where the exponential construction does not apply; for these, Gauss--Newton continuation integrates every tested direction beyond second order: on the open triangle with one external leg per vertex (the star-lattice block), the open kagome bowtie, the open diamond with three external legs per vertex, and an open strip of three triangles, a Gauss--Newton continuation finds, for random non-bond kernel directions $\delta$ and displacements $t\le0.2$, corrections $x(t)$ orthogonal to the complete kernel and to the ray with $|(I-\psi\psi)\Psi(A_0+t\delta+x)|/|\Psi|\le10{-13}$ and $|x|\approx(0.04$--$0.1)\,t2$; since $J_A$ is injective on that complement, such a curve has velocity exactly $\delta$. On the two edge-adjacent squares the same continuation leaves a residual of order $t2$ for the parallel pair and for generic directions, and none for the orthogonal pair, in agreement with the exact certificates (\nolinkurl{experiments/nongauge_integrability.py}). A proof of integrability for the triangle-based motifs, and a characterization of the tangent cone where independent and adjacent centers mix, are natural next steps.

— Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States  (2609.30066 - Cheng, 24 Sep 2026) in Appendix, Section SM-sec:obstructionchecks, paragraph following the second-order obstruction certificate