I am looking for related constructions in spectral theory / random matrix theory / dynamical systems where a triadic (three-vertex) geometric criterion acts as a sharp switch for an order–chaos transition.
The structural motif is as follows. Consider a dyad (i,j) together with a third “anchor” vertex a. Embed the three vertices into a low-dimensional coordinate (for example)
u_k = ( z_k / z_0 , cos θ_k , sin θ_k ) ∈ R^3
and define the dyadic barycentre
ū_ij = (u_i + u_j) / 2 .
The key triadic quantity is the barycentric deviation
b_{ij|a} = || u_a − ū_ij || .
A sharp transition is modelled by a tolerance threshold b_c, e.g.
S_{ij|a} = 1{ b_{ij|a} > b_c }
or, in a smoothed form,
S_{ij|a} = sigmoid( k ( b_{ij|a} − b_c ) ),
so that crossing the tolerance boundary triggers a qualitative change (interpreted as an order–chaos switch).
In my setting, this triadic switch amplifies a breakdown / diffusion mechanism (rather than pairwise stress alone), but my question is purely structural:
Are there known results or standard constructions where a triadic geometric deviation (barycentric distance, triangle area, or related invariants) acts as a threshold for instability, dephasing, or a transition between ordered and chaotic regimes?
In related literature, is it more natural to treat the threshold as context-dependent (varying with background state) rather than as a fixed global constant? If so, what frameworks capture this cleanly?
Any pointers to analogous third-vertex switching mechanisms in spectral or random-matrix contexts (beyond purely pairwise couplings) would be appreciated.
For reference, the full formulation (including notation and the motivation for triadic switching) is available as a preprint:
Zenodo: https://doi.org/10.5281/zenodo.17950100
I am not claiming any relation to the Riemann Hypothesis; this is a structural question about triadic thresholds and order–chaos switching.
