An equilibrium stress of a finite simple graph framework 
in 
assigns a real number 
to each graph
edge 
so that the forces balance at every graph vertex 
:
 |
(1)
|
Here 
is the neighborhood of 
. Equivalently, the stress vector
belongs to the null space of the transpose
of the rigidity matrix.
The stress matrix 
is the symmetric matrix with entries
 |
(2)
|
It annihilates the all-ones vector and each coordinate column of the placement. If 
points affinely span 
, its matrix
rank is at most 
. For a generic placement
with 
, a stress of this rank characterizes generic
global rigidity (Connelly 2005, Gortler et al. 2010).
A stress is nondegenerate in the sense of Jordán and Tanigawa (2019) when, for every graph vertex 
and every collection of 
pairwise disjoint nonempty subsets 
, 
, ..., 
of 
whose union is a proper subset, the partial force vectors
, 
, are linearly
independent. The condition is vacuous at a graph
vertex admitting no such collection. Cruickshank (2026) reports a proof that
every generic realization of every generically
globally rigid graph in 
, 
, has a nondegenerate equilibrium
stress satisfying
 |
(3)
|
This is the nondegenerate-stress conjecture of Jordán and Tanigawa (2019). Cruickshank attributes discovery of the new proofs to ChatGPT 6.1 Sol and reports checking the arguments. Independent specialist verification or formal certification of the complete proof has not been reported.
