The candidate complex \(Y\) is exactly the closed Vietoris-Rips complex \(\operatorname{VR}(T_{7,7};2)\) in the notation of Adams, Adetowubo, Barriga-Acosta, Feng, and Sterling. Their metric on \(T_{n,n}\) is the sum of the two circular coordinate distances, and a finite set is a simplex when its diameter is at most the scale. This agrees with the stated Lee metric and clique convention.

Theorem 5.8 of the paper proves \[ \operatorname{VR}(T_{n,n};k)\simeq T^2 \qquad\text{for }k\geq2\text{ and }n>3k. \] Here \(k=2\) and \(n=7>6=3k\). Therefore \[ \boxed{Y\simeq T^2}. \]

The proof classifies the facets in the small-scale range and realizes their complex as the nerve of a good cover of \(\mathbb R^2/(7\mathbb Z)^2\) by projected \(l^1\)-balls. Every nonempty finite intersection in the cover is contractible, so the nerve theorem gives the homotopy equivalence. A simplicial collapse is unnecessary because the published nerve equivalence settles the stated question.

Exact clique enumeration gives chain-group ranks \[ (\operatorname{rank}C_0,\ldots,\operatorname{rank}C_4) =(49,294,490,294,49). \] The boundary maps \(\partial_1,\ldots,\partial_4\) have ranks \[ (48,244,245,49). \] It follows that the Betti numbers are \[ (1,2,1,0,0). \]

The Smith normal form of each boundary matrix has only \(1\) among its nonzero invariant factors, with multiplicities \(48,244,245,49\), respectively. Thus every boundary image is primitive in its target chain group. Each homology group is a subgroup of a free quotient \(C_i/\operatorname{im}\partial_{i+1}\), so it is torsion-free. Hence \[ H_i(Y;\mathbb Z)\cong \begin{cases} \mathbb Z,&i=0,2,\\ \mathbb Z^2,&i=1,\\ 0,&i\geq3. \end{cases} \] This agrees with the published homotopy equivalence.

A focused search for Lee-metric, torus-grid, graph-power, and Vietoris-Rips formulations located a direct source. Adams, Adetowubo, Barriga-Acosta, Feng, and Sterling define \(T_{n,n}\) on \(\{0,\ldots,n-1\}^2\) with the same circular \(l^1\) metric. Their closed Rips complex contains each finite set of diameter at most \(k\), which is the clique complex in the candidate statement.

Their Theorem 5.8 covers every \(k\geq2\) and \(n>3k\), including \((n,k)=(7,2)\). The abstract states the equivalent small-scale range \(n\geq7\) and \(2\leq k\leq(n-1)/3\). The article appeared in Mediterranean Journal of Mathematics in 2025. John Sterling's 2025 Auburn thesis gives an extended exposition of the same project, with the corresponding result as Theorem 4.2 on pages 26 and 27.

The candidate's open-status claim is therefore outdated. This parameter pair has a published affirmative classification.

The program labels the 49 vertices by pairs in \(\{0,\ldots,6\}^2\). It creates an edge exactly when the sum of circular coordinate distances is at most two, then extends cliques in increasing vertex order until none remain. The resulting face vector is \[ (f_0,f_1,f_2,f_3,f_4)=(49,294,490,294,49). \] There are 49 maximal tetrahedra and 49 maximal four-simplices.

For each oriented simplex, the program builds the integral boundary by alternating deletion of one vertex. SymPy computes exact Smith normal forms over \(\mathbb Z\). The boundary shapes are \[ 49\mathbin{\times}294,\quad294\mathbin{\times}490,\quad490\mathbin{\times}294,\quad294\mathbin{\times}49, \] and their ranks are \(48,244,245,49\). Every nonzero Smith entry equals one. The report therefore certifies the Betti numbers and absence of integral torsion stated in the homology claim.