Sharp Heisenberg eigenvalue and minimal profile

For each $r=1,\dots,n$, let $S_r$ be a strip of $n$ pentagons labelled by the relator $[x,y]z^{-1}$ filling $x^n y x^{-n} y^{-1} z^{-n}$. Glue these strips to form the source funnel $A_n$, let $B_n$ be its reverse-oriented copy, and let $C_n$ be the $n\times n^2$ rectangular grid of $[x,z]$-squares described above. The resulting van Kampen diagram $\Delta_n=A_n\cup C_n\cup B_n$ has boundary word $W_n=[x^n,[x^n,y^n]]$ and area $n^3+2n^2$. Here $\widetilde\mu_1$ denotes the unweighted face-dual Dirichlet eigenvalue, and $\Lambda^\ast_{\mathcal P_{H_3}}$ denotes the minimal weighted profile for the Heisenberg presentation.

Sharp Heisenberg eigenvalue and minimal profile. The explicit family satisfies

$$\widetilde\mu_1(\Delta_n)\asymp n^{-2}$$

and the minimal weighted profile satisfies

$$\Lambda^\ast_{\mathcal P_{H_3}}(n)\asymp n^{-2}.$$

The upper bound for $\widetilde\mu_1(\Delta_n)$ follows from the rectangular corridor and a sine test function. A path-Poincaré argument is suggested for the lower bound, but the supplied text does not establish it; the sharp asymptotic and the corresponding minimal-profile claim therefore remain to be verified.