Parity of a 1d Ising model, and with higher order terms

Question

I don't know if this should be asked here or in a math stack exchange, but I'll try here first.

Consider the classical 1d Ising model with periodic boundary condition: \begin{equation} H_2 (\vec{\sigma}) = \sum_{i=1}^n \sigma_i \sigma_{i+1} \end{equation} where $\sigma_i=\pm1$, and we refer to the collective $(\sigma_1,\sigma_2,\ldots,\sigma_n)$ as $\vec{\sigma}$.

Now, if there exists a configuration $\vec\sigma$ such that $H_2(\vec\sigma)=0$, that actually means $n$ is divisible by $4$. Mathematically, \begin{equation} \exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_2(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 4. \end{equation} This is because all two-body terms multiplies to $1$, i.e. $G:=\prod(\sigma_i\sigma_{i+1})=1$, and therefore there are even number (say, $2k$) of terms that are $\sigma_i\sigma_{i+1}=-1$. The LHS condition says there should be the same amount of -1 terms as the +1 terms. This means the total number of terms $n$ must be double of the number of -1 terms, i.e. $n=4k$.

This was an innocent exercise that resulted from the PBC, but maybe we can also view it as an "extension" from the following more trivial fact. Consider the 1-body version of the Hamiltonian: \begin{equation} H_1 (\vec{\sigma}) = \sum_{i=1}^n \sigma_i . \end{equation} Now, we have a more trivial statement that looks similar to the previous one. \begin{equation} \exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_1(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 2. \end{equation}

If we observe the trend here, it is tempting to guess that the following is true. Define \begin{equation} H_4 (\vec{\sigma}) = \sum_{i=1}^n \sigma_i\sigma_{i+1}\sigma_{i+2}\sigma_{i+3} . \end{equation} Then, the following holds? \begin{equation} \exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_4(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 8. \end{equation} However, this turns out to be false! The correct statement actually is \begin{equation} \exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_4(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 4 ~~~ \wedge ~~~ n\neq 4 , \end{equation} which seems rather random.

All of this seems to be potentially connected to deeper ideas like gauge symmetry etc. because for example, the first mod $4$ result heavily relies on the PBC, which is about the topology of the system. Also, in papers that talk about connections between stat-mech models and gauge redundancies, Kramers-Wannier dualities, the $G=1$ seems to be crucial (e.g. in https://arxiv.org/pdf/2310.16032.pdf appendix B-I they discuss that).

My question is this:

1. Do people talk about Hamiltonians like $H_4$? It looks like the 1d version of the $\mathbb{Z}_2$ lattice gauge theory to me. But is that even a thing? Is there a "better" way to generalize $H_1$ and $H_2$ so that the result is more consistent?
2. Perhaps in the math field this is a thing with a name like boolean analysis. Does anyone know any reference for those things?

Thanks!