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(I am not announcing new results or asking to verify a proof or a preprint. This post is not a a request to check the work for correctness or an announcement of results. I hope this question meets all of MathOverflow's research standards, if not, please...

In 3D bond percolation, the Sak criterion gives a threshold $s = 2 - \eta_\mathrm{SR}$ for the long-range/short-range crossover. Is there a known closed-form relationship between $s_*$, $d$, and $\eta_\mathrm{SR}$ that holds across dimensions, or is...

Let $D$ a finite semigroup, $card(D)$ odd and for all $x \in D$, $o(x)$ even.
Is it true $D$ is monogenic ?

1. Setup
Let $k$ be a field (assume $k = \mathbb{Q}$ for concreteness) and let $R = k[x_0, x_1, \ldots, x_9]$ be the polynomial ring in 10 variables, $\mathbb{Z}_{\ge 0}$-graded by $\deg(x_i) = 1$.
Let $\mathrm{CL} : \{0, 1, \ldots, 9\}^2 \to \{0, 1, \ldots, 9\}$ be the symmetric function defined by the following $10 \times 10$ table (the values $0, \ldots, 9$ label abstract operators; the arithmetic substrate is incidental to the question):...