It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some conditions, meaning the moduli space $Z$ of all such lifts is *connected*. This is mentioned in Theorem 1.1 of Lurie's "A Survey of Elliptic Cohomology" [Surv], for example.

In Remark 7.0.2 of Lurie's "Elliptic Cohomology II: Orientations" ([ECII]), Lurie says "...beware, however, that $Z$ is not *contractible*". In other words, $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is not the unique lift up to contractible choice (the gold standard of uniqueness in homotopy theory).

(Side note: in [ECII] and [Surv], Lurie is talking about the moduli stack of smooth elliptic curves, but the uniqueness up to homotopy of a derived stack $\overline{\mathsf{M}}_\mathrm{ell}^\mathrm{or}$ lifting the compactification of the moduli stack of smooth elliptic curves is also stated in the literature; for example, in Theorem 1.2 of Goerss' "Topological Modular Forms [after Hopkins, Miller, and Lurie]". I am interested in the compactified situation mostly, but both are related.)

Although I do not hope that $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ does possess this much stronger form of uniqueness, I would like to understand the reason for this failure:

**Why is the moduli space $Z$ not contractible?** and **Does a similar statement apply in the compactified case?**

To be a little more precise, let $\mathcal{O}^\mathrm{top}$ be the Goerss--Hopkins--Miller--Lurie sheaf of $\mathbf{E}_\infty$-rings on the small affine site of the moduli stack of elliptic curves $\mathsf{M}_\mathrm{ell}$. Denote this site by $\mathcal{U}$. The moduli space $Z$ can then be defined as the (homotopy) fibre product $$Z=\mathrm{Fun}(\mathcal{U}^{op}, \mathrm{CAlg})\underset{\mathrm{Fun}(\mathcal{U}^{op}, \mathrm{CAlg}(\mathrm{hSp}))}{\times}\{\mathrm{h}\mathcal{O}^\mathrm{top}\},$$ where $\mathrm{CAlg}$ is the $\infty$-category of $\mathbf{E}_\infty$-rings, and $\mathrm{CAlg(hSp)}$ is the 1-category of commutative monoid objects in the stable homotopy category. The presheaf $\mathrm{h}\mathcal{O}^\mathrm{top}$ can be defined using the Landweber exact functor theorem (at least on elliptic curves whose formal group admits a coordinate), and hence $Z$ can be seen as the moduli space of presheaves of $\mathbf{E}_\infty$-rings recognising the classical Landweber exact elliptic cohomology theories.

To prove uniqueness up to homotopy, I am aware one should use some arithmetic and chromatic fracture squares to break down the problem into rational, $p$-complete, $K(1)$-local, and $K(2)$-local parts. The $K(2)$-local part of $\mathcal{O}^\mathrm{top}$ is unique up to contractible choice by the Goerss--Hopkins--Miller theorems surrounding Lubin--Tate spectra (see Chapter 5 of [ECII] for a reference which you might already have open). The $K(1)$-local part also seems to be unique up to contratible choice, as all of the groups occuring in the Goerss--Hopkins obstruction theory vanish (this is discussed at length in Behrens' "The construction of $tmf$" chapter in the "TMF book" by Douglas et al). Similarly, the rational case also has vanishing obstruction groups; see *ibid*.

Edit: As pointed out by Tyler below, these claims about the function of Goerss--Hopkins obstruction theory above are wrong!

I'm then lead to believe that is something *interesting* (being a pseudonym for "I don't know what's") going on in the chromatic/arithmetic fracture squares gluing all this stuff together. **Are their calculable obstructions/invariants to see this? Or otherwise known examples that contradict the contractibility of $Z$?**

Any thoughts or suggestions are appreciated!