[proofplan] We prove the contrapositive. If some formula of the countable complete theory $T$ has the order property, then the instability spectrum theorem produces non-isomorphic models of $T$ in every uncountable cardinal. Applying this in the uncountable cardinal in which $T$ is assumed categorical gives a direct contradiction. [/proofplan] [step:Assume an order-property formula exists and name the categorical cardinal] Let $\kappa$ be an uncountable cardinal such that $T$ is $\kappa$-categorical. Suppose, toward a contradiction, that some formula $\varphi(x;y)$ has the order property modulo $T$. Thus there exist a model $M \models T$ and sequences $(a_i)_{i \in \mathbb{N}}$ and $(b_j)_{j \in \mathbb{N}}$ of finite tuples from $M$, with $|a_i| = |x|$ and $|b_j| = |y|$, such that for all $i,j \in \mathbb{N}$, \begin{align*} M \models \varphi(a_i;b_j) \quad \iff \quad i < j. \end{align*} [/step] [step:Apply the instability spectrum theorem in the categorical cardinal] We use the standard instability spectrum theorem for countable complete theories: if a formula of a countable complete theory has the order property, then for every uncountable cardinal $\lambda$ there exist at least two non-isomorphic models of the theory of cardinality $\lambda$ (citing a result not yet in the wiki: Instability Spectrum Theorem for the Order Property). The hypotheses of this theorem are satisfied: $T$ is complete, its language is countable, and $\varphi(x;y)$ has the order property modulo $T$. Applying the theorem with $\lambda := \kappa$, we obtain two models $N_0 \models T$ and $N_1 \models T$ such that \begin{align*} |N_0| = |N_1| = \kappa \end{align*} and $N_0$ is not isomorphic to $N_1$. [guided] The contradiction must occur at the same cardinal where categoricity is assumed, so we apply the instability spectrum theorem with $\lambda := \kappa$. The theorem requires three inputs: the theory must be complete, the language must be countable, and at least one formula must have the order property modulo the theory. These are exactly the standing hypotheses and the contradiction assumption. Therefore the theorem gives models $N_0 \models T$ and $N_1 \models T$ of the same cardinality $\kappa$ such that \begin{align*} |N_0| = |N_1| = \kappa \end{align*} but $N_0$ and $N_1$ are not isomorphic. This is the model-theoretic content of instability here: an order-property formula codes enough linear-order behavior to build distinct models in every uncountable size. [/guided] [/step] [step:Contradict categoricity and conclude no formula has the order property] The preceding step gives two non-isomorphic models of $T$ of cardinality $\kappa$. This contradicts the assumption that $T$ is $\kappa$-categorical, since $\kappa$-categoricity says that any two models of $T$ of cardinality $\kappa$ are isomorphic. Hence the supposition that some formula of $T$ has the order property is false. Therefore no formula $\varphi(x;y)$ has the order property modulo $T$. [/step]