Seetapun's theorem

We build a sequence of Mathias conditions $(\sigma_0, X_0) \geq (\sigma_1, X_1) \geq \cdots$, with $(\sigma_0, X_0) = (\varepsilon, \nat)$ and all reservoirs $Z$-computable, and let $G = \bigcup_s \sigma_s$, assuming the strings grow unboundedly. Then $G$ satisfies every condition. The steps of the construction handle, in a fixed order, three kinds of tasks.

Infinity. Given $(\sigma, X)$, let $n = \min X$, and pass to $(\tau, X \setminus \{0, \dots, n\})$, where $\tau$ extends $\sigma$ by zeros up to position $n$, and $\tau(n) = 1$. The set $G$ then contains $n$.

Cohesiveness for $R_i$. One of $X \cap R_i$ and $X \setminus R_i$ is infinite; pass to $(\sigma, Y)$ with $Y$ that set, which is $Z$-computable. Then $G \setminus \sigma \subseteq Y$, so $G \subseteq^* R_i$ or $G \subseteq^* \overline{R_i}$.

The requirement $\Phi^{Z \oplus G}_e \neq C$. Given $(\sigma, X)$, consider the following cases.

1. There are $x$ and a string $\tau$ compatible with $(\sigma, X)$ with $\Phi^{Z \oplus \tau}_e(x)\converge \neq C(x)$. Pass to $(\tau, X \setminus \{0, \dots, |\tau|\})$. Every $G$ satisfying it has $\Phi^{Z \oplus G}_e(x) \neq C(x)$, by the use principle.
2. There is $x$ such that $\Phi^{Z \oplus \tau}_e(x)$ diverges for every $\tau$ compatible with $(\sigma, X)$. Keep $(\sigma, X)$. Every $G$ satisfying it has $\Phi^{Z \oplus G}_e(x)\diverge$: a convergent computation would converge with a prefix of $G$, which is compatible.
3. Otherwise, for every $x$ some compatible $\tau$ gives a convergent computation, and all such computations give $C(x)$. Then $C \leq_T Z$: given $x$, search, with $Z$ and the $Z$-computable set $X$, for a compatible $\tau$ with $\Phi^{Z \oplus \tau}_e(x)\converge$, and output its value. This case is excluded by hypothesis.

The resulting set $G$ is infinite, cohesive, and satisfies $\Phi^{Z \oplus G}_e \neq C$ for every $e$.

Let $Z_0 = Z$, and $Z_{n+1} = Z_n \oplus P_n$ with $P_n$ PA relative to $Z_n$ and $C \not\leq_T Z_n \oplus P_n$, by the cone avoidance basis theorem (Cone avoidance basis theorem) relativized to $Z_n$. The ideal $\{ X : \exists n\ X \leq_T Z_n \}$ is a Scott ideal by Proposition 2.3 of “Weak König's lemma”, as in Theorem 2.4 of “Weak König's lemma”, and does not contain $C$.

Write $A^0 = A$ and $A^1 = \overline{A}$. Let $\mathcal I$ be a Scott ideal containing $Z$ and not $C$ (Lemma 3.3). If some infinite subset $H$ of $A^0$ or of $A^1$ belongs to $\mathcal I$, then $Z \oplus H \in \mathcal I$, so $C \not\leq_T Z \oplus H$, and we are done. Assume from now on:

(H) no infinite subset of $A^0$ or of $A^1$ belongs to $\mathcal I$.

We build two sets $G^0 \subseteq A^0$ and $G^1 \subseteq A^1$ simultaneously, with conditions $(\sigma_0, \sigma_1, X)$: $X \in \mathcal I$ is infinite with $\min X \geq |\sigma_0|, |\sigma_1|$, and the finite set $\sigma_i$ is contained in $A^i$, for $i < 2$. A condition $(\tau_0, \tau_1, Y)$ extends $(\sigma_0, \sigma_1, X)$ if each $(\tau_i, Y)$ extends $(\sigma_i, X)$ as Mathias conditions. The sets $G^i$ are the unions of the strings $\sigma_i$ along a sequence of conditions.

Infinity. Given a condition and $i < 2$, the set $X \cap A^i$ is infinite: otherwise $X$ minus finitely many elements would be an infinite subset of $A^{1-i}$ in $\mathcal I$, contradicting (H). Let $n \in X \cap A^i$, extend $\sigma_i$ by zeros and then $1$ at position $n$, and replace $X$ by $X \setminus \{0, \dots, n\}$.

The requirement $\mathcal R_{e_0, e_1}$: $\Phi^{Z \oplus G^0}_{e_0} \neq C$ or $\Phi^{Z \oplus G^1}_{e_1} \neq C$. Fix a condition $c = (\sigma_0, \sigma_1, X)$. For $x \in \nat$ and $v < 2$, say that $c$ answers $(x, v)$ if for every partition $\nat = U^0 \sqcup U^1$, there are $i < 2$ and a string $\tau$ compatible with $(\sigma_i, X \cap U^i)$ (compatibility is defined as for Mathias conditions, even if $X \cap U^i$ is finite) with $\Phi^{Z \oplus \tau}_{e_i}(x)\converge = v$.

The set of partitions, coded as $U^0$, for which this fails is a $\Pi^0_1(Z \oplus X)$ class: failure means that every $\tau$ compatible with $(\sigma_i, X \cap U^i)$, for both $i$, gives no convergent computation with value $v$, and each instance of this condition involves finitely many bits of $U^0$. By compactness, $c$ answers $(x, v)$ if and only if this class is empty, a $\Sigma^0_1(Z \oplus X)$ property of $(x, v)$. Let $W$ be the set of pairs answered by $c$; it is c.e. in $Z \oplus X$, which is a member of $\mathcal I$.

1. If $(x, 1 - C(x)) \in W$ for some $x$: apply the definition to the partition $(A^0, A^1)$. There are $i$ and $\tau$ compatible with $(\sigma_i, X \cap A^i)$ with $\Phi^{Z \oplus \tau}_{e_i}(x)\converge = 1 - C(x)$. The new elements of $\tau$ lie in $A^i$. Replacing $\sigma_i$ by $\tau$ and $X$ by $X \setminus \{0, \dots, |\tau|\}$ gives an extension forcing $\Phi^{Z \oplus G^i}_{e_i}(x) \neq C(x)$.
2. If $(x, C(x)) \notin W$ for some $x$: the $\Pi^0_1(Z \oplus X)$ class of partitions witnessing it is non-empty, and since $\mathcal I$ is a Scott ideal containing $Z \oplus X$, it has a member $U^0 \in \mathcal I$, with $U^1 = \overline{U^0}$. For some $i$, $X \cap U^i$ is infinite; it belongs to $\mathcal I$. The condition $(\sigma_0, \sigma_1, X \cap U^i)$ extends $c$, and every $G^i$ satisfying $(\sigma_i, X \cap U^i)$ has $\Phi^{Z \oplus G^i}_{e_i}(x) \neq C(x)$, divergent or different: a convergent computation with value $C(x)$ would come from a compatible string.
3. Otherwise, every $x$ has some $v$ with $(x, v) \in W$, and only $v = C(x)$ occurs. Then $C$ is computable in $Z \oplus X \in \mathcal I$, by enumerating $W$: so $C \in \mathcal I$, a contradiction.

Conclusion. Build a sequence of conditions that handles infinity for both sides and every requirement $\mathcal R_{e_0, e_1}$ in turn, and let $G^0$, $G^1$ be the limits. Both are infinite, $G^i \subseteq A^i$, and every requirement holds. If $\Phi^{Z \oplus G^0}_{e_0} = C$ for some $e_0$, then $\Phi^{Z \oplus G^1}_{e_1} \neq C$ for every $e_1$, so $C \not\leq_T Z \oplus G^1$. Otherwise $C \not\leq_T Z \oplus G^0$. Either $H = G^0$ or $H = G^1$ works.

The sequence $R_x = \{ y : f(\{ x, y \}) = 1 \}$ is $Z$-computable. By Theorem 3.2, it has a cohesive set $G$ with $C \not\leq_T Z \oplus G$. Let $p$ enumerate $G$ in increasing order, and $g(\{ i, j \}) = f(\{ p(i), p(j) \})$; it is a stable coloring (Theorem 2.4), computable in $Z \oplus G$. By Theorem 3.4 with $Z \oplus G$ in place of $Z$, applied to the limit set $A_g$, there is an infinite set $K$ contained in $A_g$ or in its complement with $C \not\leq_T Z \oplus G \oplus K$. By Proposition 2.6 (2), $g \oplus K$ computes an infinite homogeneous set $L$ for $g$, and $H = p[L]$ is homogeneous for $f$, computable in $Z \oplus G \oplus K$. So $C \not\leq_T Z \oplus H$.

Build sets $Z_0 = \emptyset \leq_T Z_1 \leq_T \cdots$ with $\emptyset' \not\leq_T Z_n$. Fix an enumeration of all pairs $(e, m)$ in which each pair appears infinitely often. At stage $n$, take the next pair $(e, m)$ with $m \leq n$. If $\Phi^{Z_m}_e$ is a coloring of $[\nat]^2$ with two colors, it is $Z_n$-computable, and Theorem 3.5 gives an infinite homogeneous set $H$ with $\emptyset' \not\leq_T Z_n \oplus H$; let $Z_{n+1} = Z_n \oplus H$. Otherwise $Z_{n+1} = Z_n$.

The class $\mathcal I = \{ X : \exists n\ X \leq_T Z_n \}$ is a Turing ideal not containing $\emptyset'$. Every coloring in $\mathcal I$ is $\Phi^{Z_m}_e$ for some $e, m$, and is handled at some stage, so it has a homogeneous set in $\mathcal I$. By absoluteness, $\mathcal I$ satisfies $\mathsf{RT}^2_2$, and it is not a jump ideal.