In Classical Recursion Theory Vol.I by P.Odifreddi, section V.5 on the Tree Method, the proof for the existence of hyperimmune-frees involves the construction of a series of trees. Some definitions first:

• Tree: a function from initial segments to initial segments (binary) s.t. (1) $T(\sigma)\downarrow \wedge \tau \subseteq\sigma \rightarrow T(\tau)\downarrow \wedge T(\tau) \subseteq T(\sigma)$; (2) if one of $T(\sigma * 0), T(\sigma * 1)$ is defined, both are defined and incompatible.
• A is a branch of tree T if $T(\sigma) \subseteq A$ for infinitely many $\sigma$'s;
• Q is a subtree of T ($Q \subseteq T$) if for every $\sigma \in ran(Q), \sigma \in ran(T)$

My question is concerning the following technical lemma:

$Lemma (Totality)$: Given $e$ and a recursive tree $T$, there is a recursive tree $Q \subseteq T$ s.t. one of the following holds:

• for every branch $A$ on $Q$, {$e$}$^A$ is not total;
• for every branch $A$ on $Q$, {$e$}$^A$ is total and $(\forall n)(\forall \sigma)(|\sigma|=n \rightarrow ${$e$}$^{Q(\sigma)}(n)\downarrow)$

Ideas in the proof: First step is to see whether the following condition is fulfilled, $(\exists \sigma \in T)(\exists x)(\forall \gamma \supseteq \sigma)(\gamma \in T \rightarrow ${$e$}$^{\gamma}(x)\uparrow)$. If it exists, the first condition the fulfilled since we can take the full subtree above $\gamma$ on $T$. If not, $Q(\emptyset)= least$ $\gamma\in T$ s.t. {$e$}$^\gamma \downarrow$, inductively, for $Q(\sigma)$ there is an extension of it $T(\gamma)$ on $T$ s.t. {$e$}$^{T(\gamma)}(|\sigma|)\downarrow$, let $Q(\sigma *i) = T(\gamma * i)$, for $i=0,1$.

But is the following assertion true? {$e$}$^A (n) \leq max_{|\sigma|=n}$ {$e$}$^{Q(\sigma)}(n)$, for any branch A on Q s.t. {$e$}$^A$ is total. The claim in the book is it is true since {$e$}$^A(n)$ is already defined on the $n^{th}$ level of the tree $Q$. However, is it reasonable to consider the possible e-splitting as well, namely, $\tau_1, \tau_2$ are two different strings s.t. {$e$}$^{\tau_1}\downarrow,$ {$e$}$^{\tau_2}\downarrow$ but they are not equal? The above construction only makes sure the existence of the value on the n$^{th}$ level, but will the value change later on?