Every limit ordinal has cofinality omega in Gitik's model

Fix an infinite regular ground cardinal $\delta$. Use the canonical name whose value is the union of the $\delta$-sections of conditions in $G$: a pair $⟨n,\xi ⟩$ enters the named graph exactly under conditions with $p(\delta )(n)=\xi$. Directedness makes this union a one-to-one partial map. For every $n<\omega$, support extension and finitely many legal successors give a dense class of conditions whose $\delta$-section contains $n$. For every $\xi <\delta$, uniformity makes each relevant successor set unbounded, so pruning above $\xi$ and taking the next $\delta$-successor is dense. Thus the value $g_{\delta }:\omega \to \delta$ is total and cofinal, but need not be onto. Every automorphism in $H_{\{\delta \}}$ fixes the $\delta$-section and permutes the witnessing conditions only outside that coordinate. Its graph entries use only check names, so the name is hereditarily symmetric with support $\{\delta \}$. F3 therefore puts $g_{\delta }$ in $N_G$.

Let $\lambda$ be a nonzero limit ordinal. In the ground model put $\delta ={\mathrm{cf}}^M(\lambda )$ and take the strictly increasing cofinal map $h:\delta \to \lambda$ supplied by F4. Its check name is fixed by every automorphism and hereditarily symmetric, so F3 puts $h$ in $N_G$. If $\delta =\omega$, it is already the required witness. If $\delta >\omega$, then $\delta$ is an infinite regular ground cardinal, so step 1.1 gives $g_{\delta }\in N_G$ and ZF in F1 forms $c=h\circ g_{\delta }:\omega \to \lambda$. Given $\xi <\lambda$, choose $\eta <\delta$ with $\xi \le h(\eta )$ and then $n<\omega$ with $\eta \le g_{\delta }(n)$. Monotonicity of $h$ gives $\xi \le c(n)$, so $c$ is cofinal. This proves the claimed omega upper bound in both cases without any surjectivity assertion.

By F4, the cofinality of a limit ordinal is infinite. Equivalently, the empty range is not cofinal and every nonempty finite set of ordinals has a maximum still below $\lambda$, so no finite map is cofinal in $\lambda$. Step 2.1 gives a cofinal map of length $\omega$; since $\omega$ is the least infinite ordinal, the least cofinal length is exactly $\omega$. The construction and this verification take place inside the transitive ZF model $N_G$.