Symmetric Bases and ${\mathrm{\ell }}^p$ Embeddings in Musielak--Orlicz Sequence Spaces under ${\delta }_2$

We study isomorphic embeddings into Musielak-Orlicz sequence spaces ${\mathrm{\ell }}_{\mathrm{\Phi }}$, where the normalized generating sequence $\mathrm{\Phi }=\{{\phi }_n\}$ satisfies the ${\delta }_2$ condition. Using admissible normalized blocks, we construct an associated limiting family ${\mathcal{C}}_{\mathrm{\Phi },\tau }$ of Orlicz functions. If a Banach space $X$ with a symmetric basis embeds into ${\mathrm{\ell }}_{\mathrm{\Phi }}$, then there is a function $\psi \in {\mathcal{C}}_{\mathrm{\Phi },\tau }$ such that the basis of $X$ is equivalent to the canonical basis of ${\mathrm{\ell }}_{\psi }$. An Orlicz sequence space ${\mathrm{\ell }}_{\psi }$ embeds if and only if $\psi$ is equivalent near zero to a member of ${\mathcal{C}}_{\mathrm{\Phi },\tau }$; in particular, ${\mathrm{\ell }}^p$ embeds exactly when such a member is equivalent to $t^p$. Assuming differentiability of the generators on a common interval near zero, we also derive the necessary bound $p\le b_{\mathrm{\Phi },(c_n),\delta }$, where the upper growth index incorporates the summable errors in the ${\delta }_2$ condition. The same index yields cotype $q$ for every $q\ge 2$ with $q>b_{\mathrm{\Phi },(c_n),\delta }$. Examples with nonuniform growth and lacunary weights illustrate this bound. For alternating Nakano exponents, the explicit limiting family gives a sharper restriction than the index alone. For general Nakano exponents, we recover the known exact cotype criterion, including attainment of the optimal exponent. For weighted Orlicz spaces, we express the cotype criterion through summable scalar errors and recover the formula using the global upper growth index when $w\in \mathcal{A}$.