If we replace the single axiom "Emergence" in the axiomatic system $T$ presented at posting [What is the set theory synonymous with this order-set theory], by the following schema. Would the resulting theory still be synonymous with $\sf PA$? If no, is there a kind of restricted synonymy notion that it would have with $\sf PA$?
$ \textbf {Emergence: } i=0,1,2,\dots; \ n=i-1, \dots, -1;\\ \max_i x < \max_i y \land (\bigwedge_n \max_n x = \max_n y ) \to x < y $
$ \max_{-1}x = \varnothing \\\max_0 x = \min y: \forall m \in x \, (m \leq y) \\ \max_{n+1} x = P_x(\max_n x); \text { if } n > -1$
Where $P_x$ is the predecessor relation on $x$. This is: $$ a=P_x(b) \iff a \in x \land b \in x \land a < b \land \not \exists c \in x: a < c < b$$
Where, "$\max_i x < \max_i y$" is defined as:
$ \exists b \, \bigl( b=\max_i y \land \forall a \, (a= \max_i x \to a < b) \bigr)$
While, " $\max_i x = \max_i y$" is defined as:
$\exists a \exists b \, (a=\max_i x \land b=\max_i y \land a=b)$