What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?

I can think of more than one natural way to generalize the notion of a rational number, but "ratio of omnific integers" is not one of them, since every real number is a ratio of omnific integers. Maybe ratios of ordinals are the thing to look at (as a generalization of the non-negative rationals). Or maybe we should look at the Field-closure of the ordinals. Or perhaps we should consider the Class containing every surreal number whose normal form involves only rational numbers at all levels.

I can also think of more than one way to generalize the notion of eventual periodicity to sign-sequences indexed by a general ordinal alpha. One of them is a variant of "Kaufman decimals" (see https://mchouza.wordpress.com/2013/08/25/kaufman-decimals/ and http://www.jefftk.com/p/decimal-inconsistency) in which the digit-set {0,...,9} is replaced by {$+$,$-$} and every over-bar is assigned an ordinal.

A seemingly different but possibly equivalent notion generalizing eventual periodicity involves a kind of symbolic dynamics I haven't seen before, where the Monoid of ordinals acts on ordinal-indexed sequences: if $s$ is a sequence indexed by some initial segment of the ordinals, and $\iota$ is some ordinal, define $T^{\iota}(s)$ to be the sequence obtained by omitting the first $\iota$ terms of $s$ (with $T^{\iota}(s)$ defined to be the empty sequence if $\iota$ is greater than or equal to the length of $s$). Then eventual periodicity (in the case where $s$ is indexed by the natural numbers) is seen to be a special case of the condition that the orbit of $s$ under the action of the Monoid of all ordinals is finite. (See Joel Hamkins' recent post, showing that constant sequences satisfy this finiteness condition: http://jdh.hamkins.org/every-ordinal-has-only-finitely-many-order-types-for-its-final-segments/.)

If my original question seems too vague (what does "right generalization" mean?), here are two very concrete ones that are relevant: does the surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) lie in the field generated by $\omega$? and, is it expressible as a ratio of ordinals? (Note that this sign-sequence does not satisfy the aforementioned finiteness property, although perhaps it satisfies some weaker regularity condition.) Here $-^{\omega}$ denotes a string of $\omega$-many $-$'s and $\cdots$ denotes that the pattern continues $\omega$ times.

I'd be interested in any implications that might hold between these various properties.

Introduction to the theory of surreal numbers, but he defines field operations using LR-forms. Do you claim that every rational surreal number has an eventually periodic sign expansion? Do you know if the class of eventually periodic sign sequences is stable under field operations? If so, the field generated by $\omega$ contains only eventualy periodic sign expansions so the answer is no. $\endgroup$