What are the implications of "Beini u'Vein B'nei Yisrael" ... ?
Mechilta: It implies that the sign is confined to Yisrael, and that a Nochri who keeps Shabbos is not subject to reward. 1
Beitzah, 16a: 'Every Mitzvah that Hashem gave Yisrael he gave them in public, 2 with the sole exception of Shabbos, where the Torah wrote "Beini u'Vein B'nei Yisrael". 3
Torah Temimah, citing Sanhedrin, 58b: In fact a Nochri who keeps Shabbos is Chayav Misah. See Torah Temimah No'ach, Bereishis, 8:22.
With reference to the reward. See Oznayim la'Torah note 39.
The Gemara actually learns it from the word "Os Hi le'Olam", which is written without a Vav' - implying that it is hidden - See Torah Temimah, note 39.
What is the significance of "Os Hi le'Olam"?
Rashi (in Machzor Vitri): Hashem made a covenant with Yisrael that if they keep the Shabbos, they will come to no harm, and this is a sign between Hashem and Yisrael that He protects them.
Beitzah, 16a: It teaches us that Shabbos will never become Bateil from Yisrael.
Oznayim la'Torah (citing the Chofetz Chayim): In former times, a shopkeeper would hang a sign outside his shop so that people would know what he was selling or what sort of service he offered. As long as the sign was up, people would know that the shop was still in operation - even if the owner was sick or on holiday. However once the sign was removed, they would know that he had closed shop. In the same way, as long as a Jew is keeping Shabbos, the sign is up that he is still in the fold - irrespective of other sins of which he may be guilty. But once he stops keeping Shabbos, the sign is down and it is clear to all and sundry that he has left the fold.
What are the implications of "Os Hi le'Olam"?
Beitzah, 16a: It implies that Yisrael will never negate Shabbos. 1
See Torah Temimah, note 41. See also Oznayim la'Torah.
What are the connotations of "Va'yinafash"?
Rashi (citing Targum Onkelos) and Targum Yonasan: It means "and He rested (desisted from work)". It also refers to refreshing of the soul that occurs when a person does so. 1
Ramban and Seforno: It refers to the Neshamah Yeseirah that Hashem places in man with the advent of Shabbos, 2 which enables a person to attain higher levels of G-dliness. 3
Beitzah 16a: An extra Neshamah comes with the advent of Shabbos, and leaves when Shabbos ends, at which point Hashem laments 'Vai Avdah Nefesh!? - the acronym of 'Vai (Avdah) Nefesh. 4
Which explains the double expression "Shavas Vayinafash ".
Da'as Zekenim, Rosh (in Pasuk 13) and Ba'al Ha'turim - The last letters of "u'va'Yom Ha'shevi'i Shavas Vayinafash" spell 'Shetayim', to hint at this.
As hinted in the words "Na'aseh Adam be'Tzalmeinu ki'Demuseinu", Bereishis, 1:26 (Seforno).
Oznayim La'torah: Hashem laments the fact that Yisrael have not yet reached the level of 'Yom She'kulo Shabbos' - at which point they will retain the Neshamah Yeseirah permanently. See Oznayim La'torah.
How can the Torah apply "va'Yinafash" to Hashem, who is not subject to fatigue?
Rashi: Even though it is not applicable to Hashem, the Torah often writes about Hashem in human terms in a way that we can relate to it.
If Hashem observes Shabbos, how can He move the wind and the rain from one domain to another?
Da'as Zekenim (citing Midrash Tanchuma): Because, since the entire world belongs to Him, 1 it is all one domain. 2
Da'as Zekenim (Ibid.): As the Pasuk writes in Yeshayah 66:1, "ha'Shamayim Kis'i ve'ha'Aretz Hadom Raglai"
Hashem brings rain and waters vegetation on Shabbos. How can we explain this? The Ra'avan in Rosh Hashanah 30b: Hashem commanded; He is not commanded (He may nullify the Torah whenever He wants.) In Midrash Tanchuma, R. Akiva answered Turnusrufus according to the latter's mindset (PF).
Why is "le'Olam" in the current Pasuk written Chaser (without a 'Vav')?
Kol Eliyahu and Divrei Eliyahu: Because without a 'Vav', it is like 'Le'alem' (to conceal). And the Pasuk continues "Ki Sheishes Yamim Asah Hashem ..." This hints that if one is in the desert and does not know which day is Shabbos, he counts six days, and calls the seventh day 'Shabbos', as the Gemara explains in Shabbos 69b.
Refer to 31:17:1.1:2*.
