What is the source that the western wall of har habayis will never be destroyed?
I remember learning it somewhere and heard it numerous times but forgot the source.
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Several Midrashim address the "Western Wall" though it's not entirely clear what those midrashim are referring to. This includes the Bamidbar Rabba 11:2 as well as Shir Hashirim Rabba 2:9 (sometimes called Midrash Chazis), the early Tanhuma to Shemos 10, and Yalkut Shimoni to Kings I, ch. 8. The language in all of those midrashim is the כותל המערבי של בית המקדש, which indicates that the midrash is referring to the actual wall of the Temple itself, either that of the Heichal, Ulam, or Azaros (or maybe even the Cheil).
However, the actual wall which is today known as the "Western Wall", or כותל המערבי, is none of these wall of the temple itself - it is the wall around the Temple Mount, the complex that was built by Herod to level off the top of the mountain (Wikipedia link). While many in the past centuries have identified the Western Wall as belonging to the Temple itself, among them a scholar as great as the Radvaz (Responsa 2:648). This article that I found online discusses the opinions of various poskim regarding the status of the contemporary "Western Wall", but almost all contemporary poskim concede to the fact that it was never part of the Temple proper, and that all of the walls of the actual Beis Hamikdash were destroyed long ago. Therefore, the midrashim cited above probably refer to the Western Wall of the Temple Mount, and their description of those walls as belonging to the 'בית המקדש' is to be taken in a more general sense. This is indeed how the phrase is understood in the linked article, as well as by R. Zalman Koren, cited below.
However, the Midrash Eicha Rabba 1:31 gives details of a story, indicating that only the Western wall remains standing, while the other walls have been destroyed. This is problematic, because the walls of the Temple Mount are still relatively intact, and not just the western wall. R. Zalman Koren, probably the leading expert in the halakhic studies of the Temple Mount, has the following answer (from the Kotel's official website):
בפועל, בצפון הר הבית, כמעט ואין למצוא כיום שרידים עתיקים. גם הכותל המזרחי העתיק נחרב כמעט לכל אורכו, וכמעט לא נשארו ממנו שרידים בחלקיו המרכזיים. באשר לשרידים שבקצוות, ניתן לקבוע כי הם היו מחוץ לתחום המקודש, ואולי משום כך לא דקדקו הדוכסים להרוס אותם לגמרי. באשר לכותל הדרומי, לכאורה גם הוא לא חרב, וקטע גדול ממנו קיים עד היום. אבל כותל זה שבימינו, ניצב מחוץ לתחום המקודש, בעוד אשר הכותל הדרומי הקדום, שהיה בקצה התחום המקודש של הר הבית, עמד צפונית לו. גם מן הכותל הדרומי המקורי לא שרד דבר שניתן לראותו בעין.
In the north of the Temple Mount, almost no ancient remains can be found today. Even the ancient eastern wall was destroyed throughout almost its entire length, leaving hardly any trace of it in its main parts. As to that which remains of the edges, we can determine that they were out of the 'sanctified' area, and maybe that's why the officials [of Rome, who were supposed to destroy the Temple's walls] didn't bother to destroy them completely. Regarding the southern wall, it too seems not to have been destroyed, and a large section of it still exists today. But this wall stands outside the 'sanctified area', while that of the ancient Southern Wall was at the edge of the sanctified area of the Temple Mount, was north of it. From the original southern wall, nothing survived that can be seen with the eye.
In other words, while other sections of the Temple Mount's walls have survived, these are either later parts of the wall or sections of the wall that did not have the holy status of the Temple Mount (see Keilim 1:8). (This small paragraph is a summary of much of R. Koren's life's work into the measurements of what and where should be considered sanctified on the Temple Mount - see his many books and articles on the topic).
There is a medrash hagadol (i'll post the maareh makom soon, when i get back to my sforim) that says the stone (stones) from Yaakov avinu's mizbe'ach was used in the binyan beis hamikdosh and kept jumping out because he didn't want to get destroyed. Untill they put him in the western wall which will never be destroyed, then he stayed put. (The medrash i mentioned is brought in the Torah Shleimah parshas vayeitzei chapter 28 vs 22 note number 157, It is not a medrash hagadol but Rabbi Kasher says it is a medrash.)