Li-zchus Moshe Yehuda ben Pesha Dina Hanus for bracha vi-hatzlacha in everything he does! 

The mishna says [Bava Metzia 25b]

מצא בגל ובכותל ישן הרי אלו שלו מצא בכותל חדש מחציו ולחוץ שלו מחציו ולפנים של בעל הבית אם היה משכירו לאחרים אפילו בתוך הבית הרי אלו שלו:

 If one found lost items in a heap of stone rubble or in an old wall, these belong to him. If one found lost items in a new wall from its midpoint and outward, they belong to him. If he found the items from its midpoint and inward, they belong to the homeowner. If the homeowner would rent the house to others on a regular basis and there was a steady turnover of residents, even if one found lost items inside the house, these belong to him. Since the owner of the lost items cannot be identified based on location, he will certainly despair of recovering his lost items.

The gemara [26a] says:

תנא אם היה כותל ממולא מהן חולקין פשיטא לא צריכא דמשפע בחד גיסא מהו דתימא אשתפוכי אישתפוך קמ"ל:

It is taught: If the hollow in the wall was filled with lost items, e.g., coins, the homeowner and the finder divide them. The Gemara asks: Isn’t that obvious? The Gemara answers: No, it is necessary to teach this only in a case where the hollow in the wall is inclined toward one side of the wall. Lest you say that all the items were initially on the elevated side, and due to the incline they slipped and filled the entire space, the tanna teaches us that the homeowner and the finder divide them.

We have to understand: Why was it so simple to the gemara that the lost items are divided? Why don't we say that the person who finds it is מוחזק and therefore gets to keep it. 

The Shittah Mekubetzes cites the Ritva who said that the reason they divide it is as follows:

"דכיון דכותל ממולא מהן, ומילתא דספיקא היא, לא שייך בה אלא חלוקה, דהא כמאן דתפסי לה תרוייהו הוי, וכעומד בחזקת שניהם, ע"כ. 

Since the wall is filled with them and it is a matter doubt, only division is possible, because it is like both [the owner and the finder] are holding on to it, it is like it is in the possession of both of them.

This is puzzling! Why is it considered as if they are both holding on to it? It is now in the possession of the finder and he is the תופס [holding on to it]?! Apparently, we can say, based on what it says in מגיד משנה [Issurei Biah 15-26] about a foundling [אסופי] who was discovered in a city that is mostly Jewish, that we return a lost object to him just as we would to a Jew. Even though if the ox of a Jew gores his ox, he wouldn't have to pay, because we don't follow the majority in monetary matters [אין הולכין בממון אחר הרוב], nevertheless we return the lost object because the finder doesn't have a חזקת ממון [a presumption of ownership]. However when it comes to paying damages there is no requirement, even if there is a majority of Jews, because we can't extract money from the Jew to give to the foundling. This is cited in the Beis Shmuel [4-56] who says that a finder of a lost object doesn't have a חזקת ממון.  

But we still can say that even though the finder doesn't have a חזקת ממון, nevertheless he has a תפיסה on the אבידה. And since he is תופס, he has more rights than the owner of the wall. However it appears, that since the אבידה was in the property of the owner, it emerges that he was מוחזק at least in half. Then, the finder came and took it out of his possession. So this is a case of תפיסה שלא ברשות - snatching away without permission after the doubt was generated [לאחר שנולד הספק] and the הגהות אשר"י in Bava Basra [126b] writes that תפיסה מספק is only valid if done before the doubt was generated or if he says ברי - I am absolutely sure that it is mine. ['ועיין בספר תקפו כהן סי' נ"ז וע]. Here is a case of תפיסה לאחר שנולד הספק - snatching away after the doubt was generated, for immediately at the time of the finding we already have a doubt of the lost object belonged to the בעל הבית or it come from an outside source. And also, there is no טענת ברי, claims of certitude that it belongs to him, because the finder doesn't know if it came from an outside source or from the בעל הבית. If so, there is no דין תפיסה and automatically the חזקה of the בעל הבית on half remains in place and thus יחלוקו - they divvy it up. 

[עפ"י נתן פריו עמ' קמ"ה-קמ"ו]