Rav Shmuel Rozovsky says that the Rambam concludes that the reason why one must leave these amounts of lines is because “perhaps he will cut the Sefer.” Rav Shmuel explains that the Rambam is elucidating that the reason why he must leave four lines between each Chumash, even though the sefer does not have the kedushah of a Sefer Torah, is because perhaps he will come to cut it off. Maybe the sefer that has the Torah written on it will attain the kedushah of a Sefer Torah. Therefore, one must leave four lines between each of the Chumashim in case it will one day become a Sefer Torah.
Accordingly, we can conclude that the Rambam indeed agrees with the Chidushei Anshei Shem that only three of the lines are left for separation purposes, and that the fourth line is left due to the general halacha that every parshah must have a line after it.
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