army military war - Were Amalek offered a peaceful exit?

Inspired by this:

The Rambam in Hilchos Melachim 6:1 writes:

אין עושין מלחמה עם אדם בעולם, עד שקוראין לו לשלום--אחד מלחמת הרשות, ואחד מלחמת מצוה

War is not waged with anyone in the world until they were offered peace...

and in 6:6 the Rambam includes Amalek in this rule:

אבל שבעה עממין ועמלק שלא השלימו, אין מניחין מהם נשמה

If the Seven Nations or Amalek do not accept peace, wipe them out.

In the story of King Saul's war on Amalek (Samuel 1 15:1-9), there is no mention of any offering of peace before Amalek is destroyed. Is there any reference to such a communication, and if not how does the Rambam know to include Amalek in this rule? I would have considered that as Hashem is in a constant state of war with Amalek (Shemos 17:16) and we are instructed to erase their memory (Devarim 25:19), there would be an objective desire to have the war with them, and making peace would not suffice.

Answer

The Chidushei HaGriz (§ 161) comments that the narrative between Shmuel and Shaul is a basis for the Rambam's opinion in 6:4. In verse 18, Sh'mu'el says that Sha'ul was told to destroy אֶת-הַחַטָּאִים אֶת-עֲמָלֵק:

And the LORD sent thee on a journey, and said: Go and utterly destroy the sinners the Amalekites, and fight against them until they be consumed. (Mechon Mamre translation)

The Brisker Rav inferred that this indicates that they did not accept the seven mitzvos or terms for peace (both of which are necessary criteria according to the Kesef Mishne's interpretation of the Rambam¹), otherwise they would not have been called "sinners":

שמואל א' (ט"ו י"ח) וישלחך ה' בדרך ויאמר לך והחרמתה את החטאים את עמלק וגו'. מדקאמר "את החטאים" מכאן מקור לשיטת הרמב"ם פ"ו מה' מלכים ה"ד, דעמלק אם קבל ז' מצוות והשלים אין הורגים אותו, ע"ש ודו"ק.

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¹_{The Kesef Mishne writes (ibid.): אפשר לטעון בעד רבינו ולומר שבכלל השלימו הוא קבלת שבע מצות שאם קבלו עליהם שבע מצות הרי יצאו מכלל שבעה עממין ומכלל עמלק והרי הם כבני נח הכשרים. This seems to put the Rambam's opinion at odds with the position of the M'chilta (end of B'shalach): ר' אליעזר אומר נשבע המקום בכסא הכבוד שלו שאם יבא אחד מכל אומות העולם להתגייר שיקבלוהו ולעמלק ולביתו לא יקבלוהו.}