dybmh
ויהי מבדיל בין מים למים
does [he] freely admit there are things [he] doesn't know?
No.
Where does it explain how God knows there's nothing [he] doesn't know [he] doesn't know?
The questions are answered in scripture, but, it requires a human to articulate the explanation.
Here is a link to a recent post where the verses were brought which describe God as absolutley literally infinite and eternal. - LINK
In regard to omniscience, look at the verses from Jeremiah which describe absolute omnipresence. That covers knowledge of all objective phenomena.
Here are some verses which describe God as all knowing of all subjective phenomena. From the Psalms of King David:
139:1
למנצח לדוד מזמור יהוה חקרתני ותדע׃
To the chief Musician, A Psalm of David. O Lord, you have searched me, and known me.
139:2
אתה ידעת שבתי וקומי בנתה לרעי מרחוק׃
You know when I sit down and when I rise up, you understand my thoughts from far away.
139:3
ארחי ורבעי זרית וכל־דרכי הסכנתה׃
You have measured my going and my lying down, and you are acquainted with all my ways.
139:4
כי אין מלה בלשוני הן יהוה ידעת כלה׃
For before a word is in my tongue, behold, O Lord, you know it all.
139:5
אחור וקדם צרתני ותשת עלי כפכה׃
You have beset me behind and before, and laid your hand upon me.
139:6
פלאיה [פליאה] דעת ממני נשגבה לא־אוכל לה׃
(K) Such knowledge is too wonderful for me; it is high, I cannot attain it.
139:7
אנה אלך מרוחך ואנה מפניך אברח׃
Where shall I go from your spirit? Where shall I flee from your presence?
Because God is omniscient of all objective and subjective phenomena from everlasting to everlasting, there is nothing omitted from God''s knowledge.
How does God know this? Because God is currently creating all of it. There is nothing but God prior to creation. So, just as you know that there is nothing additional in the post you are typing other than the letters and spaces which you are typing, there is nothing additional which is not known in the reality which God is currently creating.
There are few places in scripture which describe creation as ongoing. The easiest place to see it in in Psalms 119:
King David writes:
לעולם יהוה דברך נצב בשמים׃
For ever, O Lord, your word is fixed in heaven.
פתח דבריך יאיר מבין פתיים׃
The unfolding of your word gives light; it gives understanding to the simple.
נר־לרגלי דברך ואור לנתיבתי׃
Your word is a lamp to my feet, and a light to my path.
Because God is creating using the word, creation via divine fiat, "let there be light", this implies ongoing creation. Creation is not an event which happened. It is happening.That's how God knows that there is not any which is not known which is not known. All of it is known and nothing is left out, nothing can be left out.
There's another example:
2 Sam 22:31
האל תמים דרכו אמרת יהוה צרופה מגן הוא לכל החסים בו׃
As for God, his way is perfect; the word of the Lord is tried; he is a shield to all those who trust in him.
The word choice for perfect, also means complete. It cannot be perfect if it is incomplete. Therefore nothing is missing from God's knowledge.
I'll be offline for the sabbath. if you have further questions or comments, I won't see them till late saturday evening.