I have been intensely studying the Rosh Hashana and Yom Kippur and I have been seeing a pattern in the timing of the appointed times, however I can find no writings that provide explanation of the numeric significance of 10 days. Are there any writings that further clarify this.

The deeper meaning of the number ten being the uniting number can be found in the writings of the Maharal of Prague in his Book Gevurot Hashem chapters 34 and 57. The same idea there, is repeated in many of his writings.