This paper develops a recursive normalization framework for comparing prime-admissiblenumbers, twin-prime structures, and cousin-prime structures on the integer number line.Revision 1 - Significant rewrite for clarity, and consistency.Revision 2- Revised Final Theorem to be more clear.Revision 3 (2.2) Added a Table of calculations showing the steps taken.Revision 4 Improved Final Theorem and previous section based on feedback. Included certification of twin primes.Revision 5 : Extensive rewrite to establish a stronger connection between the three complimentary frameworks. Tightened and refined logic.Revision6: Improved langauge and clarified for brevity.Revision 7 : Improved Langauge and clarified for brevity again (down to 16 pages now)Revision8: Made connection between Phi and Twin primes more rigorous.Revision 8.1 : Clarified wording in Prime Persistence subsection.