Dedicated to my son Panha

2014 We introduce a fundamental theorem of prime sieving (FTPS) and show how it illuminates structure on numbers co-prime to a random product of unique prime numbers. This theorem operates on the transition between the set of numbers co-prime to any product of unique prime numbers and the new set when another prime number is introduced in the product. We use this to develop tools which exactly count certain gap n-tuples in sets modulo a unique product of prime numbers.