A Cumulative Prime-Reduction Principle for the Even Goldbach Conjecture

We propose a Cumulative Prime-Reduction Principle (CPRP) for the study of the even Goldbach conjecture. The method is formulated as a cumulative strong induction over the progression of even values. Its central idea is to regard the prime representations already established for smaller even integers as a cumulative mathematical patrimony. At each new even value, two previously reached even integers are selected whose sum is the new target. Their established representations as sums of two primes produce a representation of the target as a sum of four primes. The cumulative primereduction property then supplies the reduction of the relevant four-prime configuration: three of the four primes can be selected so that their sum is prime, leaving the fourth prime as the second addend. The reduction is therefore performed using configurations generated from representations already contained in the cumulative construction.The method does not require the two primes in a Goldbach representation to be distinct. Repeated primes are allowed both in the two-prime representations and in the intermediate four-prime configurations. The complete inductive property is written P(x) = P1(x) + P2(x), where P1 records the accumulated two-prime representations and P2 records the cumulative prime-reduction property attached to the configurations generated from that accumulated patrimony. A direct propagation based on twin primes is the elementary case of the construction.We provide an explicit computational table from 20 through 100. With 10 fixed as one of the two previous even addends, the required reduction is obtained throughout 20–96 and again at 100. The only exception to the fixed choice in this interval is 98; it is recovered by replacing 10+88 with 6+92.