Authors: Christoper Mututu
We introduce a deterministic construction for generating composite number pairs (A,B) from strictly isolated prime sextets, configurations of exactly six primes situated at fixed offsets {0,8,14,18,24,32} from a base value a≡9 (mod 10) with no additional prime existing anywhere within the interval [a,a+32]. We term such configurations strictly isolated prime constellations of order six.The structural constraint a≡9 (mod 10) forces the six primes to terminate in the digit pattern 9,7,3,7,3,1 respectively which is a consequence of the fixed offsets modulo 10. These six primes are arranged into a 2×4 rectangle whose columns are indexed by the digits {1,3,7,9}, the complete set of possible terminal digits of any prime greater than 5. Column wise addition and subtraction yield a Sums row and a Difference row from which the composite A and B are defined by their respective totals.We prove that this construction satisfies four universal invariants. First, the closed form identities A=6a+96 and B=2a+52 hold for every valid cluster. Second, A is always divisible by 1,2,3,5 and 6 while B is always divisible by 1,2,5 and 10, both following algebraically from a≡9 (mod 10). Third, the decimal expansion of A/B always carries a signature 2.9u2026 and B/A always carries initial signature 0.3u2026 for all a≥274 proven via closed form analysis. Fourth, both A/B and B/A always produce non terminating repeating decimal expansions guaranteed by the arithmetic structures of the reduced denominators.These four invariants are established algebraically and confirmed computationally across 17,138 valid clusters up to 100 billion with zero failures on every claim.
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