The Eight Factor Doors: From Digit Tails to a Reduced FactorScape

Every prime greater than 5, and every composite number relatively prime to 30, belongs to one of eight residue families. These families are ordinarily encountered as the survivors of a familiar sieve: numbers not divisible by 2, 3, or 5. This paper asks what becomes visible when they are treated instead as eight structured doors through which factors may enter a product. Beginning with the decimal tails 1, 3, 7, and 9, the eight families are organized into a reduced arithmetic landscape. Their multiplication relations separate the landscape into two regions; their square residues distinguish two exceptional families; and their admissible factor pairings generate explicit bilinear sheets of composite numbers. The same organization then passes into difference-of-squares factorization, where it restricts the possible centers of a Fermat search to selected lanes modulo 15. These lanes can also be translated into an additive gnomonic construction and a first-match construction based on descending squares. The result is not proposed as a new high-performance factorization algorithm. Its purpose is structural: to show that several apparently different procedures are coordinate views of one reduced FactorScape. In the spirit of WidiVision, the paper studies not only what arithmetic operations produce, but how a changed field of vision can reveal relationships among them.