(4/5 Changing the Geometry of Arithmetic) The geometry we have built is multiplicative. But many of the hardest questions about primes are additive. Consider two constraints: p + q = N and q − p = h. Neither is native to multiplicative coordinates. So rather than trying to turn addition into multiplication, keep both structures and let them interact. An additive constraint selects a family — a fibre — of states inside the multiplicative geometry. Now the question changes. We are no longer asking only whether the fibre contains a desired prime configuration. We can ask how multiplicative transport behaves inside that fibre, and what would have to prevent the expected transport from being arithmetically realized. This creates an interface between two geometries of the same integers: additive position determines the fibre; multiplicative structure determines the internal geometry. And different additive constraints produce radically different fibres. For p + q = N, the available additive width grows with N. For q − p = h, the separation remains fixed. Same multiplicative geometry. Very different additive environments. So one should not expect the same obstruction mechanism to work in both. The next question is the crucial one: Can permanent arithmetic absence be converted into a geometric obstruction — and can that obstruction survive? #Mathematics #NumberTheory #PrimeNumbers #Geometry