(5/5 Testing the Geometry) Mersenne: remove abundance We have been progressively removing resources. First, an expanding additive fibre. Then, a fixed gap with no growing width. Now consider M_q = 2^q − 1. Here the arithmetic is more rigid still. A prime divisor of a Mersenne number cannot have arbitrary form. Its possible location is constrained by the exponent q and by congruence conditions. The admissible arithmetic states are sparse. There may be no wide fibre across which to distribute transport and no abundant population of candidates on which to rely. So the obstruction problem changes again. Instead of asking whether abundance overwhelms the obstruction, ask whether an admissible state can survive the restrictions at all. Follow the first one that does. Stop at the first hit. Then ask what the geometry forces that survivor to carry. The terminal mechanism changes because the arithmetic environment has changed. A geometry is not seriously tested by repeating the same argument. It is tested by removing, one by one, the resources that made the previous argument possible. Width. Freedom. Abundance. What remains should be structure. #Mathematics #NumberTheory #Mersenne #PrimeNumbers #Geometry