We present the Gradient Foam Framework, a unified perspective on systems where a slowly evolving scaffold (foam) creates potential gradients along which a filler flows without pressure. This principle appears across scales: perfect numbers (abundancy index), the cosmic web (dark matter filaments), second sound in super-fluids, DNA hydrogen bonding, human population dynamics, technology evolution,and the topology of odd integers. We show that the odd positive integers form a M¨obius strip (a nonorientable, onesided space) in which the abundancy index can approach 2 arbitrarily closely but never reach it. The smallest odd abundant number, 945, marks the first crossing above 2 and serves as a numerical witnessto the topological obstruction. We introduce memention — the warm, directional bridge between memory and intention — as a dynamical vector in the foam, and we relate the impossibility of odd perfect numbers to a de Sitterlike 'conclusionary horizon': the odd perfect number can only exist in nonstandard models beyond thelimits of provability. The paper synthesises number theory, topology, physics, andinformation theory into a coherent cosmological picture: the foam holds, the number 2 is the guardian of twosidedness, and the nonexistence of odd perfect numbers is a necessary condition for a stable, flowing universe.