A Topological Constraint on Cosmological Dimensionality: Testing the Septenary Manifold Hypothesis via Gravitational Wave Echoes

We present a topological cosmological framework constrained by division algebra theory. Following Hurwitz’s theorem, we argue that viable unified field theories must utilize one of four normed division algebras ( ), corresponding to specific dimensional structures. We explore the consequences of an octonionic ()-dimensional spacetime manifold, organized as nested hyperspheres ( ) with our observable universe occupying the innermost layer . From this geometric ansatz, we derive three testable predictions: (1) Gravitational wave echoes arriving milliseconds after black hole mergers, distinguishable from standard ringdown modes; (2) Anomalous gravitational acceleration in wide binary stars at ; and (3) Modified galactic rotation curves consistent with observations but distinguishable from particle dark matter via orientation-dependent effects. We calculate that galactic rotation anomalies could arise from gravitational leakage from the adjacent higher-density manifold , while cosmic acceleration emerges from inter-brane topological tension. Unlike previous extra-dimensional models, we provide specific numerical predictions falsifiable with current LIGO data and upcoming Gaia measurements. This work presents a mathematical framework, not a complete theory. We acknowledge significant open questions regarding parameter values and provide honest comparisons with alternative explanations. Our aim is to contribute a testable hypothesis to the ongoing investigation of cosmological structure.