Topological Closure of State Space: A Non-Local Memory Operator via Ramanujan Cusp Forms as a Geometric Complement to Dark Matter Models

This project introduces a mathematical "toy model" that approaches the missing mass problem in galactic rotation curves through geometry rather than postulating invisible particles (Dark Matter).

Instead of adding a hypothetical mass source to Newton's and Einstein's equations, this framework models spacetime as a system with structural memory. It proposes that the dissipative history of a galaxy restricts the available degrees of freedom in its phase space (topological closure).

To make this non-local spatial memory mathematically solvable, the system's history is not integrated over time, but mapped globally using Ramanujan Cusp Forms. The attached documents demonstrate a strict mathematical isomorphism:

• At local scales (e.g., the Solar System): The phase space collapses into a cusp ($q \to 0$), the modular form vanishes, and classical Newtonian dynamics are perfectly recovered.

• At galactic scales: The expansion of the topological parameter naturally activates the first-order Fourier term, yielding an exact $1/r$ acceleration profile.

The uploaded files (a short Abstract and a theoretical Toy Paper) show how the empirical effects of an isothermal Dark Matter halo can be organically reproduced as a pure restriction of degrees of freedom in a structurally closed state space.