Here is a hypothesis: [Feedback Requested] Toy model replacing Dark Matter halos with a topological phase space restriction (using Ramanujan cusp forms)
Hi everyone,
I am an independent researcher and I've been working on an alternative mathematical approach to flat rotation curves. Instead of adding a mass source (Dark Matter), I tried modeling the "missing mass" as a structural restriction of degrees of freedom.
The basic idea: I replaced the infinite historical memory of a system's dissipation (which is uncomputable) with a global state-space operator using Ramanujan Cusp Forms, M_k(tau).
If we map the phase space to a parametric torus, the parameter q expands proportionally to 1/r as we move to the galactic edge.
Here is where the math gets interesting: In local, highly ordered systems (like our solar system), q approaches 0 (the cusp limit). The modular form vanishes entirely, leaving us with pure, uncorrected Newtonian/Einsteinian dynamics.
But at large radii (galactic edge), the first-order Fourier term (c_1 * q) takes over. Since q is proportional to 1/r, this organically yields an effective extra acceleration of a = c * 1/r. This mathematically mimics the exact profile of an isothermal Dark Matter halo without needing a single gram of extra mass.
I've written a very short, 5-point Toy Paper outlining the mechanics and uploaded it to Zenodo here: https://zenodo.org/records/20300918
I would love for someone to stress-test the math in section 4. Does this isomorphism hold up in your opinion? Thanks!