Over the last two months, I’ve developed a model where computation is realized as symbolic states embedded directly on geometric manifolds—specifically, helices with torsion, chirality, and combinatorial structure.

In a nutshell - using geometry you can encode computational logic into the rotation group of SO2 (the circle) via the torsion of the helix. Helices are always oriented shapes, so their unit vector/axis of symmetry is used to determine the logic of the system. This deterministic approach stands in contrast to quantum computing where you use SU2. Likewise, the gluing maps between helical manifolds can create what I call "computational geometric bordisms".

This approach deliberately sidesteps the classical “coordinate-free” dogma in math, instead offering a deterministic geometric substrate for computation that naturally interpolates between classical, quantum, and topological computation.

The results are honestly pretty strange: potentially foundational, deeply cross-disciplinary, and—full disclosure—I’m not a professional mathematician. My intent here isn’t to claim a “Theory of Everything,” but rather to provide a teaching tool for understanding computation through geometry.

However, in working through the details, this model seems to (a) threaten to “destroy” one of the Clay Millennium Prize Problems, and (b) force a substantial extension of Grothendieck’s Homotopy Hypothesis. That wasn’t my original goal, but it’s hard to ignore the consequences.

I’m really looking for a sanity check, constructive critique, or any pointers to researchers working along similar lines. Feedback—critical or otherwise—is genuinely appreciated.

Repo here: https://github.com/LambdaLord/Twin-Helix-Geometric-Computati...

Abstract below:

Geometric Computability: Twin Helices and the Algebra of Flow

In this work, I construct a manifold comprising of twin helices that explicitly validates the existence of computational geometric bordisms as anticipated by the project of the cobordism hypothesis. I investigate its geometric, topological, algebraic and computational properties. This work suggests the emergence of a new research direction that I call “Geometric Computability Theory”, situated at the crossroads of geometry, topology, and computability theory. I prove that encoding of arbitrary size finite state machines is possible using these “Geometrically Computable Manifolds” leading to the emergence of Turing Complete phenomena. Further constructions are made on the nature of quantum-like and n-ary/fuzzy computing behavior leading to the formation of a general metatheorem for the behavior of computational logic to explain the origin of Rice’s Theorem. I describe their behavior in terms of Kleene’s theory of partial recursive functions and use vector spaces and ordinary differential equations to model fluidic computation. This perspective opens numerous avenues for future investigation, potentially reshaping our understanding of computation in continuous spaces including applications to the infamous Yang-Mills existence and mass gap. As of writing, algebraic systems to formalize the behavior these geometrically computable manifolds appears to be absent. Future work will involve the development of formal algebraic frameworks, rigorous computational models, and experimental simulation of system behaviors.