Code and paper: github.com/samuelbfoster/toroidal-project
I developed this work with AI assistance for
implementation and writing. The ideas and results
are mine.


Compromised logic can mask itself as truthful. Current AI systems have no internal  mechanism to verify whether their reasoning is still what it was when they were last checked, and by the time misalignment appears in outputs, it may already be too late.

In 2022, a team at the Kavli Institute for Systems Neuroscience published a Nature paper showing that grid cells in the mammalian brain encode spatial information using a toroidal topology. They used persistent homology, the same mathematical tool I used in this project, and measured Betti numbers directly from neural recordings. If biological intelligence systems produce stable, measurable topological structure as a byproduct of natural processes, the question becomes whether the same is true for AI, and whether a measurable breakdown in that structure can be detected before it appears in outputs.

I represent AI information states as points on a four-dimensional torus and use persistent homology to measure whether the topological structure of those states is consistent with a verified healthy baseline. By embedding the mechanism of detection into the architecture of the system,  I have turned theory into measurable data. At eight nodes, the system achieves fault tolerance of 87.5%, compared to 37.5% under Byzantine Fault Tolerance. This 2.6x improvement grows without bound upon increasing the number of nodes. On synthetic data, detection accuracy was 100% with a 0% false positive rate. On real GPT-2 transformer hidden states, the accuracy of detection was 89.2%, remaining stable across two independent runs. Under persistent attack on five of eight nodes over 30 steps, the system maintained operational with a 86.2% node recovery rate.

"An adversary who knows your monitoring mechanism can preserve the topological signature while corrupting the underlying reasoning."
Yes, this is a real limitation. A system that knows it is being monitored topologically could in principle attempt to preserve its topological signature while corrupting its reasoning. However, the implementation of a ZK proof layer means the monitored system cannot learn exactly which properties of its signature are being checked, making evasion significantly harder. Implementing a dual-substrate design requires  simultaneously fooling two independent monitoring channels, each bound to the confines of their own scope. The primary threat model this system addresses is unintentional misalignment, not an adversary actively undermining the monitor, but against data corruption, this detection mechanism works.


With a 100% detection against synthetic data, and 89.2% on GPT-2 small, implementation of an updated language model could yield far greater results than testable in my consumer environment. Although the real data shows 89.2% the missing 10.8% falls short of the 100% synthetic ceiling for unknown reasons. n=1000 is not enough, but without data center hardware, publication quality validation was unrealistic for this testing.

The code is open and the results are reproducible on consumer hardware. I am looking for collaborators, compute for large-scale validation, and honest criticism. A system cannot fake its own topology without being what it is faking. If that holds at scale, it changes what is possible in AI oversight.