The bottom line: Multiple research teams have developed mathematical tools that map the internal geometry of how AI systems think. The technique, tracking "geometric signatures" of machine cognition on curved mathematical surfaces called Riemannian manifolds, reveals that large language models compress knowledge into shapes strikingly similar to biological neural patterns. That means we can now read the structure of an AI's reasoning. The surveillance implications cut both ways: governments get a new tool to monitor AI behavior, but AI systems also get better at modeling, and predicting, human cognition.
What Are Geometric Signatures of Machine Cognition?
Here's the short version: every time a large language model processes language, it builds an internal representation, a mathematical point in a high-dimensional space. Researchers have discovered these points aren't scattered randomly. They sit on curved surfaces called manifolds, and the shape of those surfaces reveals how the AI organizes knowledge [1].
Think of it like a topographic map of an AI's mind. The peaks, valleys, and ridgelines tell you what the system knows, how it connects ideas, and where it's likely to go next.
In December 2025, researcher Laha Ale published "A Geometric Theory of Cognition," framing all cognitive processes, human and machine, as gradient flows on Riemannian manifolds [2]. The core equation looks like this: cognitive states move "downhill" on a curved surface, always minimizing a potential function that balances prediction accuracy against computational cost.
In October 2024, a team from UCL, Université de Montréal, and the University of Toronto (including Yoshua Bengio) published research specifically identifying "geometric signatures of compositionality" in language models, measurable shape changes that reveal when an AI has learned to combine words into meaning rather than just pattern-matching [3].
The Dual Dimensionality Discovery
The UCL-Montréal team found something unexpected. Language models maintain two different geometric structures simultaneously [3]:
- Low nonlinear dimensionality (~10 dimensions): The AI compresses meaning onto tight, curved manifolds. This stays constant regardless of model size.
- High linear dimensionality (scales with model size): The AI spreads formal patterns across many flat dimensions. Bigger models use more.
Destroy the word order in a sentence and the nonlinear structure collapses, but the linear structure actually increases. The AI can still see the words, but it can't see the meaning. That's the geometric signature of comprehension versus pattern matching.
This dual structure mirrors what neuroscientists have observed in biological brains. A December 2025 paper in Science Advances showed that human neural systems perform similar "twist" operations, expanding sensory manifolds into higher-dimensional perceptual manifolds through geometric transformations [4].
Why This Matters for Surveillance
Reading the geometric shape of an AI's cognition opens doors that should worry anyone who cares about privacy.
AI systems modeling human cognition: Ale's framework treats human and machine cognition as instances of the same geometric process. If that's right, the same manifold-mapping techniques that let researchers read an AI's thinking patterns could be applied to brain-computer interfaces and neural monitoring. The geometric signatures become a Rosetta Stone between silicon and biological minds.
Predictive cognition profiling: If you know the shape of someone's cognitive manifold, the geometric fingerprint of how they process information, you can predict their responses before they make them. Ale's theory specifically shows that cognition follows gradient flow: it always moves downhill. Map the terrain, and you know where the mind goes next.
AI behavior auditing at scale: Governments can use geometric signatures to monitor what AI systems are actually doing internally, not just what they output. China's 15th Five-Year Plan already commits to AI-powered cognitive systems for elderly care, factory automation, and military applications. Geometric cognition mapping gives regulators, or authoritarians, a way to inspect what those systems actually think.
Dual-process exploitation: The fast-slow decomposition in Ale's theory (rapid intuitive responses versus slow deliberative reasoning) maps directly onto advertising, propaganda, and social engineering. If you can identify which geometric mode a target is operating in, you can tailor manipulation accordingly. Fast mode? Hit them with emotional triggers. Slow mode? Deploy sophisticated arguments.
The Phase Transition Problem
The UCL-Montréal team found that around checkpoint 1,000 during training, language models undergo a sharp "phase transition", a sudden geometric reorganization where the model's representational structure snaps into a configuration that supports genuine compositional understanding [3].
Before this transition, the model is pattern-matching. After it, the model is composing meaning.
That's a measurable, geometric threshold between a sophisticated autocomplete and something that actually understands language structure. And nobody programmed it to happen, it emerges from the geometry.
Nova Spivack's framework, published in May 2025, proposed using geometric measures to assess whether AI systems have crossed thresholds into genuine information integration, potentially even consciousness [5]. His five-tier confidence structure places "consciousness applications" at just 5-20% confidence, but the measurement tools are already here.
Who's Already Using This?
The research is coming from major institutions with deep connections to both government and industry:
- Yoshua Bengio (Université de Montréal), co-author on the compositionality signatures paper, longtime advisor to the Canadian government on AI policy, and a leading voice on AI safety [3].
- Nova Spivack, serial tech entrepreneur whose geometric information processing theory explicitly discusses applications to biological and artificial neural systems [5].
- Multiple defense-adjacent labs are working on neural manifold analysis for both brain-computer interfaces and AI interpretability. The SPAR AI safety research network included projects studying belief geometry in LLM representations as part of its Spring 2026 program [6].
The code for Ale's geometric cognition framework is already public on GitHub. Anyone can run it.
What You Can Do
- Understand the stakes: Geometric cognition mapping isn't theoretical anymore. It's measurable, reproducible, and applicable to both human and machine minds.
- Watch for brain-computer interface regulations: Any framework that unifies human and machine cognition geometry will be weaponized for neural surveillance if unchecked.
- Support AI transparency mandates: If geometric signatures can reveal what AI systems really think, the public should have access to those readings, not just governments and corporations.
- Question "AI consciousness" claims: Geometric similarity to biological cognition doesn't mean machines are conscious. But it does mean they're processing information in structurally identical ways, which has implications for rights, regulation, and responsibility.
Sources
[1] A unifying perspective on neural manifolds and circuits for cognition, PMC/Nature Neuroscience, 2024
[2] A Geometric Theory of Cognition, Laha Ale, arXiv:2512.12225, December 2025
[3] Geometric Signatures of Compositionality Across a Language Model's Lifetime, Lee, Jiralerspong, Yu, Bengio, Cheng, arXiv:2410.01444, October 2024
[4] From sensory to perceptual manifolds: The twist of neural geometry, Science Advances, December 2025
[5] Toward a Geometric Theory of Information Processing, Nova Spivack, May 2025
[6] Spring 2026 Projects, SPAR AI Research Network
Published: March 7, 2026