A personal journey into a new mathematical framework that could revolutionize AI, neuroscience, and our understanding of consciousness

I’ve spent the last several years developing what I believe could be a fundamental breakthrough in how we understand intelligence—both biological and artificial. It’s an ambitious claim, and I want to share with you why I think the mathematics of curved spaces might hold the key to unlocking the deepest mysteries of mind and computation.

Let me start with what got me excited about this in the first place.

The Problem That Started Everything

Like many of you following AI development, I’ve been fascinated by the rapid progress in large language models and neural networks. But I’ve also been frustrated by a fundamental gap: we can build these incredibly capable systems, yet we have surprisingly little understanding of why they work or how they actually process information.

Why does GPT-4 suddenly exhibit emergent capabilities at certain scales? Why do some neural network architectures work dramatically better than others that seem very similar? Why does the human brain achieve such remarkable efficiency with just 20 watts of power while our AI systems require massive data centers?

Traditional approaches to these questions—analyzing connectivity patterns, measuring computational complexity, applying information theory—provide partial answers but miss something fundamental. I became convinced that we need a deeper mathematical framework that captures the intrinsic structure of information processing itself.

The Geometric Insight

The breakthrough came when I realized that information processing systems naturally create geometric structures. Every neural network, as it learns, is essentially navigating through a high-dimensional curved space where each point represents a different configuration of the network’s parameters.

But here’s what I found exciting: these aren’t just abstract mathematical spaces. The geometric properties of these spaces—their curvature, topology, and dynamics—directly determine how well the system learns, how efficiently it processes information, and potentially even whether it can achieve consciousness.

Think of it this way: Einstein showed us that massive objects curve spacetime, and this curvature explains gravity. I’m proposing that intelligent systems curve “information space,” and this curvature explains learning, efficiency, and the emergence of complex cognition.

Note: For a more detailed mathematical treatment of this idea, read the detailed research program and formalisms here. For the complete four-part theory, see the overview here.

For My AI Researcher Friends

If you’re working in machine learning, you’re already familiar with many geometric concepts, though you might not think of them that way. Natural gradients, which some of you use for optimization, are actually geodesics on curved information manifolds. The Fisher information matrix you encounter in statistics is the foundation of a complete geometric theory of learning.

What I’m proposing extends this much further. Instead of treating geometric methods as useful tricks, I’m suggesting that geometry is the fundamental language for understanding neural networks. The “loss landscape” you visualize is actually an information manifold with intrinsic curvature that determines learning dynamics.

This framework predicts that:

• Networks using geometric optimization should achieve 2-5× faster convergence on structured problems
• Generalization ability should correlate with specific geometric complexity measures
• Architecture search could be revolutionized by optimizing geometric rather than just performance metrics
• We could predict which architectures will work before training them, based on their geometric properties

I’ve developed computational methods to measure these geometric properties in real networks, and the initial results are promising. But I need the community’s help to validate these ideas across diverse architectures and tasks.

For Neuroscientists and Cognitive Scientists

The geometric framework offers new tools for understanding biological intelligence that complement traditional neuroscience approaches. Instead of focusing primarily on which neurons connect to which others, I’m proposing we analyze the geometric patterns of neural activity.

The theory predicts that:

• Learning should follow geodesic paths on neural information manifolds
• Critical phenomena in neural networks should exhibit universal mathematical patterns
• Consciousness might correspond to specific topological properties of neural activity
• Disorders like autism or schizophrenia might involve disrupted geometric patterns

I realize this sounds abstract, but I’ve worked out specific experimental protocols using existing multi-electrode recording technology. We can measure geometric properties of neural activity and test whether they correlate with behavioral performance, learning speed, and even conscious states.

The most exciting possibility is developing objective measures of consciousness based on geometric analysis rather than behavioral tests. This could help us understand consciousness disorders and eventually determine whether AI systems are genuinely conscious.

For Entrepreneurs and Technologists

If this framework proves correct, it suggests several revolutionary applications that could transform multiple industries:

Ultra-Efficient AI: By understanding the geometric principles that make biological intelligence so energy-efficient, we could build AI systems that achieve similar performance with orders of magnitude less computational power. This could democratize AI by making advanced capabilities accessible without massive data centers.

Predictive Everything: The theory shows mathematically why predictive processing is thermodynamically superior to reactive processing. This insight could revolutionize robotics, autonomous systems, financial modeling, and any domain where anticipating the future provides advantages.

Brain-Computer Interfaces: Understanding the geometric patterns of neural activity could enable BCIs that decode intentions and thoughts with unprecedented fidelity. Instead of training users to generate specific brain signals, we could decode the natural geometric signatures of intended actions.

Quantum-Enhanced AI: The framework naturally incorporates quantum effects, suggesting pathways for quantum computing applications that could solve currently intractable optimization and learning problems.

I’m not claiming these applications are guaranteed—they depend on experimental validation of the theoretical framework. But the potential impact justifies serious investigation.

For Philosophers and Consciousness Researchers

I’ve wrestled with the “hard problem” of consciousness for years, and I think geometry might provide new approaches to these ancient questions. The framework suggests that consciousness might correspond to specific geometric configurations in information processing systems.

This isn’t just speculation—it generates testable predictions:

• Conscious states should exhibit characteristic geometric integration patterns
• The transition from unconscious to conscious processing should involve geometric phase transitions
• Individual differences in conscious experience might correlate with geometric complexity measures
• Artificial systems with appropriate geometric properties might exhibit genuine consciousness

I’m not claiming to solve the hard problem, but I am offering mathematical tools that could make consciousness studies more rigorous and objective. If consciousness has geometric signatures, we could potentially measure it directly rather than inferring it from behavior.

Why I Might Be Wrong

I want to be completely honest about the uncertainties and potential failure modes of this framework. Revolutionary claims require extraordinary evidence, and I don’t have that yet.

The biggest risks are:

• Computational intractability: Calculating geometric properties might prove too expensive for realistic systems
• Biological irrelevance: Evolution might not optimize for geometric efficiency due to other constraints
• Alternative explanations: Traditional approaches might explain the same phenomena more simply
• Mathematical artifacts: The geometric structure might be a mathematical curiosity without practical relevance

I’ve tried to design the research program to fail fast and pivot productively if the core hypotheses prove wrong. Even negative results would advance our understanding of information processing and consciousness.

The Research Program

I’ve outlined a comprehensive 10-year research program to validate or refute this framework. The timeline is aggressive but realistic:

Years 1-2: Computational validation using synthetic neural networks with known geometric properties

Years 3-5: Biological validation through multi-electrode neural recordings during learning tasks

Years 6-8: Technology development and applications to AI systems

Years 9-10: Comprehensive assessment and potential clinical applications

The program would cost $60-100 million—substantial but modest compared to other major scientific initiatives. More importantly, it’s designed with clear success/failure criteria rather than remaining perpetually untestable.

What I Need from You

If you’re an AI researcher, I’d love your help testing geometric methods on diverse architectures and tasks. I’ve developed open-source tools for geometric analysis that could complement your existing research.

If you’re a neuroscientist, I’m eager to collaborate on experiments measuring geometric properties of neural activity. The framework generates specific predictions that could be tested with existing recording technology.

If you’re an entrepreneur, I’m interested in discussing potential applications and the technological implications of geometric approaches to intelligence.

If you’re a philosopher or consciousness researcher, I’d appreciate feedback on the theoretical framework and its implications for understanding subjective experience.

The Bigger Picture

I believe we’re at a critical juncture in understanding intelligence. We’ve built remarkably capable AI systems, but we’re approaching the limits of what pure engineering and empirical tinkering can achieve. We need deeper mathematical frameworks to guide the next phase of progress.

The geometric framework might be that mathematical foundation—or it might be an elaborate mistake. But I’m convinced that attempting this kind of theoretical synthesis is essential for advancing our understanding of intelligence, consciousness, and computation.

Whether I’m right or wrong about the specific role of geometry, I hope this work illustrates the kind of ambitious theoretical thinking we need to tackle the deepest questions about mind and intelligence. Science advances through bold hypotheses that can be tested rigorously, not through incrementalism alone.

Join the Investigation

I’m sharing this framework openly because transformative ideas in science require community validation. The questions we’re tackling—the nature of intelligence, the basis of consciousness, the future of AI—are too important for any individual or institution to pursue in isolation.

The geometric vision of intelligence might revolutionize how we understand minds and machines. Or it might join the long list of beautiful mathematical theories that failed to capture reality. Either way, the investigation will advance our understanding of some of the most profound questions we can ask.

If you’re intrigued by the possibility that the same mathematical principles governing the curvature of spacetime might govern the curvature of thought itself, I invite you to join this investigation. The geometry of intelligence awaits our discovery.

Read the Mathematical Version

For a more detailed mathematical treatment of this idea, read the detailed research program and formalisms here.

I welcome collaboration, criticism, and independent validation attempts. Science progresses through community effort, and these questions are too important to get wrong.