Topological Cognitive Diffusive Emergence (TCDE)

TCDE: When Geometry Thinks

What if intelligence isn't about processing symbols, but about shapes evolving in space?

TCDE (Topological Cognitive Diffusive Emergence) reimagines artificial intelligence from first principles. Instead of training neural networks on massive datasets, TCDE creates continuous fields that flow across adaptive geometric surfaces—and something remarkable happens: cognition emerges naturally from the mathematics itself.

The core insight is deceptively simple. Traditional AI treats information as discrete tokens shuffled through statistical pipelines. TCDE treats information as a living field Φ(x) spreading across a Riemannian manifold whose very geometry adapts to what it learns. When the field encounters new patterns, the underlying space curves. When it recognizes something familiar, the geometry smooths. Learning becomes literal shape-shifting.

This geometric foundation produces capabilities that shouldn't exist in such a minimal system. With just 3 examples and zero pre-training, TCDE achieves 70-80% accuracy on pattern recognition—genuine few-shot learning emerging from pure mathematics. The system demonstrates measurable self-reflection (Φ operating on Φ yields 0.997 coherence), anticipation of future states, and autopoietic self-maintenance. These aren't programmed behaviors; they're geometric inevitabilities.

The numbers challenge conventional assumptions. Sub-millisecond inference (0.8-5.1 ms). Memory footprint under 16 KB. 100% passage rate across 50 rigorous validation tests. No GPU required. No training phase. No massive parameter counts. Just differential equations on curved spaces, producing emergent intelligence.

TCDE represents a fundamental question made concrete: Can continuous geometry be a more natural substrate for cognition than discrete computation? The experimental evidence suggests yes. The mathematical framework—built on Ricci curvature, adaptive metrics, and topological diffusion—provides rigorous foundations. The implementation proves it works.

This isn't incremental improvement to existing AI. It's a different answer to what intelligence might be.

Developed February-December 2025 by Mehdi Wahbi, Move37 Initiative DOI: 10.5281/zenodo.17907427