The Condensate Theorem: Transformers are O(n), Not $O(n^2)$

We present the Condensate Theorem: attention sparsity is a learned topological property, not an architectural constraint. Through empirical analysis of trained language models, we find that attention mass concentrates on a distinct topological manifold -- and this manifold can be identified dynamically without checking every position. We prove a general result: for any query, projecting attention onto the Condensate Manifold (Anchor + Window + Dynamic Top-k) achieves 100% output equivalence with full $O(n^2)$ attention. This is not an approximation -- it is lossless parity. We validate this across GPT-2, Pythia, Qwen2, TinyLlama, and Mistral, demonstrating bit-exact token matching on 1,500+ generated tokens. By mapping this topology to hardware, our Topological Attention kernel achieves a 159x measured speedup at 131K tokens (3.94ms vs 628ms) and a projected >1,200x speedup at 1M tokens, reducing inference costs by >99.9% compared to Flash Attention. We conclude that the quadratic bottleneck is an artifact of naive implementation, not intelligence.

Warp-Cortex: An Asynchronous, Memory-Efficient Architecture for Million-Agent Cognitive Scaling on Consumer Hardware

Current multi-agent Large Language Model (LLM) frameworks suffer from linear memory scaling, rendering "System 2" parallel reasoning impractical on consumer hardware. We present Warp Cortex, an asynchronous architecture that theoretically enables million-agent cognitive scaling by decoupling agent logic from physical memory. Through Singleton Weight Sharing and a novel Topological Synapse--inspired by hybrid landmarking techniques from Topological Data Analysis (TDA)--we reduce memory complexity from O(N * L) to O(1) for weights and O(N * k) for context, where k << L. By treating the KV-cache as a point cloud in latent space, we apply witness-complex-inspired sparsification to preserve persistent homological features of the context manifold. On a single NVIDIA RTX 4090, we empirically demonstrate 100 concurrent agents at 2.2 GB total VRAM, with theoretical capacity exceeding 1,000 agents before compute latency becomes the bottleneck. We further introduce Referential Injection, a non-intrusive KV-cache update mechanism that allows asynchronous sub-agents to influence primary generation without stream disruption.