Appreciate the thoughtful digging — seriously. Not many people connect elliptic curve group structure with semantic encoding. A few clarifications though: 1. ECAI is not a “proposal” or theoretical direction. There is a concrete implementation of EC-based knowledge encoding. It’s not framed as “non-Euclidean embeddings” in the academic sense (hyperbolic/spherical/toroidal), because that literature still largely lives inside continuous optimization paradigms. 2. ECAI does not treat ECs as geometric manifolds for embeddings. It treats them as deterministic algebraic state spaces. That’s a very different design philosophy. 3. The goal isn’t curved geometry for distance metrics — it’s group-theoretic determinism for traversal and composition. Most embedding research (including TorusE etc.) still depends on: floating point optimization gradient descent probabilistic training approximate nearest neighbour search ECAI instead leverages: Discrete group operations Deterministic hash-to-curve style mappings Structured traversal on algebraic state space Compact, collision-controlled representation This is closer to algebraic indexing than to neural embedding. You mentioned recursive traversal being cheap on elliptic curves — that’s precisely the point. Group composition is constant-structure and extremely compact. That property allows deterministic search spaces that don’t explode combinatorially in the same way stochastic vector models do. Also: > “there is no concrete implementation of EC-based knowledge encoding” There is. You can explore the live implementation here: 👉

ECAI Search

Specifically look at: The search behaviour Deterministic output consistency Algebraic composition paths This is not a ZKP play. It’s not an embedding paper. It’s not a geometry-of-meaning academic experiment. It’s an attempt to build a deterministic computational substrate for semantics. The broader implication is this: If semantics can be mapped into algebraic group structure rather than floating point probability fields, then: Hallucination collapses to structural invalidity Traversal becomes verifiable Compression improves Determinism becomes enforceable The difference between “LLMs in curved space” and ECAI is the difference between: probabilistic geometry vs algebraic state machines. Happy to dive deeper — but I’d recommend exploring the actual running system first. Numbers still have a lot left in them.