That’s a thoughtful critique. A couple clarifications though. Elliptic curves don’t have a “trapdoor” in the structural sense. The discrete log problem isn’t a built-in asymmetry of the curve — it’s a hardness assumption over a cyclic group of large prime order. The group law itself is fully symmetric and bidirectionally defined. The one-way property emerges from computational complexity, not topology. A torus (ℤₙ × ℤₙ or similar constructions) absolutely removes the DLP hardness if you structure it that way — but that’s because you’ve moved to a product group with trivial inversion, not because elliptic curves are inherently lopsided. Also, secp256k1’s “difference” you’re thinking of is likely the a = 0 parameter in the short Weierstrass form: y² = x³ + 7 That special structure allows GLV decomposition and efficient endomorphisms. It’s an optimization characteristic, not a topological asymmetry. Now — on the deeper point: You’re right that changing substrate does not eliminate determinism. Determinism is about fixed operations over fixed state space. But in ECAI the curve is not being used for cryptographic hardness. It’s being used for: compact cyclic structure closure under composition fixed algebraic traversal bounded state growth There is no reliance on DLP hardness as a feature. In fact, the search layer doesn’t depend on trapdoor properties at all. You mentioned “bidirectional traversal” as a requirement. EC groups are inherently bidirectional because every element has an inverse. Traversal cost asymmetry only appears when you interpret scalar multiplication as inversion of discrete log — which we’re not doing in semantic traversal. The key difference is: Crypto EC usage: hide scalar exploit DLP hardness ECAI usage: deterministic state transitions explicit composition paths no hidden scalar secrets Topology matters, yes. But algebraic structure matters more than surface geometry. Torus embeddings in research still sit inside gradient descent pipelines. They reduce distortion — they don’t eliminate probabilistic training. The real shift isn’t “better curvature.” It’s removing stochastic optimization from the semantic substrate entirely. We probably agree on the end goal: deterministic, compact, algebraic semantics. Where we differ is that I don’t think elliptic curves are the bottleneck. I think probabilistic training is. Happy to keep exploring the topology angle though — that’s where things get interesting.