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.

also, i would like to see the code so i can play with it. playing with your application is not enough, and i won't go into the mechanical turk problem underlying that too far just to point it out (substitute LLM with fixed seeds, for example). i understand the principles you are working with, especially the use of deterministic composition and mapping knowledge to hash functions, the problem i see is defining that hash function exactly. simple enough with text, to assign a coordinate to each word, but word relations are key to semantics, and produce entirely distinct and not reversible relations between them. there are other types of hash functions that are deliberately collision-prone, like nilsimsa, and blurhash, and others, i think blurhash works via structural decomposition, which finds the common parts and creates a near neighbour situation between related, similar data points. i think with EC acting as the number space, you have to use a proximity hash of some kind, not a cryptographic hash. it's still deterministic, but the key difference, as i see it, is the geometry of the group. so, partly it's the generator point and partly it's the curve function. i hope i'm impressing upon you that while i may not be literally qualified in that i barely understand the multi-parameter matrix math involved, i am very strong on geometry, topology and recognising analogs between things. i could help, but i can't do anything without seeing the algorithms.