Three peer-reviewed proofs power every recall. The Fisher-Rao information metric, Riemannian lifecycle management, and information-theoretic compression — all verifiable, all open.
Cosine similarity treats all dimensions equally. A memory stored with confidence 0.12 — nearly a guess — gets the same geometric weight as a memory stored with confidence 0.99. That is not a retrieval system. That is noise.
Confidence is information. When an agent stores a fact it observed once versus a fact it verified twelve times, those are not the same thing. Any distance metric that ignores this distinction will surface the wrong memories under load — precisely when correctness matters most.
The Fisher-Rao metric lives on the statistical manifold — the curved space where probability distributions actually live. Distance there is measured along geodesics, not straight lines. High-confidence memories are geometrically closer to their neighbors. Low-confidence memories are pushed outward. The geometry does the work.
slm recall "Where does Alice work?"Where p and q are probability distributions over memory confidence scores. The sum Σ is taken over all memory dimensions i. The arccos maps the result to angular distance on the unit sphere — the natural geometry of probability distributions under the Fisher information metric.
Memory lifecycle follows geodesic paths on the Riemannian manifold. ∇γ ∂tγ = 0 is the geodesic equation — it defines the "straightest possible path" between memory states, parallel-transporting the tangent vector along its own trajectory. Expp(v) is the Riemannian exponential map: starting at memory state p with velocity v, it gives you the memory state after one unit of geodesic travel — the mechanism by which memories consolidate toward related facts.
Memories that cohere — that have short geodesics to related facts in the graph — strengthen through consolidation. Memories that are isolated — with long geodesics from everything else — decay. The geometry decides. Not a scheduler, not an arbitrary timer, not a recency window.
These are Shannon's fundamental information inequalities. H(X|Y) is the conditional entropy of prompt X given prior context Y — the irreducible information content that cannot be compressed away without information loss. I(X;Y) is mutual information — the quantity of information that X and Y share, which is always non-negative.
The compression algorithm is bounded by these inequalities. It cannot reduce the prompt below H(X|Y) without losing information. Everything above that floor is redundancy — and redundancy is what SLM strips before forwarding to the LLM.
Information geometry, developed by Shun-ichi Amari and C. R. Rao, is the study of probability distributions as geometric objects. The key insight: probability distributions do not live on a flat plane. They live on a curved manifold — the statistical manifold — where the natural notion of distance is the Fisher information metric, not Euclidean distance. Amari's 1985 monograph unified differential geometry and statistics into a single framework that lets you reason about distributions the same way classical geometry reasons about shapes.
When an agent stores a fact, it does not store a crisp value — it stores a probability distribution over possible values, weighted by confidence at observation time. A memory with confidence 0.97 and one with confidence 0.14 are not points separated by a number. They are distributions separated by a geodesic on the statistical manifold. Treating them as Euclidean points — as cosine similarity does — throws away exactly the information that matters most for reliability. Information geometry keeps it. That is why SLM retrieval improves as the agent operates: the manifold accumulates evidence and the geometry tightens around verified facts.
Every claim on this page is traceable to a peer-reviewed arXiv preprint. Read the proofs, reproduce the results, cite the work.
Confidence-weighted geodesic on the statistical manifold. Replaces cosine similarity in all recall paths. 3 formal properties. 1 tight bound.
arXiv:2603.14588 →Geodesic consolidation and geometric decay. Memories strengthen along short geodesics to related context. No TTL, no scheduler — geometry decides.
arXiv:2603.02240 →Shannon-bounded extractive compression. 60–95% on structured payloads. Byte-exact reversible. The bound is proven. The ratio is measured.
arXiv:2604.06392 →