Knowledge Object
K = (G, T, V, S)
G: geodesic coordinate (manifold)
T: type signature
V: vector embedding ∈ ℝⁿ
S: sheaf section
Composite Distance (Def 3.4)
d(Kₐ,K_b) =
α·d_geo(a,b)
+ β·(1 − TVS(Tₐ,T_b))
+ γ·‖Sₐ ∘_π S_b‖
α,β,γ ≥ 0 are weighting coefficients. When the sheaf is globally consistent, this is a true metric space (ε = 0).
Chip Assignment (Voronoi)
chip(q) = argmin_c ‖q − μ_c‖₂
q is assigned to the chip whose centroid μ_c minimises Euclidean distance — the Voronoi cell.
Cosine Similarity (Crumb Rank)
sim(a,b) = (a·b)/(‖a‖·‖b‖) ∈ [−1,1]
Within a chip's Voronoi cell, crumbs are ranked by cosine similarity to the query vector.
Expected Complexity
T(n) = O(log n) + O(|c|)
At K = √n → O(√n)
O(log n) for chip navigation (HNSW). O(|c|) for linear scan within the Voronoi cell.
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