sketch_theta.nx

sketch_theta.nx -- Theta sketch (Beyer/Haas 2014). Extends KMV with EXPLICIT theta and SET INTERSECTION + DIFFERENCE. Where KMV's theta is always implied by kth_smallest, Theta carries theta as a first-class field so intersected sketches (which have smaller-than-natural theta) remain valid for further composition. Theta semantics: theta ∈ (0, HASH_MAX]: the bound below which hashes are sampled. Initially HASH_MAX (entire domain). After K hashes seen, theta = kth_smallest. After intersection, theta = min of inputs'. Cardinality estimator (theta-corrected, unbiased per Beyer 2007): est = |hashes < theta| * HASH_MAX / theta For a freshly-filled KMV this reduces to (K-1)*HASH_MAX/kth which matches the KMV formula. COMPLETES THE SET-OP TRIO: - union: runtime/sketch_kmv.nx::nx_kmv_union - intersection: nx_theta_intersect (THIS FILE) -- new capability - difference: nx_theta_difference -- new capability This is the family DataSketches ships as Theta + Tuple sketches. We ship Theta first; Tuple (attaching auxiliary data to each hash) is queued for v2 once the core sketches stabilize. Reference: DataSketches Theta paper -- arXiv:1306.5664 "Beyond HyperLogLog: A Survey of Sketches for Cardinality Estimation"