Protein unit cells, their selling reduced representation (one of them), the associated COB, and associated Kurlin roots

Abstract

Single-file DuckDB table of 206,214 crystallographic PDB entries. This is a new version of 10.5281/zenodo.22986222. Same entries and the same RCSB source. This file adds the sorted linear search key, in columns l0–l5, and a schema_meta table (schema_version = 2, search_key = sorted_linear_key). The deposited cells, space groups, volumes, square-root columns, similarity columns, Selling-reduced cells, and deposited-to-reduced changes of basis are the rows from that record. Each row is one deposited conventional cell: pdb_id a, b, c, alpha, beta, gamma, volume — the RCSB conventional cell (Å, degrees, ų) sg_number, sg_hm — space-group number and Hermann–Mauguin symbol r0…r5 — Kurlin’s sorted square-root products sort(√p_ij) (Å), computed on the primitive lattice. Retained from the previous record. Not the search key. l0…l5 — the sorted linear search key sort(p_ij / sqrt(T)) (Å), computed on the same primitive lattice. p_ij are the six Selling conorms of one obtuse superbase and T is their sum. T is the same for every obtuse superbase of a lattice, so each lattice has one key. The sorted distance is a rearrangement lower bound on the labelled distance, the key is Lipschitz in the metric, and it scales with the cell edges. A cube of edge a has nonzero slots a/√3. s0…s5 — the square-root products r0…r5 divided by V^(1/3) (dimensionless). Not recomputed in this version. red_a, red_b, red_c, red_alpha, red_beta, red_gamma — one Selling-reduced cell of that lattice cob00…cob22 — the change of basis from the deposited basis to that reduced cell, stored as exact rationals numerator/denominator, row-major Columns of that matrix P are the reduced basis vectors in the deposited basis, so Pᵀ G P equals the reduced metric. P is the primitive-centering matrix times an integer unimodular Selling reduction. Its determinant is ±1 for a primitive lattice and a rational of absolute value 1/m for a centred lattice. One reduced cell is stored per row. The Selling orbit is not stored. Centred conventional cells are reduced to primitive using the lattice letter of the deposited symbol before the roots, the linear key, and the Selling reduction are computed. The stored cell parameters and volume stay the RCSB conventional values. Built from the RCSB Data API (holdings + GraphQL). Only the identifier, unit cell, and space group were fetched — no coordinates, structure factors, or other archival content. This version was made by copying 10.5281/zenodo.22986222 and filling l0–l5. It is not a new RCSB harvest. Intended for nearest-neighbour and radius search on l0–l5 (Å), for example with the agentsg CellDatabase API. A small linear distance means the same lattice for search, not a proof of identity for every Voronoi type. The square-root key in r0–r5 is only Hölder-½ at a vanishing conorm, which is why search moved off it. The stored matrix takes the deposited cell to the reduced cell. The operator between a query cell and a deposited cell is composed at query time.