Certified rank 0 fibres of the Pythagorean family y^2 = x(x-U^2)(x+V^2): census and verification data

Data and code accompanying the paper "The 2-Selmer group of the Pythagorean family in closed form" by D. H. Rodriguez (preprint, 2026). The paper gives a closed form for the 2-Selmer group of the non-isotrivial family E(a,b): y^2 = x(x-U^2)(x+V^2), where (U,V,W) = (a^2-b^2, 2ab, a^2+b^2) runs over primitive Pythagorean triples. Membership is decided by five conditions on Legendre symbols of the primes dividing U, V and W, together with one congruence modulo 8, with no residual case distinctions. These repackage into a square alternating matrix over F_2 with three vanishing diagonal blocks, whose corank bounds the rank. Contents. The certified census: 10,982,837 fibres of rank 0 with a at most 9,999,922, obtained by using the five-primes certificate of the paper as a sieve over p, q < 5*10^6, with no descent computation performed on them. The verification runs quoted in the paper: the closed-form criterion against complete 2-descent, compared as sets and not only as cardinalities (4,582 fibres); the matrix corank against the criterion (18,281 fibres); the rank bound against the true rank (21,682 fibres of certified rank); the five-primes certificate against the actual Legendre symbols (167,248 quintuples, meeting all 256 classes modulo 120); and the Cassels-Tate pairing on the corank 3 locus (5,058 fibres). No mismatch in any of them. The source code that produced and checked them: 34 Python modules and 5 PARI/GP scripts, including a complete 2-descent that does not call ellrank and an exact p-adic solvability procedure for pairs of binary quadratic forms. A companion note of 16 pages with the diagnostics of the density model. See README.md for the file-by-file description, the reproduction commands and the checksums. The census file carries the SHA-256 recorded by the run that produced it. Version 2.0 supersedes the preliminary version 1 files: it adds the main article PDF and the moments article, and replaces the README, the manifest, the code archive and the diagnostic supplement with the final revised versions.