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 "An explicit 2-Selmer criterion for the Pythagorean family y^2 = x(x-U^2)(x+V^2)" 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 four conditions on Legendre symbols of the primes dividing U, V and W, the dyadic one entering as (2/uw) = 1, with no residual case distinctions. These repackage into a square alternating matrix N over F_2 with three vanishing diagonal blocks, and corank N = rank E(Q) + dim Sha(E/Q)[2]. 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 symbolic local images against an exact p-adic decision procedure (4,521,856 comparisons); 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 locus where corank N = 2 (5,058 fibres). No mismatch in any of them. The source code that produced and checked them: 36 Python modules and 8 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 21 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 and the code are pinned by the two SHA-256 values printed in the data-availability statement of the article. Version 3.0 supersedes version 2.0. The article is the revised 34-page version and the diagnostic supplement is 21 pages. The code archive adds three PARI/GP scripts, which exhibit a prime quintuple in each of the 32 residue classes of the five-primes theorem with APR-CL primality proofs, check the symbol reductions of its proof on 24,650 quintuples, and verify the three-primes criterion on the even-exponent branch excluded from the earlier sample. final_census.py gains a --fresh mode that regenerates the census from scratch. A second manifest, MANIFEST-code.sha256, pins the code and the archived outputs independently of the PDFs. The census file itself is unchanged.