A certified census of spiral knots: no satellites and no coincidences in a finite range

The spiral knot S(p,q,eps) is the closure of the braid (sigma_1^eps_1 ... sigmaₚ₋₁^epsₚ₋₁)^q. Blackwell, Das, Mayer, Moyar, Quraishi and Stees (arXiv:2506.17889) asked whether spiral knots can be satellite knots (Question 5.1) and when two spiral knots coincide (Question 5.2), and conjectured that two equal spiral knots have the same parameters (p,q) unless they are torus knots (Conjecture 5.5). We check all three in the range of coprime p, q with 3 <= p <= 16 and braid length q(p-1) <= 55, together with all coprime 3 <= p <= 9, 2 <= q <= 11: 20,428 parameter triples up to the two symmetries of the sign vector. Every one of the 20,357 triples with a non-constant sign vector gives a knot whose complement is certified hyperbolic by interval arithmetic (SnapPy), and any two of these knots are distinct, even up to mirror image, by a certified volume interval or by the determinant (364 pairs of volume intervals overlap; in each the determinants differ). Inside the range this settles the three questions; outside it the computation says nothing. The deposit contains the note (PDF and LaTeX source), the record of every triple (determinant, signature, certified volume interval, numerical isometry signatures), the programs, the counter-cases run before the census, the independent checks (enumeration against the Burnside count, determinant from the Alexander polynomial formula of the paper, pair resolver, a recount, from the data file alone, of every count over the final data) and the logs. STATUS: a finite computer check, not a proof outside the range; not refereed. This work was developed in collaboration with the AI system Claude Code (Anthropic): the choice of problem, the direction of the work and the computing hardware are due to the author (a sculptor); the formalisation, the algorithms and the proofs were developed by Claude Code (Anthropic) working with the author. Details are in the section on tool use at the end of the note. Licences: text CC BY 4.0; code GNU General Public License v3.0 or later; data CC BY 4.0.