Theta on constructed Conway-mutant pairs: thirty separated, one not

Thirty pairs of knots related by a Conway mutation, each pair given with both planar diagrams and with the value of the Bar-Natan/van der Veen Theta invariant on both members. NO THEOREM IS CLAIMED AND NONE OF THE MATHEMATICS IS OURS: Theta is theirs (arXiv:2509.18456, arXiv:2109.02057), mutation is classical, and we do not know what is already published about Theta and mutation. What is deposited is a set of computed cases, built because the published supply of mutant test cases for this invariant is small. Construction: each pair is a pretzel knot P(2,b,c,d) against P(2,c,b,d), the two middle tangles exchanged, which is a Conway mutation by construction. From 31 parameter sets, 30 survived a three-stage sieve in which every stage can reject - both members hyperbolic with a geometric solution; DIFFERENT ISOMETRY SIGNATURES, so the pair really is two knots rather than one knot presented twice (mutation frequently returns the same knot, and five of the eight mutant pairs already in our corpus are of that kind, where agreement of Theta is correct behaviour rather than blindness); and equal Alexander, Jones, HOMFLY and determinant, so that no classical invariant we compute distinguishes the members. The pairs run from 13 to 29 crossings and from determinant 123 to 1171, involving 32 distinct knots. Theta separates all thirty at three rational evaluation points, and this is verified symbolically for those computed at deposit time. The symbolic test has two stages and the second is the one that matters: the stored Theta is the Delta-multiplied form, Delta is defined only up to a unit, so the difference must be non-zero AND the quotient must fail to be a unit before a separation is recorded; a control pair which is one knot presented two ways returns difference exactly zero. Diagram independence was checked separately: nine of the knots occur under two different parameter sets with the same isometry signature, and Theta returns the same value in 28 of 28 such comparisons. A COUNTEREXAMPLE IS INCLUDED DELIBERATELY: 13a_1231 and 13a_1237, a genuine mutant pair from Stoimenow's 13-crossing table, carry byte-identical Theta in our corpus; this was reported to Bar-Natan and van der Veen on 15 June 2026 and is recorded in the data file so the positive count is never quoted alone. Theta is also blind on the Kanenobu pair K(0,4)/K(2,2), a different mechanism - not mutants, distinct Alexander modules, distinct volumes, and Lobb (arXiv:1105.3985) proved sl(n)- and HOMFLY-homology do not separate them either. The deposit also records a limitation of our own database found while doing this: our distinctness key is assembled from hyperbolic volume, cusp shape, exact algebra and HOMFLY, every one of which is mutation-invariant, so a mutant pair necessarily collapses to one entry - and 13a_1231/13a_1237 does collapse there, although each record separately carries its own correct KnotInfo identification. It surfaced only because both members have table names; between two machine-generated knots the merge would have been silent, so our distinct-knot counts should be read as lower bounds. HONESTY NOTE: no invariant in this deposit is computed by an AI. Every number comes from established open tools - SnapPy/spherogram, SageMath, and the authors' own Theta.sage - and is recomputable from the planar diagrams included here. A dependency-free reproduce.py re-checks the internal coherence of the data file and prints what it cannot verify. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the geometry, the questions and the hardware are Stanggassinger's; the pipeline driving the tools and the machines that generate the knots are the joint contribution.