Two new lower bounds for the size of spherical designs: no 4-design with 21 points on S^4 and no 6-design with 33 points on S^3

Let N(Sⁿ⁻¹, t) be the smallest number of points of a spherical t-design on Sⁿ⁻¹, with equal weights. We show that there is no spherical 4-design with 21 points on S^4 and no spherical 6-design with 33 points on S^3, so N(S^4, 4) ≥ 22 and N(S^3, 6) ≥ 34. The best published lower bounds we found are 21 and 32 (Boyvalenkov and Nikova, 1998), and the smallest known designs have 29 points (Hardin and Sloane, 1992) and 42 points (Sloane, Hardin and Cara, 2003). So 22 ≤ N(S^4, 4) ≤ 29 and 34 ≤ N(S^3, 6) ≤ 42. Each proof is an exact rational certificate for a three-point linear program in which the triple moments are fixed by conditioning on two design points (the method of our earlier note, doi:10.5281/zenodo.23615049). Positivity of the three-variable part on the Gram region is proved by exact integer Bernstein subdivision. The supplement contains both certificates, two separately written exact checkers, tamper tests, controls on known designs (a 160-point design on S^4, the 600-cell on S^3), and the literature check. As far as we checked, both bounds are new; pointers to earlier work are welcome. AI use: the computations, the certificates, the checkers and the text were produced with Claude (Anthropic) under the author's direction; the second checker was written by a separate session of the same AI system, which is not independent human review. Not yet peer reviewed. Contact: gameruchral71@gmail.com.