BK-Fewnomial Incompatibility, a CSEP Correction, and the Auxiliary-Existence Bottleneck

In the real irrational rank-one case, BK-degree and exact fewnomial complexity are directly incompatible: for real λ ≠ 0 and irrational s no nonzero polynomial with BK-degree < N² vanishes on X_s(2N) (it needs ≥ 4N² + 1 monomials, while the BK budget allows ≤ N² + 1), so Dasgupta–Kakde's condition (m) fails for every N in the hypothetical real state; it also corrects R23's union cardinality (card Y(M) = 2M² − 1) and reduces the route to a pure auxiliary-existence problem (BAEP).