Multiplier Obstructions for Legendre Pairs of Length 333

ABSTRACT A Legendre pair of length 333 would yield a Hadamard matrix of order 668. We study the structured case in which both sequences are fixed by a common subgroup acting by coordinate multiplication. We prove that such a pair can exist only when . After a mod‐3 compression reduces the problem to an order‐108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order‐9 subgroup is eliminated analytically: its orbit structure restricts the 9‐compressed entries to ; the Legendre equations force a pair in one compressed sequence, and a single‐shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full‐image compression, a row‐sum congruence, exact meet‐in‐the‐middle enumeration, and proof‐carrying pseudo‐Boolean encodings. The solver‐assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common‐multiplier symmetry only; it does not decide the unrestricted existence of a Legendre pair of length 333.