Multiplier obstructions for Legendre pairs of length 333

A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture.

We study the structured case in which both sequences are fixed by a common subgroup $H\leq(\mathbb Z/333\mathbb Z)^\times$ acting by coordinate multiplication.

We prove that such a pair can exist only when $|H|\leq 6$.

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 $\{\pm1,\pm17,\pm19,\pm35,\pm37\}$; the Legendre equations force a $+17,-17$ 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; the unrestricted existence problems remain open.