No Circulant Hadamard Matrix of Order 4·11715²

We exclude infinite families of circulant Hadamard orders, among them parameters left open by the published field-descent criteria: there is no circulant Hadamard matrix of order 4u^2 for u in {11715, 550605, 3877665, 27143655}. If s is odd and q is an odd prime with (q,s)=1 which is self-conjugate modulo 4s, and if h is odd with h^2 < q^b for some b >= 1, then no circulant Hadamard matrix has order 4(h s^a q^c r)^2 for any a >= 0, any c >= b, and any r coprime to and self-conjugate modulo 4 s^(2a) q^b. The four parameters above come from (s,q) = (3,71) and (71,331). The proof keeps a large odd-fibre quotient during ramified descent, so that every coefficient of the projected word is odd and the coefficient at the identity outruns the remaining norm; a scalar-division lemma removes any factor coprime to, and self-conjugate modulo, the terminal modulus. For 11715, a single ramified step at 71 leaves an odd word on C_2556 of squared norm 121*900, and scalar division by 11 reduces it to squared norm 900, which is impossible because 2556 odd squares cannot sum to 900. The applications use only quadratic reciprocity and elementary order calculations after the standard theorems of Turyn and Ma; no enumeration or class-group computation is required.