Signature Deficiency and Subfield Unit Indices in Totally Real Biquadratic Extensions

Let K/F be a totally real biquadratic extension with intermediate fields L1, L2, L3, and let q(K/F) = [E_K : E_L1 E_L2 E_L3] be the classical Kuroda unit index. We relate this index to the real signatures of the four fields and prove an upper bound for v_2(q(K/F)) in terms of the signature image of the subfield units, the Galois action on the totally positive unit square classes of K, and the signature deficiencies of the L_i. In particular, the usual dimension bound for q(K/F) improves by at least one as soon as one intermediate field has signature deficiency at least two, or the Galois action on the totally positive unit square classes of K is nontrivial. We also give a layer-only consequence, and the construction and inequalities extend to totally real elementary abelian 2-extensions of rank at least two. We also introduce a primitive signature-relation quotient obtained by removing relations induced from the L_i. It is a trivial F_2[Gal(K/F)]-module of dimension at most delta(K) minus the largest delta(L_i), and delta(K) is at most [K:Q]/2 plus the smallest delta(L_i). For octic extensions over real quadratic fields these bounds explain a quartic wall where all three quartic intermediates have deficiency two. We give unconditionally certified examples attaining the primitive bound, an equality example for the unit-index bound, and a counterexample showing that the one-step improvement is false without either hypothesis. A deduplicated PARI/GP experiment on 21,725 wall records reports computed top deficiencies 2, 3, 4, 5, 6, with no value above the proved ceiling six. Companion note to "Primitive Unit-Signature Relation Modules in Degree-Eight Fields".