Primitive Unit-Signature Relation Modules in Degree-Eight Fields

For a totally real number field K, the gap between the narrow and ordinary class numbers is governed by the rank of the unit-signature map, equivalently by the signature deficiency def_sig(K) = n - rank sgn(E_K). We study this deficiency in degree eight through an F_2[Gal(K/Q)]-module invariant: the primitive quotient P(K) = S_K^perp / Q_4(K) of the unit-signature relation space by the relations induced from quartic subfields, which measures the relation space not explained by quartic subfields. The theoretical input is an elementary structural observation, independent of the number-field census, which applies known modular representation theory (for D_4, the special biserial structure of the dihedral group algebra) to unit signatures. When Gal(K/Q) is generated by involutions (in degree eight, D_4 or C_2^3), a norm-descent lemma shows that P(K) is a trivial module, and an analysis of the augmentation ideals gives dim P(K) <= 2: unconditionally for D_4, and for def_sig(K) in {5, 6} when Gal = C_2^3. We apply this to a reproducible PARI/GP-assisted finite census of degree-eight unramified 2-class subextensions over real quadratic fields whose 2-class group has order 64, extending the unit-signature and narrow 2-class-tower analysis of Benjamin and Snyder to the 2-rank-three regime. All 58 census rows of signature deficiency at least 5 (23 absolute isomorphism classes) satisfy dim P(K) <= 2 with trivial primitive action. A larger computation (roughly 22,000 V4 rows whose three quartic layers all have deficiency two) shows that a pattern seen in the census window, rank Q_4(K) = 4 at high deficiency (which we had formulated as a conjecture of our own), fails beyond that window: PARI-computed fields realize def_sig(K) = 5 and 6 with dim P(K) = 3 and 4. The deficiency-five witness with dim P(K) = 3, over Q(sqrt 2), is fully certified by bnfcertify. The deficiency-six witness over Q(sqrt 55) is GRH/PARI-conditional. When all three quartic layers have deficiency two, inflation bounds give def_sig(K) <= 6 and dim P(K) <= 4; their sharpness is computational (unconditional at deficiency five, conditional on GRH and PARI at deficiency six), and the full deficiency spectrum {2, ..., 6} is realized. A genus-character orientation conjecture remains open. The source archive contains the LaTeX source, bibliography, verification scripts and their outputs.