Obstruction Hierarchy for the Hodge Conjecture on Degree-7 Fermat Fourfolds

We investigate the Hodge conjecture for Fermat fourfolds of degree d in projective 5-space via the character decomposition of the primitive middle cohomology under the natural cyclic symmetry group. A complete census for d up to 13 shows that d=6 is the last fully-split case; for d=7, exactly 12 non-split characters appear in two orbits of types (1,4,4,4,4,4) and (3,3,3,3,3,6). We establish a seven-level obstruction hierarchy proving that these characters resist all standard algebraic constructions: split products, Tate-Galois arguments, algebraic correspondences, complete intersections, Segre products, ambient intersections (the Lefschetz wall), and Newton-Vieta determinantal factorizations (the Hilbert-Burch obstruction). We introduce a dihedral signature filter that cleanly separates Hodge-type characters from non-Hodge-type characters for odd prime degree. We detect the target Hodge motive on the open torus over finite fields (q=29, 43, 71) using character sums with proved closed forms, and explain how the arithmetic 7th-power sieve keeps it visible over finite fields while complex projective closure destroys it. The sharp remaining question is to produce a polynomial identity among six 7th powers with 1+5 asymmetry whose elimination ideal is genuinely non-complete-intersection. Supplementary materials include Python verification scripts and Macaulay2 code reproducing all computational results.