arXiv2026
In this work, we study the symmetric hit problem in four variables by analyzing the relative cohit space, defined as the kernel of the inclusion-induced map from symmetric to ordinary cohits. We establish that this relative space is isomorphic to a kernel in the first group cohomology of the permutation group and, equivalently, to an $\operatorname{Ext}^1$ kernel for the general linear group, equipped with a canonical action of the normalizer quotient. We use explicit Lucas--Cartan identities to construct unconditional fixed Steenrod preimages for nontrivial stabilizers, resolving infinite families of monomials strictly above the minimal-spike weight. Using the ordinary and symmetric Kameko isomorphisms, we reduce the determination of complete bases in two generic degree families to exact calculations in degrees one, three, six, ten, and sixteen. The relative space first becomes nonzero in degree fifteen, where it is one-dimensional, and it is an indecomposable two-dimensional normalizer module in degree eighteen. In the family $5\cdot2^t-4$, the symmetric cohit dimension stabilizes at six and the relative dimension at one for $t\geq2$; in the family $7\cdot2^t-4$, they are four and zero for $t\geq1$. We give complete bases by explicit exponent formulas and prove Janfada's monomial implication [7, Conjecture 1] in every degree of both families. We extend the known optimal bound for ordinary Kameko isomorphisms to every permutation-invariant cohit space, prove its optimality there, and transport the entire symmetric hit basis by a direct-sum formula.