Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy
We develop the cohomological structure of a family of linearly constrained Kloosterman-type exponential sums in arbitrary dimension. The phase is analyzed through its Newton polytope and the polytope at infinity, whose distinct normalized volumes govern respectively the critical-point geometry and compactly supported cohomology. After a face-by-face non-degeneracy argument, we determine concentration, rank, boundary contribution, Swan conductors, lissité, and the full weight filtration; the top-weight rank is $2^n-\binom{n}{\lfloor n/2\rfloor}$. We then prove the coordinate-boundary tameness needed for middle convolution by a local Fourier-transform argument, compute the tame unipotent Jordan blocks, and identify the restriction of the top-weight sheaf with the middle convolution. A complete fourth-moment calculation based on the Cayley cubic gives $M_4=2$ for $n\ge3$; together with the nontrivial boundary unipotent and Larsen's alternative this yields $G^0_{\mathrm{geom}}(\mathcal W_n)=\mathrm{SL}_{r(n)}$ for every $n\ge2$ in the stated characteristic range. The argument avoids finite-group classification.