Guralnick's 1985 conjecture holds that the dimension of first cohomology H^1(G,M) is bounded by an absolute constant for all finite groups G and faithful absolutely irreducible modules M. Earlier computations of affine Kazhdan–Lusztig mu-coefficients pointed to growth with rank, but they were too costly to extend.

Annular webs serve as graphical guidance for computing affine Kazhdan–Lusztig coefficients, with web Hom spaces identified with Hecke and Schur maps. An explicit affine family of elements indexed by m is constructed, and its mu-coefficients are read off from canonical webs. A Lean development encodes the chosen web as a labeled graph and proves that it evaluates to the six-map morphism used. It also formalizes the row-sum identity (declarations row_sum and named_row_and_lower_bound).

The mu-coefficients in the family sum to m(m+1)/2, which gives dim H^1 ≥ ⌈(m+1)/2⌉. First cohomology dimensions are therefore unbounded, already for finite simple groups of type A, which disproves Guralnick's conjecture.