Projective Units and Witt Vanishing for Stable Module Categories

Let G be a finite group and let k be any field of odd positive characteristic. We prove that the symmetric and skew-symmetric Cartan homomorphisms from the Witt groups of projective kG-modules to those of all finite-dimensional kG-modules are isomorphisms. The proof constructs a projective symmetric form whose module Witt class is the trivial line. An explicit middle-layer calculation supplies this form for groups with normal cyclic Sylow subgroups; Dress's classical 2-local induction theorem then gives it for arbitrary G. Tensoring by a projective lift of the unit inverts both Cartan maps. Balmer's kernel-cokernel formulas consequently imply W^i(kG-stab)=0 for every integer i. No splitting-field hypothesis is required. The scope is the ordinary contragredient duality in odd positive characteristic, not characteristic-two variants, Grothendieck-Witt groups, support restrictions or endotrivial twists. This is an AI-assisted, self-audited, unrefereed preprint. No independent expert review, proof-assistant verification or absolute priority is claimed.