Pure Spikes: Lefschetz Obstructions, Contact Modules and Big Algebras

We study trace-free multiplication by the infinitesimal representation on polynomial covariants of irreducible symplectic representations with one-dimensional weight spaces. Surjectivity is equivalent to a single-monomial identity for generalized exponents, called a pure spike, and to the vanishing of a signed Lefschetz obstruction. We classify the pure representations as principal-nilpotent-preserving restrictions of non-type-A contact modules. A least-denominator construction gives the polynomial kernel, while a Kostant-section matrix, primitive degrees and weight-line data determine minimal resolutions and divisorial support of the cokernel. Examples with identical generalized-exponent differences have different support schemes. The same kernel constructs symmetric invariants of associated Heisenberg extensions and separates symmetric from unrestricted tensor detection. Beyond weight multiplicity one, the first two invariant-derivative layers generate Hausel's adjoint big algebra in every simple type; the Saito metric proves independence of the second-layer primitives and produces a polynomial primitive projection. Shift-of-argument methods give universal containment through derivative order three and counterexamples at order four. A Casimir-corrected quantum operator lifts pure exact sequences and is injective on the negative signed part of every simple adjoint big algebra. We compute the signed cokernels in types A₂, C₂ and A₃, prove equality of the big and full quantum family algebras for weight-multiplicity-free representations, and establish signed preservation for the 26-dimensional F₄ module. Finally, higher invariant-gradient operators yield explicit rank-one and rank-two cokernels with contrasting support, while the minuscule cases admit an interpretation through equivariant cohomology and hard Lefschetz theory.