Algebraic Shifting Can Destroy Every Polynomial HVC Bound

Kalai asked whether algebraic shifting preserves homological VC dimension, a uniform polynomial bound on total Betti numbers in a family closed under induced subcomplexes and links. We give a full negative answer to the unrestricted assertion in Oberwolfach Report 32/2004, p1730, Conjecture 3, recorded as OWR-734-007 in UnsolvedMath v1.6.0. A strongly hereditary family of minimum polynomial HVC exponent exactly 1, with constant 3, has after either exterior or symmetric algebraic shifting over the rationals a strongly hereditary closure with no finite polynomial HVC exponent. For a shift of the r-dimensional cross-polytope boundary, the link of the first s vertices has 2r-s vertices and top reduced Betti number binom(r,s); central links therefore grow exponentially. The proof uses established shifting properties and the classical shifted facet interpretation of h-vectors, which are credited. The example has unbounded simplicial dimension and does not resolve a fixed-dimensional modification or neighboring fractional-Helly and nerve conjectures. This four-page English preprint includes a standard-library checker with 4,317 exact checks. Developed with AI assistance and self-audited, it is unrefereed, without independent review, proof-assistant verification or absolute-priority certification.