bibkey: lerario2012convex authors: Antonio Lerario year: 2012 title: Convex pencils of real quadratic forms doi: null url: https://arxiv.org/abs/1106.4678v3 claim: Theorem 18 expresses reduced Betti numbers of spherical intersections of two homogeneous real quadratic constraints through relative Betti numbers of the positive-inertia filtration of their symmetric pencil. strata_touched: [] license: citation-only triage: anchor

Topology from a pencil of two quadratic forms

The cited version is arXiv:1106.4678v3, updated 6 September 2012; the first version was posted 23 June 2011. Theorem 18 in the appendix concerns Y = {x in S^n : q(x) in K}, where q has two homogeneous quadratic components and K is a closed convex cone. It uses homology over Z/2. For k < n−2, it expresses the reduced kth Betti number of Y as the sum of a relative zeroth and a relative first Betti number of the positive inertia sets on the polar circle. For K = {0}, the whole circle is used.

Quantum-volume section 219 applies the sphere formula with n = 4 and k = 0. Reduced b0 = 3 means four sphere components. The sign of the nonzero kernel-coordinate separates them into two equal gauge copies. The theorem permits singular intersections and does not itself establish regularity; section 219 proves regularity from simple pencil roots.

The source provides the topological counting tool, not the particular rational pencil certificate, its perturbation neighborhood, the channel-specific rank-one-defect processor, or the analytic sharp preparation with exact two-jets. The real-fiber counterexample in section 219 also does not establish an obstruction to complex preparations.