The Mirror Clause of the Geometric Selection Conjecture Is Empty
The Geometric Selection Conjecture asks for a functional on the heterotic landscape, invariant under mirror symmetry, whose unique minimum is a phenomenologically viable vacuum. It was shown to be equivalent to the existence of a viable vacuum fixed by the mirror involution wherever that involution is defined. We prove that no viable vacuum is ever fixed: a fixed vacuum is isomorphic to its mirror, so its threefold has Euler number zero, the third Chern class of its bundle vanishes on its component, and by Hirzebruch-Riemann-Roch it has no net families. The conjecture is therefore equivalent…
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