  The genuine Kaluza-Klein-like theories--with no fields in addition to
gravity--have difficulties with the existence of massless spinors after the
compactification of some space dimensions \cite{witten}. We proposed (Phys.
Lett. B 633 (2006)771) such a boundary condition for spinors in 1+5
compactified on a flat disk that ensures masslessness of spinors in d=1+3 as
well as their chiral coupling to the corresponding background gauge field
(which solves equations of motion for a free field linear in the Riemann
curvature). In this paper we study the same toy model: M^{(1+3)} x M^{(2)},
looking this time for an involution which transforms a space of solutions of
Weyl equations in d=1+5 from the outside of the flat disk in x^5 and x^6 into
its inside, allowing massless spinor of only one handedness--and accordingly
assures mass protection--and of one charge--1/2--and infinitely many massive
spinors of the same charge, chirally coupled to the corresponding background
gauge field. We reformulate the operator of momentum so that it is Hermitean on
the vector space of spinor states obeying the involution boundary condition.
