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Manuscript VI@1 · VI.tex · figure · b5b60afb

Plateau-switch obstruction

An exact compact-switch obstruction › A two-representative witness with a plateau switch · lines 1591–1609

VI@1#fig:VI-plateau-witness

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted

schematic nested supports

supp⁡χξ\operatorname{supp}\chi_\xi
χξ=1\chi_\xi=1 near supp⁡g\operatorname{supp} g
supp⁡g\operatorname{supp} g, with g=1g=1 near supp⁡f\operatorname{supp} f
supp⁡f\operatorname{supp} f
ξ=χξ∂1\xi=\chi_\xi\partial_1. Derivatives of χξ\chi_\xi may be nonzero outside a neighborhood of supp⁡g\operatorname{supp} g.
gKξ=0,ϕ1,g[Kξ,ϕ0]=0,w[Kξ]=ξgK\xi=0,\quad \phi_{1,g}[K\xi,\phi_0]=0,\quad w[K\xi]=\xi
∂1ϕ0≥12,f≥0,∫f=1\partial_1\phi_0\geq\tfrac12,\quad f\geq 0,\quad \int f=1A(1)(Kξ)−A(1)(0)=−∫f∂1ϕ0≤−12\mathcal A^{(1)}(K\xi)-\mathcal A^{(1)}(0)=-\int f\partial_1\phi_0\leq -\tfrac12same f,g,ϕ0f,g,\phi_0 for both representatives

Plateau-switch obstruction. The gauge parameter is a constant translation near the entire switch support, so and both switched forced responses vanish. The unswitched coordinate compensator nevertheless changes by . A unit-integral smearing where gives a gap at most , despite near the observation.

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