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Manuscript V@1 · V.tex · figure · e7ddf37d

Residual witness and generic failure

Classification on the spacelike-compact solution spaces › Compact-data witness and generic failure · lines 1475–1507

V@1#fig:residual-generic-failure

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
Initial data
ϕ(0)=0, ϕ˙(0)=F\phi(0)=0,\ \dot\phi(0)=Fξ1(0)=0, ξ˙1(0)=∂1F, □ξ1=0\xi^1(0)=0,\ \dot\xi^1(0)=\partial_1 F,\ \Box\xi^1=0

Other vector components vanish. Data are compact and smooth.

Short-time witness
schematict = 0w > 0
w=ξ1∂1ϕ=t2(∂1F)2+O(t4)w=\xi^1\partial_1\phi=t^2(\partial_1 F)^2+O(t^4)
Positive response
Braw(Kξ)=BR(Kξ)=B0(Kξ)=∫fw>0B_{\mathrm{raw}}(K\xi)=B_R(K\xi)=B_0(K\xi)=\int fw>0

Nonnegative future smearing on the region where the witness is positive.

In DQ\mathscr D_Q, the kernel of f↦∫fwf\mapsto\int fw is a closed proper subspace. Its complement is open dense.

Fixed specially constructed scalar. Not a theorem for every scalar background.

Residual witness and generic failure. Compact smooth Cauchy data give a residual pure gauge with positive response for a suitable future smearing. For the fixed scalar and vector wave, the nonzero continuous functional supplies an open dense family of obstructed smearings in the stated Fréchet test space. No genericity claim is made for every scalar background.

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