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

Raw-current reconstruction

Raw current and complete gauge defect › Distributional reconstruction · lines 729–760

V@1#fig:raw-current

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
Truncated causal support
schematict = 0tK_fsupp f
u=−GAf,V=Θ(t)uu=-G_Af,\quad V=\Theta(t)u
Braw(h)=∫0Tu Lhϕ=⟨Jraw,h⟩B_{\mathrm{raw}}(h)=\int_0^T u\,L_h\phi=\langle J_{\mathrm{raw}},h\rangleJrawμν=∂(μV ∂ν)ϕ−12ημν∂α(V∂αϕ)J_{\mathrm{raw}}^{\mu\nu}=\partial^{(\mu}V\,\partial^{\nu)}\phi-\tfrac12\eta^{\mu\nu}\partial_\alpha(V\partial^\alpha\phi)
Jraw=Θ(t)Jbulk+δ(t)jΣJ_{\mathrm{raw}}=\Theta(t)J_{\mathrm{bulk}}+\delta(t)j_\SigmajΣ00=12u0ϕ˙∣0,jΣ0i=−12u0∂iϕ∣0,jΣij=12δiju0ϕ˙∣0j_\Sigma^{00}=\tfrac12 u_0\dot\phi|_0,\quad j_\Sigma^{0i}=-\tfrac12 u_0\partial_i\phi|_0,\quad j_\Sigma^{ij}=\tfrac12\delta^{ij}u_0\dot\phi|_0
Compact order-zero distribution. Not yet a smooth conserved test tensor.

Raw-current reconstruction. Truncation of the advanced scalar solution at the initial surface produces a compact bulk density and a slice measure. The latter is part of the fixed-time response and cannot be omitted. Compactness and order zero alone do not establish conservation or smooth FH representability.

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