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

Harmonic prescriptions and boundary pairing

Classification on the spacelike-compact solution spaces › Section current and its boundary pairing · lines 1036–1071

V@1#fig:harmonic-boundary

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
wR=G1,RD(h),sreth=h−KwRw_R=\mathsf G_{1,R}D(h),\quad \mathfrak s_{\mathrm{ret}}h=h-Kw_Rdeclared retarded domain
□w0=D(h),Data0w0=0,s0h=h−Kw0\Box w_0=D(h),\quad \mathrm{Data}_0 w_0=0,\quad \mathfrak s_0 h=h-Kw_0

Both yield a harmonic image.

Residual wave KξK\xi, □ξ=0\Box\xi=0, passes through both lanes unchanged:sret(Kξ)=s0(Kξ)=Kξ\mathfrak s_{\mathrm{ret}}(K\xi)=\mathfrak s_0(K\xi)=K\xiHarmonic prescriptions, not complete gauge-quotient sections.
A=G1,AH,Jsec=Jraw−D∗A,K∗Jsec=0A=\mathsf G_{1,A}\mathbf H,\quad J_{\mathrm{sec}}=J_{\mathrm{raw}}-D^*A,\quad K^*J_{\mathrm{sec}}=0⟨Jsec,θh⟩+⟨Aν,(∂μθ)hˉμν⟩\langle J_{\mathrm{sec}},\theta h\rangle+\langle A_\nu,(\partial_\mu\theta)\bar h^{\mu\nu}\rangle

The second pairing is a boundary term. Conservation alone does not remove it. Advanced tails may remain.

Harmonic prescriptions and boundary pairing. Compensators remove the harmonic defect on their stated domains but leave residual vector-wave gauges unchanged. The conserved section distribution can have advanced tails. Its cutoff pairing contains an explicit boundary term; conservation alone does not justify discarding that term.

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