λEnter Calculus CompanionRegistered manuscripts, kept distinct

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

Sourced solution-jet family

The linearized interface and a sourced nonempty domain › A sourced family in the coefficient space · lines 813–834

VI@1#fig:VI-sourced-family

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
ϕ0\phi_0
Q=14T(η,ϕ0)Q=\tfrac14 T(\eta,\phi_0)
(ρ,Ji)(\rho,J_i)
a=−2Δ−1ρ,Vi=2Δ−1Jia=-2\Delta^{-1}\rho,\qquad V_i=2\Delta^{-1}J_iBij=∂iVj+∂jVi−δij∂ℓVℓB_{ij}=\partial_i V_j+\partial_j V_i-\delta_{ij}\partial_\ell V_\ell
prescribed trace-reversed data□ηjˉ=−2Q\Box_\eta \bar j=-2Q
D(j)(0)=0,∂tD(j)(0)=0,□ηD(j)=0⇒D(j)=0D(j)(0)=0,\quad \partial_t D(j)(0)=0,\quad \Box_\eta D(j)=0 \Rightarrow D(j)=0
G(1)(j)=Q,Pϕ2=−Ljϕ0,h=ϕ1=0G^{(1)}(j)=Q,\quad P\phi_2=-L_j\phi_0,\quad h=\phi_1=0

schematic Newton tails, not compact metric support. Explicit family, not a general constrained Cauchy theorem.

A nonempty sourced solution-jet family. Conserved scalar stress determines Newton initial data and the second-order metric forcing. The harmonic constraint has zero Cauchy data and propagates homogeneously, giving . The construction permits Newton tails and establishes a family rather than a general constrained existence theorem.

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