λEnter Calculus CompanionRegistered manuscripts, kept distinct

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

Nonzero sourced classical response

A nonzero sourced classical second-order response › Meaning of the example · lines 1821–1842

VI@1#fig:VI-sourced-positive

Back to manuscript
metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
ϕ0(0)=F,ϕ˙0(0)=0,F=A>0 near 0\phi_0(0)=F,\quad \dot\phi_0(0)=0,\quad F=A>0\text{ near }0
ρ=18(∣∇F∣2+m2F2),a=−2Δ−1ρ,a(0)>0\rho=\tfrac18(|\nabla F|^2+m^2 F^2),\quad a=-2\Delta^{-1}\rho,\quad a(0)>0
h=ϕ1=w=v=0,Pϕ2=−Ljϕ0h=\phi_1=w=v=0,\qquad P\phi_2=-L_j\phi_0
ϕ2(t,0)=14a(0)m2A t2+O(t3)\phi_2(t,0)=\tfrac14 a(0)m^2 A\, t^2+O(t^3)

schematic positive future neighborhood, not computed data

f≥0⇒Of[2]>0f\geq 0\Rightarrow \mathcal O_f^{[2]}>0 for a suitable nonzero smearing
source-off branch
zero source, plus zero homogeneous metric and scalar-correction data
j=ϕ2=0j=\phi_2=0

Special classical family, not a universal quantum cubic identity.

Nonzero sourced classical response. For the chosen harmonic family, the relational coefficient reduces to , with a positive short-time leading term. The metric is determined by scalar stress rather than prescribed independently. Setting the source and the specified homogeneous data to zero gives the exact source-off control; no quantum or radiative identification is implied.

Nearby in this section