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Manuscript IV@1 · IV.tex · figure · 351cb2d7

Coefficients and numerical generator

Model 1: the fixed vacuum TT plane › The numerical self-adjoint generator · lines 1068–1100

IV@1#fig:scalar-to-generator

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gravitationalscalaroperator / instrumenterror / restrictionsupport / scope
ℓa,f=∫fSt0(a)ϕ0\ell_{a,f}=\int f S_{t_0}(a)\phi_0
vf(s)=∫sTsin⁡((t−s)Ωm)Ωmf(t) dtv_f(s)=\int_s^T \frac{\sin((t-s)\Omega_m)}{\Omega_m}f(t)\,dt
ba,f=−∂i∂j(aijvf),ga,f=I{t≥t0}ba,fb_{a,f}=-\partial_i\partial_j(a^{ij}v_f),\quad g_{a,f}=I_{\{t\ge t_0\}}b_{a,f}
(αg,βg)(\alpha_g,\beta_g)
ug=α^g2ω+iω/2 β^gu_g=\frac{\widehat\alpha_g}{\sqrt{2\omega}}+i\sqrt{\omega/2}\,\widehat\beta_g
active scalar space Vf=Span⁡C{u0,ue,ur}V_f=\operatorname{Span}_{\mathbb C}\{u_0,u_e,u_r\}
Formal, dashed bookkeeping
F^[1]=Φ(u0)+κ[QΦ(ue)+Π0Φ(ur)],κ2=0\widehat F^{[1]}=\Phi(u_0)+\kappa[Q\Phi(u_e)+\Pi_0\Phi(u_r)],\quad \kappa^2=0
Numerical, solid generator
A(1)(λ)=Φ(u0)+λ[QΦ(ue)+Π0Φ(ur)]A^{(1)}(\lambda)=\Phi(u_0)+\lambda[Q\Phi(u_e)+\Pi_0\Phi(u_r)]
same coefficients; no evaluation homomorphism κ↦λ≠0\kappa\mapsto\lambda\neq 0

Coefficient construction and numerical representation. The fixed-time response determines effective scalar smearings and three one-particle vectors. These coefficients enter both a first-order formal expression and a separately defined numerical self-adjoint polynomial. Numerical interpolation does not evaluate the nilpotent formal parameter.

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