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

Pair and number coefficient ideals

Model 2: spectral removal at fixed number cutoff › Pair and number estimates · lines 1881–1915

IV@1#fig:model2-kernel-ideals

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gravitationalscalaroperator / instrumenterror / restrictionsupport / scope
Hϵδ(p,q)=γΠ(p)(q⊗q)∣p∣EqJ(ϵp+δq,ϵ∣p∣+δEq)H^{\epsilon\delta}(p,q)=\gamma\frac{\Pi(p)(q\otimes q)}{\sqrt{|p|E_q}}J(\epsilon p+\delta q,\epsilon|p|+\delta E_q)
TT gain Π(p)(q⊗q)=Π(p)(k⊗k)\Pi(p)(q\otimes q)=\Pi(p)(k\otimes k)
time gain ∣J(k,ν)∣≤Cn⟨k⟩−n(1+∣ν∣)−1|J(k,\nu)|\le C_n\langle k\rangle^{-n}(1+|\nu|)^{-1}
initial endpoint retained
Pair branch
ν=−(∣p∣+Eq)\nu=-(|p|+E_q)
P∈HS,∥PR−P∥HS≤CPR−1/2P\in\mathrm{HS},\quad \lVert P_R-P\rVert_{\mathrm{HS}}\le C_P R^{-1/2}

Hilbert–Schmidt ideal. Asymptotic label only.

Mixed branch
transfer near q=pq=p; possible frequency cancellation
LL compact and bounded, ∥LR−L∥op≤CLR−1\lVert L_R-L\rVert_{\mathrm{op}}\le C_L R^{-1}
f≠0: ∥LR∥HS2≍R,L∉HSf\neq 0:\ \lVert L_R\rVert_{\mathrm{HS}}^2\asymp R,\quad L\notin\mathrm{HS}

No numerical convergence curve is drawn.

Different coefficient topologies in Model 2. TT projection and finite-time integration provide transfer gains. The pair coefficient converges in Hilbert–Schmidt norm. The mixed coefficient converges in operator norm to a compact operator, but for nonzero detector smearing its Hilbert–Schmidt norm diverges with the spectral cutoff.

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