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

Initial-time-edge removal

Model 1: the fixed vacuum TT plane › Removing the initial-time edge · lines 1195–1229

IV@1#fig:edge-removal

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gravitationalscalaroperator / instrumenterror / restrictionsupport / scope
(a)Temporal smoothing
t0t0+ε
schematic multiplier profiles, not computed data

dashed step I{t≥t0}I_{\{t\ge t_0\}}; solid ϑ((t−t0)/ε)\vartheta((t-t_0)/\varepsilon)

(b)Coefficient and unitary estimates
∥ua,f,ε−ua,f∥≤Ba,fε\lVert u_{a,f,\varepsilon}-u_{a,f}\rVert\le B_{a,f}\varepsilon
∥(Uε−U)Ψ≤n∥≤2Dε∣τ∣(n+1)(1−2C‾∣τ∣)n+2∥Ψ≤n∥\lVert(U_\varepsilon-U)\Psi_{\le n}\rVert\le \frac{2D_\varepsilon|\tau|(n+1)}{(1-2\overline C|\tau|)^{n+2}}\lVert\Psi_{\le n}\rVert
small-time condition 2C‾∣τ∣<12\overline C|\tau|<1
(c)Coherent truncation
Poisson tail
finite-sector edge error
proof order: choose nn, then remove ε\varepsilon
arbitrary fixed time: finite subdivision
convergence is strong, not operator norm

No gravitational-sector limit is taken.

Initial-time-edge removal in Model 1. Smoothing changes the scalar coefficients by . Finite-sector analytic estimates and coherent Poisson tails yield strong unitary convergence. The displayed estimate is a small-time bound; arbitrary fixed times are handled by subdivision. No gravitational-sector limit is taken.

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