Preparation and prewitness comparison

21 Preparation and prewitness comparison

Appendix B proves the preparation estimates used here. The writer's presence during preparation is paid by a remote-support Duhamel estimate and weighted oscillator hierarchy, from the original stock. It gives the handoff norm change below $2.768\times10^{-30}$, first moving-annihilator norm below $5.050\times10^{-24}$, and second annihilator norm below $1.229\times10^{-17}$. The strengthened handoff clock-error derivative is

The proof uses ordinary fourth spatial derivatives of the smooth preparation error, with nonzero Gaussian tails and cutoff derivatives retained. It never assumes the capped reference entrance belongs to a fourth radial Hamiltonian graph domain.

The preparation history itself is rederived with the original explicit Gaussian material rank $K(S,r)$. Its $Z$ derivative is zero, but its clock and radial currents are those of the new exact wave. Both phase restoration corrections are included. For $N=256001$ rank cuts, including the exterior ray, the preparation event and handoff conditional-CDF bridge are respectively below

For the later prefix, use the constant Galilean frame for both waves: $\Psi_b=e^{-\ii MvS/\hbar+\ii Mv^2t/(2\hbar)}\Psi$ and $H_b=H-\ii\hbar v\partial_S$, with the identical scalar terms. Let $\psi_{\rm old}$ denote the exact old wave in that frame, with $\psi_{{\rm old},i}=\Pi_i\psi_{\rm old}$ including its qubit coefficient. Define the normalized Weyl comparison $G_*=V(S)(\psi_{\rm old}g_0)$, with sector-one factor

and sector-zero factor $g_0$. Direct differentiation gives the exact residual

It is supported only on the pulse. The displaced stationary plateau has no residual source. Keeping the entire old clock tail and the leading edge of its characteristic stock gives

This norm estimate is not used as a trajectory-agreement assertion.

For $\delta=\Psi_{\rm new,b}-G_*$, two fixed left localizations and the oscillator moment hierarchy give

The differentiated residual includes the coefficients $1,5/2,2,1/2$ on $\gamma_S\psi_{SSS}$, $\gamma_{SS}\psi_{SS}$, $\gamma_{SSS}\psi_S$, and $\gamma_{SSSS}\psi$. Its old-wave input is

On the remote preparation-force support the characteristic helper is identically zero, so propagation of this third clock derivative uses the already bounded old error derivatives there. All activation terms elsewhere remain.