# Section 7: The writer's exact driven comparison

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<a id="section-7"></a>

## 7 The writer's exact driven comparison

 Remove the common Galilean phase and set $\xi=S-v\lambda t$, $a=\lambda t+\xi/v$. For sector one let 

$$

 q''+\Omega^2q=\Omega^2b(a)+b''(a),\quad q(0)=q'(0)=0,
 \qquad\Omega=1/\mu.

$$

 The normalized comparison is 

$$

 G_1=\varphi(\xi)e^{-it/(2\mu)}
 \exp\{i[\mu q'(y-q/2)+\vartheta]\}\gamma(y-q),

$$

 

$$

 \vartheta'=\frac\mu2[\Omega^2b+b''](q-b).

$$

 Sector zero has $q=\vartheta=0$. Direct substitution proves the fibre Schrödinger equation when the clock's residual kinetic term is omitted. For $\lambda=1$, $q=b$ and $\vartheta=0$ exactly.

The difference $e=q-b(a)$ solves 

$$

 e''+\Omega^2e=(1-\lambda^2)b''(a).

$$

 Sine/cosine integration gives $|e|<.001$, $|e'|<.07$ on the complete finite rectangle. Derivatives obey 

$$

 |q|<25,\ |q'|<1630,\ |q''|<170000,\ |q'''|<36000000.

$$

 The clock-envelope derivative norms are below $6,112,4000,190000$. Appendix [A](/quantum-measurement/research/nonequilibrium-records/appendix-a-explicit-estimates-for-the-periodic-apparatus#app:periodic) gives the polynomial derivative and envelope integrals explicitly. Keeping the Gaussian polynomial before taking norms gives 

$$

 \boxed{{\left\lVert{\partial_\xi^2G_i}\right\rVert}<110000,\qquad
 {\left\lVert{p_Z\partial_\xi^2G_i}\right\rVert},\ 
 {\left\lVert{(Z-z_0)\partial_\xi^2G_i}\right\rVert}<3\times10^6.}

$$

 Appendix [A](/quantum-measurement/research/nonequilibrium-records/appendix-a-explicit-estimates-for-the-periodic-apparatus#app:periodic) derives these jet estimates from the two coherent-Gaussian derivative polynomials, with explicit coefficient bounds. Harmonic transport inverse engineering is established prior art [[12](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-Tor)].