Section 7 4 October 2026

The writer's exact driven comparison

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131081

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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 gives the polynomial derivative and envelope integrals explicitly. Keeping the Gaussian polynomial before taking norms gives

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

Appendix A derives these jet estimates from the two coherent-Gaussian derivative polynomials, with explicit coefficient bounds. Harmonic transport inverse engineering is established prior art [12].