Section 8 4 October 2026

Finite interacting clock, not an infinite-mass limit

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

Original PDF ↗Section Markdown Full Markdown All contents

Reading position 10 of 37

8 Finite interacting clock, not an infinite-mass limit

The exact moving-frame Hamiltonian retains $-\partial_\xi^2/(2M)$ . Consequently the comparison residual is $\partial_\xi^2G_i/(2M)$ . Its propagator is the full interacting $S,Z$ propagator.

Lemma 8.1 (Full-time auxiliary errors)

For $0\le t\le3$ and $M_-=.99\,10^{18}$ ,

\norm{\Xi_i-G_i}\le\frac{3(110000)}{2M_-}<2\times10^{-13},

\norm{p_Z(\Xi_i-G_i)} \le\frac{3[6\times10^6+15000(110000)]}{2M_-} <3\times10^{-9}.

Proof

The norm estimate is Duhamel with zero initial defect. For the derivative estimate solve the oscillator operator equations. The remaining $Z$ force has norm below $5000$ , so

\norm{p_ZU(t,s)r}\le\norm{p_Zr}+\norm{(Z-z_0)r} +(t-s)5000\norm r.

Apply this to the residual and integrate. Clock recoil is in $U$ , not prescribed away. No exponential in $\Omega t$ is paid, and no source phase enters these $S,Z$ derivatives.

□