# Section 15: Whole-interval retention, including later rotor evolution

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<a id="section-15"></a>

## 15 Whole-interval retention, including later rotor evolution

 The outer regions are $\mathcal R_0=[-10,10]$ and $\mathcal R_1=[14,34]$. Four width-two bands separate them from the corresponding inner regions. After capture, the auxiliary comparison center stays within $.012$ of its intended value and has speed below $.07$. Its band wave and momentum norms are below $10^{-14}$ and $10^{-13}$.

The exact rotor waves are used during retention; no harmonic comparison is continued through a caustic. Their unit norms suffice. With $e_A=2\times10^{-13}$, $p_A=3\times10^{-9}$, $\mu_-=.00999$, the total actual probability of leaving after starting in an inner region is bounded by 

$$

 16\cdot4\cdot\frac{15}{16}\cdot\frac{43}{25}
 \left[.07\,10^{-28}
 +\frac{e_A10^{-13}+10^{-14}p_A+e_Ap_A}{\mu_-}\right]
 <10^{-16}.

$$

 This uses band-integrated absolute current and is simultaneous over $[1.28,3]$; Appendix [A](/quantum-measurement/research/nonequilibrium-records/appendix-a-explicit-estimates-for-the-periodic-apparatus#app:periodic) supplies the cutoff and Gaussian-tail calculation. Later rotor revival or source phase evolution is not frozen. The same-support actual-law neighbourhood adds $10^{-4}$ once to the complete history event.