# Section 23: Copy, whole hold, and the direct path bound

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## 23 Copy, whole hold, and the direct path bound



Appendix [E](/quantum-measurement/research/nonequilibrium-records/appendix-e-copying-and-retention-for-the-exact-loaded-wave#hold:main) proves the complete copy and holding inequalities. The exact moving annihilator is 

$$

 A=\partial_Z+Z-\Pi_1\bigl(b+{\mathrm i}\mu_Zb'\bigr).

$$

 With $c=b+{\mathrm i}\mu_Zb'$ and the characteristic variables, its forced equation has the sole finite-clock source 

$$

 ({\mathrm i}\partial_\tau-\widetilde H-1/\mu_Z)A\Psi
 =-\epsilon\Pi_1(c'\partial_s+c''/2)\Psi,
 \qquad \epsilon=\frac{\hbar}{Mv^2T_0}.

$$

 In the completed-clock region the pointer current is controlled by $\operatorname{Im}(\Psi^\dagger A\Psi)/\mu_Z$. Positive inner pointer regions $[-8,8]$ and $[16,32]$, followed by soft barriers to $R_0,R_1$, give an entire-interval estimate; recrossings are charged by absolute variation. A soft radial classifier on the same exact trajectories connects the conserved-sector proxy to the actual earlier position label. It does not replace that label by a sampled logical bit.

The copy proof retains its baseline radial band current, finite-clock radial correction, positive pointer-inner event, entire hold current, and the shared clock-core, clock-deviation and radial-exit exceptions. The hold-current term is below $3.272\times10^{-9}$; the full sum is [(10)](/quantum-measurement/research/nonequilibrium-records/observable-and-theorem#rad:eq:copy). Every term concerns the new Hamiltonian and original entrance law.

In the union of failed earlier-label transfer and failed record history, the same three exceptional events need be charged only once. Their identity is explicit in both proofs; this is a union of events, not subtraction of unrelated numerical upper bounds. The outward arithmetic in Section [26](/quantum-measurement/research/nonequilibrium-records/outward-arithmetic-and-the-complete-event-sum#sec:events) gives 

$$

 p_{\rm union}<0.003879184824.

$$

 The preserved reference earlier-label estimate, including both original $0.0001$ allowances, is $e_{\rm label,26}=0.002207585289$. Coupling first to the correct constant record of $L_{\mathrm{per}}$ and then to its target Bernoulli label proves 

$$

 {\operatorname{TV}}(\mathcal L(Y),\mathcal L(B,(B)_{t\in I}))
 \leq p_{\rm union}+e_{\rm label,26}<0.006086770113.

$$

 This establishes the whole observable of Theorem [19.1](/quantum-measurement/research/nonequilibrium-records/observable-and-theorem#rad:thm:main).

The former comparison to the entire imperfect periodic reference apparatus has the different sufficient upper expression 

$$

 p_{\rm union}+h_{\mathrm{per}}<0.006206439621,\qquad
 h_{\mathrm{per}}=0.00232725479707784720002304.

$$

 That expression does not meet the old target $0.00556515991392215279997696$. Its failure is not an actual error lower bound. The theorem uses the expressly permitted direct ideal joint-path alternative.