Section 23 4 October 2026

Copy, whole hold, and the direct path bound

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

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23 Copy, whole hold, and the direct path bound

Appendix E proves the complete copy and holding inequalities. The exact moving annihilator is

A=\partial_Z+Z-\Pi_1\bigl(b+\ii\mu_Zb'\bigr).

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

(\ii\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). 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 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

\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.

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.