Copying the actual earlier label

14 Copying the actual earlier label

Let $B_X$ be the union of the intervals at relative distances $h/5$ to $3h/10$ between adjacent zero and one wells. Let $H_X\in[0,1]$ be the quintic soft classifier, equal to the binary label outside these bands, with $|H_X'|\le20/h$.

During $[t_a,t_b]$, both $\Phi_i$ and $\partial_X\Phi_i$ vanish in these bands. Exact sector factorization gives

For clarity $Q$ in the last product denotes the fixed number $.003983$; $\E_Q$ denotes wave-reference expectation.

Use $I_0=[-8,8]$, $I_1=[16,32]$ and define

For any continuous paths,

The comparison writer has completed its drive for every retained offset at $t_b$ and its wrong-inner-region amplitude is below $10^{-12}$. With the exact auxiliary error $e_A<2\times10^{-13}$,

Theorem 14.1 (Actual historical copy)

Uniformly over the original actual class and all qubit inputs,

$$\boxed{\epsilon_{\rm copy} =16\left[D^2+\frac65DQ+(D+2\times10^{-13}+10^{-12})^2\right] =.00222725479707774720002304.}$$

This bounds failure to copy the actual coupled relative rotor label at $1.22$ into the pointer at $1.28$.

The internal sector in this proof is an orthogonal wave component, not a sampled actual spin. On the full wave the pointer velocity depends on the actual relative rotor coordinate through its branch weights.