Exact reset with its correlations retained

Expanding the positive integrand in (7.2) gives

It is continuous and strictly increasing on $[0,1/2]$. At zero it is $1$. At $1/2$ its first four terms alone exceed $3/2$ by $11015/8388608$. This proves existence and uniqueness. Since every sine moment is at most one, the remainder after $k=N$ is bounded by

Positive rational sums and bisection therefore locate the exact coefficient to any required finite accuracy. The theorem uses the exact root; measured calibration errors are a separate comparison.

For positivity write $u=\sin^2(\tau/2)$, so $(\sin^8(\tau/2))''=2u^3(7-8u)$. If $b''\leq0$, then $\nu^2\geq1$ immediately. Otherwise $u>7/8$, $b''\leq2\alpha$, and

This again gives $\nu^2\geq1$. Also $b\geq1/2$ and $|b''|\leq1$, so $\nu^2\leq16+2=18$. The function $1-b$ vanishes to order eight at both endpoints, so $\nu^2-1$ vanishes to order six. Static zero extensions give the asserted $C^5$ interaction.

In normal coordinates $q_\pm=(x\pm r)/\sqrt2$, the plus mode has frequency $1$, while the minus mode has frequency $\nu$. The exact Ermakov relation is $b''+\nu^2b=b^{-3}$. Let $\theta'=b^{-2}$ and $\theta(0)=0$. For an oscillator eigenfunction $f_k$, the minus-mode solution is

Direct substitution verifies this formula. At the endpoint $b=1$, $b'=0$ and $\theta=3\pi$, so the minus operator is $i$ times parity. The plus-mode duration $2\pi$ operator is $-I$. Relative parity exchanges $x$ and $r$, proving (7.4) on the oscillator basis and hence on all $L^2$ by unitarity. Tensoring with any passive Hilbert space proves the entangled-input assertion.