All moving coordinates and finite parameters

3 All moving coordinates and finite parameters

Let $x,R\in\T_2$ be a detector rotor and a new reference rotor, and set $X=x-R\pmod2$. Let $Z,S,C\in\R$, and retain the original 255 differential source coordinates $u\in\one^\perp\subset\R^{256}$. The unknown normalized qubit/reference vector is $\psi$, with exactly conserved projectors $P_0+P_1=I$. Set $p=\norm{P_1\psi}^2$. The target projector $P_1^{\rm tar}$ is an orthogonal projector with $\norm{P_1-P_1^{\rm tar}}\le10^{-4}$, and $p_{\rm tar}=\norm{P_1^{\rm tar}\psi}^2$. Both act as the identity on any admitted inaccessible reference factor. That reference is an internal Hilbert-space factor and introduces no additional guidance coordinates. Thus $|p-p_{\rm tar}|\le10^{-4}$. The periodic classifier uses the centred cell coordinate $u=((X-o)/h)\bmod1\in[-1/2,1/2)$: it is zero for $|u|\le1/4$ and one otherwise, with a fixed boundary convention.

Set

Let $m= m_xm_R/(m_x+m_R)=\alpha K/h^2$. Independently choosing

ensures $\alpha\in[.999,1.001]$. Nominally $m=2^{52}$ and $m_x=m_R=2^{53}$. The other masses are

The retained source has $\kappa_s=2.5\,10^{43}$ and frequencies $\Omega_k=10^{13}\sin(\pi k/256)$. It is a spectator Hamiltonian, not a field generator in this redesign.

Let $S_5(z)=10z^3-15z^4+6z^5$. The even period-one $U_{\eta_p}$ equals $z^2$ for $|z|\le1/2-\eta_p$ and has positive-cap derivative

It is $C^3$, nonnegative, and bounded by $1/4$. Define

The common origin $o$ is arbitrary. The relative sector offsets and gains can vary independently in these intervals; arbitrary nonperiodic site defects are not admitted.

For $B_9(a)=126a^5-420a^6+540a^7-315a^8+70a^9$ on $[0,1]$, with constant extensions, put

The finite writer family is

Its launch momentum is $Mv\lambda$. Spring coefficient $1/\mu$ and the scalar compensation are matched to the selected $\mu$. General symmetry-breaking or independently mismatched compensation errors are outside this theorem.