Section 19 4 October 2026

Observable and theorem

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

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19 Observable and theorem

Let $L$ be the frozen radial detector classifier evaluated at the actual radius at $t_a=0.0366\,\mathrm s$ , with the inherited cemetery outcome for a failed earlier event. Set

R_0=[-10,10],\qquad R_1=[14,34],\qquad R(Z)=\begin{cases}0&Z\in R_0,\\1&Z\in R_1,\\ \dagger&\text{otherwise.}\end{cases}

The complete holding interval is $I=[0.0384,0.09]\,\mathrm s$ and the observable is

Y=\bigl(L,(R(Z_t))_{t\in I}\bigr).

An incorrect or undefined record at any holding time is a failure. An undefined earlier label is a failure even if a cemetery record could be assigned the same symbol. No time grid replaces the interval.

Theorem 19.1 (Effective joint-path theorem)

For Hamiltonian (7), the full entrance stock above, every normalized qubit $\alpha_0|0\rangle+\alpha_1|1\rangle$ , and every admitted original reference/joint auxiliary law, there is the unchanged compatible reference coupling for which

\begin{align}\Prob(L\neq L_{\mathrm{per}})&<0.003331233001,\tag{9}\\ \Prob\bigl(L\text{ fails or }\exists t\in I:R(Z_t)\neq L\bigr) &<0.000552421956. \tag{10}\end{align}

Writing $p=|\alpha_1|^2$ and $B\sim\mathrm{Bernoulli}(p)$ only as an ideal comparison variable,

\TV\!\left(\mathcal L(Y), \mathcal L\bigl(B,(B)_{t\in I}\bigr)\right) <0.006086770113<0.01.

(11)

The microscopic dynamics contains no sampled sector variable $B$ .

Theorem 19.1 is about one effective scalar Hamiltonian. It is not a claim that the finite source candidate realizes this Hamiltonian. The difference to $0.01$ is an unachieved sufficient allocation for a future compatible microscopic comparison, not a measured or proved error of added hardware.