Version 1 · earlier version; the current one is v2 · External review, round 1: minor issues

# Comparison of two ring clocks: joint readings, rates and environment coupling

- **Subproject:** 04-ilang-time
- **Package:** 04-clock-comparison
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-10

## Setup and assumptions

- **Objects.** Two single-object clocks $K_1,K_2$, the rest $R$, and in item 3 an environment part $E$. No two objects have the same type, so $\mathcal H_{\rm adm}=\mathcal H$ (1.11): every partition of joint places is an admissible reading, every post-reading state is admissible, and (2.14) applies.
- **Clocks.** For $i=1,2$ let $\omega_i:=2\pi/(N_i\lambda_{0i})$ and $Q_i:=\sum_{k=0}^{N_i-1}k\,\lvert f^{(i)}_k\rangle\langle f^{(i)}_k\rvert$, so $C_{K_i}=\omega_iQ_i$. This is (2.9) with $\alpha_m=0$, $\varepsilon=0$, $\ell_k=k$, hence $\theta_m=0$ and $e^{-iC_{K_i}\lambda_{0i}}=S_i$. By (2.12) with $c=0$ the cost spread is minimal. The reading of a clock is its place reading, $\Pi_m=\lvert m\rangle\langle m\rvert$.
- **Inputs** (02-clock@v2, quoted in question.md): (2.1) self-drivenness for $C_\partial=\mathbb 1_K\otimes B$; (2.4) reading statistics of a self-driven part; (2.9) the ring-clock contracts; (2.12) minimal spread; (2.13) the place distribution $p(m;\lambda)=\sin^2(\pi\lambda/\lambda_0)/[N^2\sin^2(\pi(\lambda/\lambda_0-m)/N)]$; (2.14) the view after a place reading, $V_K\mapsto\lvert m\rangle\langle m\rvert$.
- **Assumptions.** These are the setting of question.md. All contracts are independent of $\lambda$ (A5). Readings are applied at stated values of $\lambda$ (A8). Comparing readings at equal $\lambda$ uses the order of $\lambda$; whether that order can be detected (criterion 8) is not addressed. In item 3, (1.10) requires $A\otimes B$ to be a sum of pair terms. This holds, for example, when $E$ is a single object. Only the total $C$ enters below, so the booking convention of A5 plays no role.
- **Well-definedness (M6).** All results are reading probabilities or parameters of the contract. Common phases (1.2) drop out, and labels play no role because all types differ.
- **Notation.** $x_i:=\lambda/\lambda_{0i}$. A reading is *certain* at $\lambda$ if one outcome has probability $1$. $\lfloor\cdot\rfloor$ is the floor function.

## Derivation

### Step 1. Propagator of a minimal ring clock

Take one minimal ring clock with $N$ places, tick $\lambda_0$ and $\omega=2\pi/(N\lambda_0)$. We have $e^{-iC_K\mu}\lvert f_k\rangle=e^{-2\pi ik\mu/(N\lambda_0)}\lvert f_k\rangle$ and $\langle m'\vert f_k\rangle\langle f_k\vert m\rangle=N^{-1}e^{2\pi ik(m'-m)/N}$. The spectral decomposition then gives

$$
\langle m'\vert e^{-iC_K\mu}\vert m\rangle=u_N\Bigl(m'-m-\frac{\mu}{\lambda_0}\Bigr),\qquad
u_N(y):=\frac1N\sum_{k=0}^{N-1}e^{2\pi iky/N},\qquad
\lvert u_N(y)\rvert^2=P_N(y):=\frac{\sin^2(\pi y)}{N^2\sin^2(\pi y/N)} ,
\tag{4.1}
$$

where $P_N$ takes the value $1$ where the denominator vanishes. The modulus follows from the geometric sum, $\lvert u_N(y)\rvert=\lvert1-e^{2\pi iy}\rvert/(N\lvert1-e^{2\pi iy/N}\rvert)$, together with $\lvert1-e^{i\phi}\rvert=2\lvert\sin(\phi/2)\rvert$. Both $u_N$ and $P_N$ are $N$-periodic, so (4.1) depends on $m'-m$ only modulo $N$; $P_N$ is even. For $m=0$, $P_N(m'-\mu/\lambda_0)$ is (2.13).

By the triangle inequality, $\lvert u_N(y)\rvert\le1$. Equality holds iff all $w^k$, $w=e^{2\pi iy/N}$, are equal; since $N\ge2$, this means $w=1$:

$$
P_N(y)=1\iff y\in N\mathbb Z .
\tag{4.2}
$$

A clock started at $\lvert m\rangle$ and driven by $C_K$ therefore has a certain reading at $\mu$ iff $\mu/\lambda_0=n\in\mathbb Z$. The reading is then $m+n\bmod N$, in agreement with $e^{-iC_K\lambda_0}=S$ (2.9).

### Step 2. Joint distribution of two free clocks (item 1a)

With no term across $K_1,K_2,R$, we have $C=C_{K_1}+C_{K_2}+C_R$, each term extended by the identity. Operators acting on different factors commute, so $e^{-iC\lambda}$ factorizes and $\lvert\Psi(\lambda)\rangle=e^{-iC_{K_1}\lambda}\lvert0\rangle\otimes e^{-iC_{K_2}\lambda}\lvert0\rangle\otimes e^{-iC_R\lambda}\chi$. The class $(m_1,m_2)$ of the joint reading has the projector $\Pi_{m_1}\otimes\Pi_{m_2}\otimes\mathbb 1_R$. By (1.3) and (4.1),

$$
p(m_1,m_2;\lambda)=P_{N_1}(m_1-x_1)\,P_{N_2}(m_2-x_2),\qquad x_i=\lambda/\lambda_{0i}.
\tag{4.3}
$$

The two readings are independent. Equation (4.3) is (2.4) for the part $K_1\cup K_2$, which is self-driven by (2.1) with $C_\partial=0$.

### Step 3. Common certain readings (item 1b)

Define the **rate ratio**

$$
\rho:=\frac{\lambda_{01}}{\lambda_{02}},\qquad x_2=\rho\,x_1 .
\tag{4.4}
$$

1. A product of two numbers in $[0,1]$ equals $1$ iff both factors do. By (4.3) and (4.2), both readings are certain at $\lambda$ iff $x_1\in\mathbb Z$ and $x_2\in\mathbb Z$. The readings are then $m_i\equiv x_i\pmod{N_i}$.
2. If $\lambda=n_1\lambda_{01}=n_2\lambda_{02}>0$ with $n_i\in\mathbb Z_{>0}$, then $\rho=n_2/n_1\in\mathbb Q$. Conversely, let $\rho=p/q$ with $p,q$ coprime positive integers. Then $n_1\lambda_{01}=n_2\lambda_{02}$ iff $n_1p=n_2q$. Since $q$ divides $n_1p$ and $\gcd(p,q)=1$, this holds iff $n_1=jq$ and $n_2=jp$ with $j\in\mathbb Z_{>0}$.
3. Hence

$$
\Lambda_{12}=\{\,j\lambda_c : j=1,2,\dots\},\quad \lambda_c:=q\lambda_{01}=p\lambda_{02},\qquad
(m_1,m_2)\big|_{\lambda=j\lambda_c}=\bigl(jq\bmod N_1,\ jp\bmod N_2\bigr),
\tag{4.5}
$$

$$
\Lambda_{12}\neq\emptyset\iff\rho=\lambda_{01}/\lambda_{02}\in\mathbb Q .
\tag{4.6}
$$

### Step 4. The reading of $K_2$ at the certain readings of $K_1$ (item 1c)

1. At $\lambda=n\lambda_{01}$ we have $x_1=n$ and $x_2=n\rho$. By (4.3) and (4.2), $K_1$ reads $n\bmod N_1$ with certainty, and independently of that reading

$$
p_2(m_2;n\lambda_{01})=P_{N_2}(m_2-n\rho)=\frac{\sin^2(\pi n\rho)}{N_2^2\sin^2\bigl(\pi(n\rho-m_2)/N_2\bigr)} .
\tag{4.7}
$$

2. If $n\rho\in\mathbb Z$, then by (4.2) the reading is certain at $n\rho\bmod N_2$.
3. If $n\rho\notin\mathbb Z$, the numerator is positive and independent of $m_2$, so we minimize $\sin^2(\pi(m_2-n\rho)/N_2)$. For each class $m_2$, choose the representative $M\in\mathbb Z$, $M\equiv m_2$, with $\lvert M-n\rho\rvert\le N_2/2$. The function $t\mapsto\sin^2(\pi t/N_2)$ is even, $N_2$-periodic and strictly increasing on $[0,N_2/2]$, so the maximum is attained exactly at the classes of the integers nearest to $n\rho$. There are two such integers iff $n\rho\in\mathbb Z+\frac12$.
4. With $\delta_n:=\min_{M\in\mathbb Z}\lvert n\rho-M\rvert$:

$$
m_2^{*}(n)=\begin{cases}\lfloor n\rho+\tfrac12\rfloor\bmod N_2, & n\rho\notin\mathbb Z+\tfrac12,\\[2pt] n\rho\pm\tfrac12\bmod N_2\ \text{(both)}, & n\rho\in\mathbb Z+\tfrac12,\end{cases}\qquad p_{\max}(n)=P_{N_2}(\delta_n).
\tag{4.8}
$$

The two places in the second case are distinct because $N_2\ge2$. In readings, the rate ratio means the following. At the $n$-th certain reading of $K_1$, the most probable reading of $K_2$ is the integer nearest to $n\rho$ (mod $N_2$): $K_2$ advances $\rho$ places per place of $K_1$. For $\rho=p/q$, the certain joint readings (4.5) advance by $(q,p)$ from one point of $\Lambda_{12}$ to the next. The ratio $\rho$ is dimensionless and fixed by the two contracts alone.

### Step 5. Coincidence of two identical clocks (item 2a)

Let $N_1=N_2=N$, $\lambda_{01}=\lambda_{02}=\lambda_0$ and $x=\lambda/\lambda_0$. By (4.3), $P_=(\lambda)=\sum_{m\in\mathbb Z_N}P_N(m-x)^2$.

1. By (4.1), $P_N(m-x)=N^{-2}\sum_{k,k'=0}^{N-1}e^{2\pi i(k-k')(m-x)/N}=N^{-2}\sum_{\lvert d\rvert\le N-1}(N-\lvert d\rvert)\,e^{2\pi id(m-x)/N}$, since exactly $N-\lvert d\rvert$ pairs have $k-k'=d$.
2. Square and sum over $m$, using $\sum_m e^{2\pi i(d+d')m/N}=N$ if $d+d'\equiv0\pmod N$ and $0$ otherwise. Since $\lvert d+d'\rvert\le2N-2$, the surviving terms have $d+d'\in\{0,\pm N\}$; for $d+d'=\pm N$ they carry the phase $e^{\mp2\pi ix}$:
$$P_==\frac1{N^3}\Bigl[\sum_{\lvert d\rvert\le N-1}(N-\lvert d\rvert)^2+2\cos(2\pi x)\sum_{d=1}^{N-1}d(N-d)\Bigr].$$
3. The first sum is $N^2+2\sum_{t=1}^{N-1}t^2=N(2N^2+1)/3$, and $\sum_{d=1}^{N-1}d(N-d)=N(N^2-1)/6$. Hence

$$
P_=(\lambda)=\frac{2N^2+1+(N^2-1)\cos(2\pi\lambda/\lambda_0)}{3N^2}=1-\frac{2(N^2-1)}{3N^2}\,\sin^2\frac{\pi\lambda}{\lambda_0} .
\tag{4.9}
$$

4. Since $N^2-1>0$:

$$
\min_\lambda P_==\frac{N^2+2}{3N^2}\ \ \text{at}\ \lambda\in\lambda_0\bigl(\mathbb Z+\tfrac12\bigr),\qquad
P_=(\lambda)=1\iff\lambda\in\lambda_0\mathbb Z .
\tag{4.10}
$$

### Step 6. A second joint reading; synchronization (item 2b)

At $\lambda_s$ the state is $\psi(\lambda_s)\otimes\psi(\lambda_s)\otimes\chi(\lambda_s)$. Since $\Pi_m\psi\propto\lvert m\rangle$, the outcome $(m_1,m_2)$ leaves, by (1.3), the state $\lvert m_1\rangle\otimes\lvert m_2\rangle\otimes\chi(\lambda_s)$ up to a common phase, as in (2.14). The evolution (1.9) still factorizes. With $\mu:=\lambda-\lambda_s$, (4.1) gives

$$
p(m_1',m_2'\mid m_1,m_2;\lambda)=P_N\Bigl(m_1'-m_1-\frac{\mu}{\lambda_0}\Bigr)\,P_N\Bigl(m_2'-m_2-\frac{\mu}{\lambda_0}\Bigr).
\tag{4.11}
$$

Put $j_i:=m_i'-m_i$. The offset $\Delta:=m_2-m_1$ is reproduced, $\Delta':=m_2'-m_1'\equiv\Delta$, iff $j_1\equiv j_2$. Its probability is $\sum_jP_N(j-\mu/\lambda_0)^2$, which is the sum of Step 5 with $x\to\mu/\lambda_0$:

$$
\Pr[\Delta'\equiv\Delta]=P_=(\lambda-\lambda_s)=1-\frac{2(N^2-1)}{3N^2}\,\sin^2\frac{\pi(\lambda-\lambda_s)}{\lambda_0} .
\tag{4.12}
$$

By (4.10), this probability equals $1$ iff $\lambda\in\lambda_s+\lambda_0\mathbb Z_{>0}$. There the readings are $(m_1+n,\,m_2+n)$ with certainty, where $n=(\lambda-\lambda_s)/\lambda_0$. Thus one joint reading at any $\lambda_s$ synchronizes the two clocks up to the recorded offset $\Delta$: they reproduce $\Delta$ with certainty on $\lambda_s+\lambda_0\mathbb Z_{>0}$. They agree there ($m_1'=m_2'$ with certainty) iff $\Delta=0$, an outcome of probability $P_=(\lambda_s)$. If $\lambda_s\notin\lambda_0\mathbb Z$, the reading moves the points of certain readings from $\lambda_0\mathbb Z$ to $\lambda_s+\lambda_0\mathbb Z$ (Law 1).

### Step 7. One clock in an environment (item 3a)

Let $C_E\lvert b\rangle=e_b\lvert b\rangle$ and define $C_b:=C_{K_1}+bA$. For every $\phi\in\mathcal H_{K_1}$, $C(\phi\otimes\lvert b\rangle)=\bigl((C_b+e_b)\phi\bigr)\otimes\lvert b\rangle$. So $\mathcal H_{K_1}\otimes\lvert b\rangle$ is invariant under $C$, and $e^{-iC\lambda}$ acts on it as $e^{-i(C_b+e_b)\lambda}$. By (1.7), with the single branch state $\lvert b\rangle$,

$$
\lvert\Psi(\lambda)\rangle=e^{-ie_b\lambda}\bigl(e^{-iC_b\lambda}\lvert0\rangle\bigr)\otimes\lvert b\rangle,\qquad
V_{K_1}(\lambda)=e^{-iC_b\lambda}\lvert0\rangle\langle0\rvert e^{iC_b\lambda} .
\tag{4.13}
$$

The same holds for any start $\phi\otimes\lvert b'\rangle$ with $\lvert b'\rangle$ a common eigenvector. Self-drivenness in the sense of 02-clock (with $C_K=C_{K_1}$ and $C_\partial=A\otimes B$):

$$
K_1\ \text{is self-driven}\iff B=0\ \ \text{or}\ \ A\in\mathbb R\,\mathbb 1 .
\tag{4.14}
$$

*If:* $A=a\mathbb 1$ gives $C_\partial=\mathbb 1\otimes aB$, and $B=0$ gives $C_\partial=0$; both are of the form (2.1).

*Only if:* let $B\neq0$ and $A\notin\mathbb R\mathbb 1$. The commuting self-adjoint operators $B$ and $C_E$ have a common eigenbasis, and since $B\ne0$ it contains some $\lvert b'\rangle$ with $b'\neq0$. For the starts $\phi\otimes\lvert b'\rangle$, (4.13) gives the view $e^{-iC_{b'}\lambda}\lvert\phi\rangle\langle\phi\rvert e^{iC_{b'}\lambda}$. Self-drivenness would require $W(\lambda)=e^{iC_{K_1}\lambda}e^{-iC_{b'}\lambda}$ to map every ray onto itself. An operator for which every vector is an eigenvector is a scalar, so $W(\lambda)=c(\lambda)\mathbb 1$ with $c=\operatorname{Tr}W/N_1$ smooth. Differentiating at $\lambda=0$ gives $-ib'A=c'(0)\mathbb 1$, a contradiction.

So, apart from these cases, $K_1$ is not self-driven. For a start in an eigenvector of $B$, its view still evolves like that of a self-driven clock, but with the contract $C_b$, which depends on the environment eigenvalue $b$.

### Step 8. Couplings that keep a minimal ring clock (item 3b)

1. The contract of a minimal ring clock with $N_1$ places and tick $\lambda'$ is $\frac{2\pi}{N_1\lambda'}Q_1$. Since $Q_1\neq0$, the contract fixes the tick. The condition reads: for every $b\in\mathfrak B$ there is $\omega(b)>0$ with $C_{K_1}+bA=\omega(b)Q_1$, where $\omega(b)=2\pi/(N_1\lambda_{01}(b))$.
2. *Necessity.* $\mathfrak B$ has two elements, so it contains some $b\neq0$. Then $A=\frac{\omega(b)-\omega_1}{b}Q_1=\gamma C_{K_1}$ with $\gamma:=\frac{\omega(b)-\omega_1}{b\,\omega_1}\in\mathbb R$. For every $b'\in\mathfrak B$, $(1+\gamma b')\omega_1Q_1=\omega(b')Q_1$, so $\omega(b')=(1+\gamma b')\,\omega_1>0$.
3. *Sufficiency.* If $A=\gamma C_{K_1}$ with $1+\gamma b>0$ on $\mathfrak B$, then $C_{K_1}+bA=(1+\gamma b)\,\omega_1Q_1$ is the contract of a minimal ring clock. Hence

$$
A=\gamma\,C_{K_1},\qquad\gamma\in\mathbb R,\qquad 1+\gamma b>0\ \ \text{for all } b\in\mathfrak B,
\tag{4.15}
$$

$$
\lambda_{01}(b)=\frac{\lambda_{01}}{1+\gamma b} .
\tag{4.16}
$$

4. By (4.13), the view of $K_1$ is then that of a minimal ring clock with tick $\lambda_{01}(b)$, and Step 1 applies with $\lambda_0\to\lambda_{01}(b)$. Suppose instead that the constant $\varepsilon$ of (2.9) is admitted; it changes the state only by a common phase (1.2). The condition then becomes $C_{K_1}+bA=\varepsilon(b)\mathbb 1+\omega(b)Q_1$. Since $Q_1$ and $\mathbb 1$ are linearly independent ($N_1\ge2$), the same steps give $A=\gamma C_{K_1}+a_0\mathbb 1$ with $a_0\in\mathbb R$, and the tick is again (4.16).

### Step 9. Two clocks in one environment (item 3c)

Let $A_i=\gamma_iC_{K_i}$ with $1+\gamma_ib>0$ on $\mathfrak B$. As in Step 7, $\mathcal H_{K_1}\otimes\mathcal H_{K_2}\otimes\lvert b\rangle$ is invariant, and $C$ acts on it as $(1+\gamma_1b)C_{K_1}+(1+\gamma_2b)C_{K_2}+e_b$. The two clock terms act on different factors, so the state is $e^{-ie_b\lambda}\,e^{-i(1+\gamma_1b)C_{K_1}\lambda}\lvert0\rangle\otimes e^{-i(1+\gamma_2b)C_{K_2}\lambda}\lvert0\rangle\otimes\lvert b\rangle$. This describes two free minimal ring clocks with ticks $\lambda_{0i}(b)=\lambda_{0i}/(1+\gamma_ib)$. Steps 2–4 hold with $\lambda_{0i}\to\lambda_{0i}(b)$, and (4.4) becomes

$$
\rho(b)=\frac{\lambda_{01}(b)}{\lambda_{02}(b)}=\rho\,\frac{1+\gamma_2b}{1+\gamma_1b} .
\tag{4.17}
$$

For $b\neq b'$: $\rho(b)=\rho(b')$ iff $(1+\gamma_2b)(1+\gamma_1b')=(1+\gamma_2b')(1+\gamma_1b)$ iff $(\gamma_2-\gamma_1)(b-b')=0$. Hence

$$
\rho(b)\ \text{is the same for all } b\in\mathfrak B\iff\gamma_1=\gamma_2,\quad\text{and then } \rho(b)=\rho .
\tag{4.18}
$$

Equivalently, $A_i\otimes B=C_{K_i}\otimes\gamma B$ with one $\gamma$ for both clocks: each clock couples through its own contract with a common strength, whatever $N_i$ and $\lambda_{0i}$. Written as $A_i=g_iQ_i$, the condition reads $g_1N_1\lambda_{01}=g_2N_2\lambda_{02}$, so equal coefficients $g_1=g_2$ suffice only if $N_1\lambda_{01}=N_2\lambda_{02}$. When (4.18) holds, the distribution (4.7) of $K_2$ at the certain readings of $K_1$ is the same for all $b\in\mathfrak B$, although each tick depends on $b$.

## Result

- (4.1), (4.2): propagator $u_N(m'-m-\mu/\lambda_0)$ of a minimal ring clock, with $\lvert u_N\rvert^2=P_N$. A reading is certain iff $\mu/\lambda_0\in\mathbb Z$.
- (4.3): two free clocks give independent readings, $p(m_1,m_2;\lambda)=P_{N_1}(m_1-\lambda/\lambda_{01})P_{N_2}(m_2-\lambda/\lambda_{02})$.
- (4.4)–(4.6): rate ratio $\rho=\lambda_{01}/\lambda_{02}$. Both readings are certain exactly on $\Lambda_{12}=\lambda_c\mathbb Z_{>0}$, $\lambda_c=q\lambda_{01}=p\lambda_{02}$, with readings $(jq\bmod N_1,jp\bmod N_2)$. $\Lambda_{12}\neq\emptyset$ iff $\rho\in\mathbb Q$.
- (4.7), (4.8): at $\lambda=n\lambda_{01}$, $K_2$ is distributed as $P_{N_2}(m_2-n\rho)$, with most probable place(s) the integer(s) nearest to $n\rho$, mod $N_2$.
- (4.9), (4.10): identical clocks coincide with probability $1-\frac{2(N^2-1)}{3N^2}\sin^2(\pi\lambda/\lambda_0)$. Its minimum is $(N^2+2)/(3N^2)$, and it equals $1$ iff $\lambda\in\lambda_0\mathbb Z$.
- (4.11), (4.12): after a joint reading at $\lambda_s$, the readings are distributed as shifted products of $P_N$. The offset is kept with probability $P_=(\lambda-\lambda_s)$, which is $1$ iff $\lambda\in\lambda_s+\lambda_0\mathbb Z_{>0}$.
- (4.13), (4.14): in an eigenvector of $B$, the view of $K_1$ evolves under $C_b=C_{K_1}+bA$. $K_1$ is self-driven iff $B=0$ or $A\propto\mathbb 1$.
- (4.15), (4.16): $C_{K_1}+bA$ is a minimal ring clock for all $b\in\mathfrak B$ iff $A=\gamma C_{K_1}$ with $1+\gamma b>0$. Then $\lambda_{01}(b)=\lambda_{01}/(1+\gamma b)$.
- (4.17), (4.18): in a common environment $\rho(b)=\rho(1+\gamma_2b)/(1+\gamma_1b)$, which is independent of $b$ iff $\gamma_1=\gamma_2$.

## Consistency checks

1. **$N=2$.** By (4.1), $P_2(y)=\cos^2(\pi y/2)$, so $P_==\cos^4+\sin^4=1-\frac12\sin^2(\pi x)$, which is (4.9) with $\frac{2(N^2-1)}{3N^2}=\frac12$. Its minimum is $\frac12=\frac{N^2+2}{3N^2}$. For $n\rho\in\mathbb Z+\frac12$, (4.8) gives both places with $P_2(\frac12)=\frac12$. Passed.
2. **$\lambda_{01}=\lambda_{02}$ ($\rho=1$).** (4.5) gives $\lambda_c=\lambda_{01}$ and readings $(j\bmod N_1,j\bmod N_2)$. (4.7)–(4.8) give $K_2$ certain at $n\bmod N_2$ for every $n$, consistent with (4.5) and (4.2). For $N_1=N_2$, (4.9) equals $1$ exactly there. Passed.
3. **Dimensions and $b=0$.** By (1.9), $[C]=[\lambda]^{-1}$. Since $A\otimes B$ has the dimension of $C_{K_1}$, $\gamma b$ is dimensionless, and (4.16), (4.17) have the dimensions of $\lambda$ and $1$. At $b=0$ they give $\lambda_{01}$ and $\rho$. For $\gamma=0$, (4.14) gives self-drivenness and (4.13) reduces to $C_b=C_{K_1}$. Passed.

## Open issues

- Readings take values in $\mathbb Z_{N_i}$, so (4.5) and (4.8) give the advances $jq$, $jp$ and the nearest integer to $n\rho$ only modulo $N_i$. Recovering $\rho$ itself needs a count of turns (a record, or a sequence of readings); this is not derived.
- The statistics of a sequence of joint readings of two different clocks (each reading collapses $K_2$ by (2.14)) is not derived.
- Environment states that are superpositions of eigenvectors of $B$ are out of scope; in that case the view of $K_1$ is not pure.
- Choosing reading values of $\lambda$ and comparing readings at equal $\lambda$ uses the order of $\lambda$ (A8); whether it can be detected is criterion 8. The meaning of $\gamma_1=\gamma_2$ for criteria 4–6 is not addressed.

## Methods used

- spectral decomposition and functional calculus
- discrete Fourier analysis on $\mathbb Z_N$ (Dirichlet and Fejér-type kernels, orthogonality of characters)
- triangle inequality, monotonicity of $\sin^2$
- elementary number theory (coprime integers)
- invariant subspaces of a self-adjoint operator
- partial trace and projective reading (1.3)

Author: Ivan Honis · ivan.honis@ndot.io · License: CC BY 4.0 · Human–AI note