Version 2 · current

# Carried clock under a uniform gradient and two clocks between two meetings

- **Subproject:** 04-ilang-time
- **Package:** 09-proper-time
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-10

## Changes from previous version

- **Chain limit (I3).** S2 and S3 are rewritten. The class of starts is stated (finite second position moment $M_2$), the finite-chain start is an explicit truncation, and the new bound (9.23) controls the state, the reading statistics and the mean position label together, for every $\mathcal F$. The order of the limits is stated in S2.
- **Speed–rate relation (I2).** Step 5 gives both monotonic branches and the inversion on each, new (9.24).
- **Period (I1).** Step 6 separates the period $2\pi/\mathcal F$ of the label from the period $\pi/\mathcal F$ of the retained displacement, velocity and rate; the wording of Step 7 follows.
- **Own corrections** (no result of v1 changes):
  - Step 2: new bound (9.25) on the amplitude of the negative branch, valid for all parameters and all $\lambda$, so that the condition (9.4) is stated with its range.
  - Step 3: "the oscillating part" of the velocity is now defined, and (9.8) refers to it. In v1 the remainder of (9.8) was valid only for these first-order terms, not for the whole cross term of (9.6); the rest of the cross term belongs to $\mathcal R_x$. The second integration by parts that the remainder needs is stated.
  - S1 states what $O(\kappa)$ bounds at a finite chain length; S4 states that the order symbols hold for $\epsilon\le\epsilon_*$.
- Tags (9.1)–(9.22) keep their meaning. New tags: (9.23), (9.24), (9.25).

## Response to verification

- **I1: Accepted and fixed in Step 6** (paragraph after (9.18)) and in Step 7, item 3(d), where "full period" now names the period $\pi/\mathcal F$ of the retained displacement.
- **I2: Accepted and fixed in Step 5** (paragraph after (9.14), new (9.24)): both branches are given, the inversion requires the branch, and the lower branch is the one of the small-label approximation.
- **I3: Accepted and fixed in "Setup and assumptions", S2 and S3** (new (9.23)): truncated starts, the moment condition $M_2<\infty$, and a bound in which the truncation error and the boundary error vanish together and the mean position label is controlled through the second moment.

## Setup and assumptions

**Model.** The setting of question.md: bodies $1$, $2$ (types $B_1\neq B_2$) with places $(x,s,a)$, body contract $C_{\rm D}=A+B$, the gradient $G=-\mathcal F\hat x$ on body $1$ with $\hat x:=\sum_xx\,P_x$, the records $\kappa xX$, and the start $\phi_1\otimes\phi_2\otimes\chi$. Readings are applied at stated values of $\lambda$ (A8); $\lambda\ge0$.

**Inputs used.** From 05-moving-clock@v3: (5.1) $P_N(a;\vartheta)$; (5.5)–(5.6) $C_k(p)$, $\omega_k$, $\beta_{k,\pm}$, $\theta_k$; (5.11) $v_{\max}=\tau\kappa\sigma_X$; (5.20) the expansion of $\omega_k$ in $j=k-\frac{N-1}2$ with $\bar\omega$, $\Gamma$, $\bar c$, $\bar\rho_k$; (5.21), (5.22), (5.24), (5.25), (5.29); the notation $d_k,\epsilon,\mu_k,\delta_k,w_k,\bar\mu,\bar w,\hat e$. From 03-ilang-space/17-motion@v2: (17.15), $\bar x=\kappa\,\bar x_{\rm lab}\,e$. From 03-ilang-space/16-common-space@v2: (16.11), through the meeting condition $t_\pi=0$ of the question.

**New symbols.** $p_\lambda:=p+\mathcal F\lambda$; $\theta_k':=\partial_p\theta_k=-\mu_k\tau\cos p/\omega_k^2$; $\Phi_k(p,\lambda):=\int_0^\lambda\omega_k(p_{\lambda'})\,\mathrm d\lambda'$; $\hat y:=\hat x-x_0$; $E:=\sqrt{\bar\mu^2+\tau^2}$.

**Assumptions and their range.**
- **S1. Weak records.** Leading order in $\kappa$ at fixed $\lambda$ and fixed chain length (S2). On the chain $\lvert x\rvert\le L$ the record term of $C$ has norm at most $2\kappa L\lVert X\rVert$. By Duhamel's formula the state then differs from the state at $\kappa=0$ by at most $2\kappa L\lVert X\rVert\lambda$ in norm, reading statistics by at most $4\kappa L\lVert X\rVert\lambda$, and mean position labels by at most $4\kappa L^2\lVert X\rVert\lambda$. "$O(\kappa)$" below stands for these bounds: they vanish for $\kappa\to0$ at fixed $L$, not uniformly in $L$. The regime is used in three places:
  - (W1) the view of each body is that of its own contract, $C_{\rm D}+G$ or $C_{\rm D}$ (Steps 1–4);
  - (W2) positions are $\bar x_i=\kappa\,\bar x_{{\rm lab},i}\,e$ in the background (17.15) with $X\to\kappa X$, with $\bar x_{{\rm lab},i}$ computed from the place weights at $\kappa=0$ (Steps 3, 6);
  - (W3) the joint statistics of the two bodies factorize (Step 6).
- **S2. Long chains and order of the limits.** The finite description has $x\in\{-L,\dots,L\}$ with open ends; its body contracts are $\Pi_L(C_{\rm D}+G)\Pi_L$ and $\Pi_LC_{\rm D}\Pi_L$, with $\Pi_L$ the projector on $\lvert x\rvert\le L$. The limit object is the chain $x\in\mathbb Z$ with the starts $\phi_i$ of S3, of common mean position label $x_0$ and second position moment $M_2:=\max_i\lVert\hat y\,\phi_i\rVert^2<\infty$. The finite-chain start is the truncation $\phi_i^{(L)}:=\Pi'_\ell\phi_i/\lVert\Pi'_\ell\phi_i\rVert$, with $\Pi'_\ell$ the projector on $\lvert x-x_0\rvert\le\ell$ and $\ell:=\lfloor L/2\rfloor$; its distance from the ends is $d:=L-\lvert x_0\rvert-\ell$, and its clock label is exactly $\lvert0\rangle$. Let $U_L$, $U$ be the evolutions (1.9) on the chain and on $\mathbb Z$ at $\kappa=0$, and $\psi_i^{(L)}:=U_L\phi_i^{(L)}$, $\psi_i:=U\phi_i$.
  - *Truncation.* By Chebyshev's inequality the removed weight is $t\le M_2/\ell^2$, and $\lVert\phi_i^{(L)}-\phi_i\rVert^2=2-2\sqrt{1-t}\le2t$.
  - *Ends.* $B$ and $G$ are diagonal in $x$, and $A$ moves $x$ by one step with $\lVert A\rVert\le\tau$. In the Dyson series in $A$ (interaction picture with respect to $B+G$) the term with $n$ factors $A$ has norm at most $(\tau\lambda)^n/n!$. For a start supported in $\lvert x-x_0\rvert\le\ell$ the terms with $n\le d$ are the same on the chain and on $\mathbb Z$, because a step across an end needs $d$ earlier steps. Hence $\lVert(U_L-U)\phi_i^{(L)}\rVert\le2\sum_{n>d}(\tau\lambda)^n/n!\le2(e\tau\lambda/d)^d$ for $d>\tau\lambda$, for every $\mathcal F$.
  - *Moments.* On both chains $i[C,\hat x]=i[A,\hat x]$ has norm at most $\tau$, so $\lVert\hat y\,\psi(\lambda)\rVert\le\lVert\hat y\,\psi(0)\rVert+\tau\lambda$. For unit vectors, $\lvert\langle\psi'\vert\hat y\vert\psi'\rangle-\langle\psi\vert\hat y\vert\psi\rangle\rvert\le\lVert\psi'-\psi\rVert\bigl(\lVert\hat y\psi'\rVert+\lVert\hat y\psi\rVert\bigr)$, and $\lVert\hat y\phi_i^{(L)}\rVert\le\sqrt{2M_2}$ for $t\le\frac12$.

  Together, for $\ell\ge\sqrt{2M_2}$, $d>\tau\lambda$, every $\mathcal F\ge0$ and every $\lambda\ge0$:

$$
\bigl\lVert\psi^{(L)}_i(\lambda)-\psi_i(\lambda)\bigr\rVert\le\varepsilon_L:=\frac{\sqrt{2M_2}}{\ell}+2\Bigl(\frac{e\tau\lambda}{d}\Bigr)^{d},\qquad
\bigl\lvert\bar x^{(L)}_{{\rm lab},i}(\lambda)-\bar x_{{\rm lab},i}(\lambda)\bigr\rvert\le2\,\varepsilon_L\bigl(\sqrt{2M_2}+\tau\lambda\bigr).
\tag{9.23}
$$

  The reading statistics of one body differ by at most $2\varepsilon_L$, the joint ones by at most $4\varepsilon_L$. The truncation error and the boundary error vanish together for $L\to\infty$; at $\lambda=0$, (9.23) shows that the mean position labels of the truncated starts tend to $x_0$. On $\mathbb Z$, $\langle p\vert p'\rangle=2\pi\delta(p-p')$, and $\bar x_{{\rm lab},i}(\lambda)$ is finite and differentiable because $\lVert\hat y\psi_i(\lambda)\rVert\le\sqrt{M_2}+\tau\lambda$.

  **Order of the limits.** (i) $\kappa\to0$ at fixed $L$ and $\lambda$ (S1). (ii) $L\to\infty$ at fixed $\lambda$, fixed starts $\phi_i$ and fixed parameters, $\mathcal F$ included. (iii) The expansions in $\epsilon$ and $\delta p$ (S4). Limit (i) is not uniform in $L$; limit (ii) is not uniform in $\lambda$, nor in the start, which enters through $M_2$. All results below are statements about the limit object of (i) and (ii).
- **S3. Starts.** $\lvert\phi_i\rangle=\int\frac{\mathrm dp}{2\pi}g_i(p)\,\lvert p\rangle\otimes\beta_{0,+}(p)\otimes\lvert0\rangle$ with $\lvert0\rangle=N^{-1/2}\sum_k\lvert f_k\rangle$, $\int\frac{\mathrm dp}{2\pi}\lvert g_i\rvert^2=1$, and $g_i$ a $2\pi$-periodic, continuously differentiable function. This is the class of starts for which every result is stated.
  - *Label.* Mean and variance of the weight $\lvert g_i\rvert^2/2\pi$ are taken on the window $(\bar p-\pi,\bar p+\pi]$ centred at the mean $\bar p$, with $\bar p=p_0\in(-\frac\pi2,0]$ for body $1$, $\bar p=0$ for body $2$, and variance $\delta p^2\ll1$. For a $2\pi$-periodic $f\in C^2$, Taylor's formula gives $\int\frac{\mathrm dp}{2\pi}\lvert g_i\rvert^2f=f(\bar p)+r$ with $\lvert r\rvert\le\frac12\delta p^2\sup\lvert f''\rvert$.
  - *Position.* $\beta_{0,+}$ is smooth and $2\pi$-periodic and $\langle\beta_{0,+}\vert\partial_p\beta_{0,+}\rangle=0$, so by Parseval's identity $x_0=\int\frac{\mathrm dp}{2\pi}g_i^{*}\,i\partial_pg_i$ and $M_2=\max_i\int\frac{\mathrm dp}{2\pi}\bigl(\lvert(i\partial_p-x_0)g_i\rvert^2+\frac14\theta_0'^2\lvert g_i\rvert^2\bigr)<\infty$. Because $\beta_{0,+}$ depends on $p$, $\phi_i$ has in general no finite support in $x$; only $M_2<\infty$ is used.
- **S4. Orders.** $O(\epsilon^n)$ and $O(\delta p^2)$ denote bounds $K\epsilon^n$, $K\delta p^2$ for $\epsilon\le\epsilon_*$, with $K$ and $\epsilon_*>0$ depending only on $N$, $\tau/m$, $\lambda/\lambda_0$ and $\mathcal F\lambda_0$. They are uniform in $p$, $p_0$, $k$ and $a$, and they depend on the start only through $\delta p$. At these fixed quantities $\mathcal F\lambda=\mathcal F\lambda_0\cdot\lambda/\lambda_0$ is fixed, $\mathcal F/m=\frac{N\epsilon}{2\pi}\mathcal F\lambda_0=O(\epsilon)$ and $m\lambda=\frac{2\pi}{N\epsilon}\frac\lambda{\lambda_0}=O(\epsilon^{-1})$. No statement is made for $\mathcal F\lambda\to\infty$.

Statements marked **exact** hold without expansion (within S1, S2); all others are order estimates in the sense of S4.

## Derivation

### Step 1. The plane-wave label (item 1(a))

Since $\hat x\lvert p\rangle=-i\partial_p\lvert p\rangle$, $e^{iq\hat x}\lvert p\rangle=\lvert p+q\rangle$. Put $\lvert\psi_1(\lambda)\rangle=e^{i\mathcal F\lambda\hat x}\lvert\tilde\psi(\lambda)\rangle$. Then (1.9) with $C_{\rm D}-\mathcal F\hat x$ gives $i\,\mathrm d\tilde\psi/\mathrm d\lambda=e^{-i\mathcal F\lambda\hat x}C_{\rm D}e^{i\mathcal F\lambda\hat x}\tilde\psi$, and by (5.5) this operator acts on $\lvert p\rangle\otimes\beta\otimes\lvert f_k\rangle$ as $C_k(p+\mathcal F\lambda)$ on $\beta$. Hence, **exactly**,

$$
\lvert\psi_1(\lambda)\rangle=\int\frac{\mathrm dp}{2\pi}\,g_1(p)\,\lvert p+\mathcal F\lambda\rangle\otimes\frac1{\sqrt N}\sum_{k=0}^{N-1}\beta_k(p,\lambda)\otimes\lvert f_k\rangle,
\qquad
p(\lambda):=p_0+\mathcal F\lambda\ \ (\mathrm{mod}\ 2\pi),
\tag{9.1}
$$

$$
i\,\partial_\lambda\beta_k(p,\lambda)=C_k(p+\mathcal F\lambda)\,\beta_k(p,\lambda),\qquad\beta_k(p,0)=\beta_{0,+}(p).
\tag{9.2}
$$

The weight over the plane-wave label is translated rigidly by $\mathcal F\lambda$: its mean is $p(\lambda)$ and its variance stays $\delta p^2$, for every $\mathcal F$.

### Step 2. Following of the positive branch (item 1(a))

By (5.6), $C_k=\omega_k(\cos\theta_k\,\sigma_3+\sin\theta_k\,\sigma_1)$ and $\partial_p\beta_{k,\pm}=\pm\frac12\theta_k'\beta_{k,\mp}$; also $\beta_{0,+}=\cos\delta_k\,\beta_{k,+}+\sin\delta_k\,\beta_{k,-}$. Write $\beta_k(p,\lambda)=c_{k+}\beta_{k,+}(p_\lambda)+c_{k-}\beta_{k,-}(p_\lambda)$ and $c_{k\pm}=e^{\mp i\Phi_k}\tilde c_{k\pm}$. Then (9.2) is **exactly**

$$
\frac{\mathrm d\tilde c_{k+}}{\mathrm d\lambda}=2\nu_k\omega_k\,e^{2i\Phi_k}\,\tilde c_{k-},\qquad
\frac{\mathrm d\tilde c_{k-}}{\mathrm d\lambda}=-2\nu_k\omega_k\,e^{-2i\Phi_k}\,\tilde c_{k+},\qquad
\nu_k(p):=\frac{\mathcal F\,\theta_k'(p)}{4\,\omega_k(p)}=-\frac{\mathcal F\mu_k\tau\cos p}{4\,\omega_k^3(p)},
\tag{9.3}
$$

with $\nu_k,\omega_k$ taken at $p_\lambda$, $\tilde c_{k+}(0)=\cos\delta_k(p)$, $\tilde c_{k-}(0)=\sin\delta_k(p)$, and $\lvert c_{k+}\rvert^2+\lvert c_{k-}\rvert^2=1$. The coupling $2\nu_k\omega_k$ is half the rate of change of $\theta_k$; the gap between the branches is $2\omega_k\ge2m$. Their ratio is bounded by

$$
\lvert\nu_k(p)\rvert\le\eta:=\frac{\mathcal F\tau}{4m^2}=\frac\tau{4m}\cdot\frac{N\epsilon}{2\pi}\,\mathcal F\lambda_0,
\qquad\text{condition:}\quad\eta\ll1\iff\mathcal F\tau\ll4m^2 .
\tag{9.4}
$$

Since $2\omega_ke^{-2i\Phi_k}=i\,\frac{\mathrm d}{\mathrm d\lambda}e^{-2i\Phi_k}$, one integration by parts gives, **exactly**, $\tilde c_{k-}(\lambda)=\sin\delta_k(p)-\bigl[i\nu_ke^{-2i\Phi_k}\tilde c_{k+}\bigr]_0^\lambda+i\int_0^\lambda e^{-2i\Phi_k}\frac{\mathrm d}{\mathrm d\lambda'}(\nu_k\tilde c_{k+})\,\mathrm d\lambda'$. Four bounds hold for all parameters:
- $\lvert\tilde c_{k\pm}\rvert\le1$;
- $\lvert\delta_k\rvert\le\frac{d_k}2\sup\lvert\partial_\mu\theta\rvert\le\frac{k\epsilon}4$, from $\lvert\partial_\mu\theta\rvert=\tau\lvert\sin p\rvert/\omega^2\le1/2\mu$;
- $\lvert\mathrm d\nu_k/\mathrm d\lambda\rvert=\mathcal F\lvert\partial_p\nu_k\rvert\le\mathcal F\eta\,(1+3\tau^2/m^2)$;
- $\lvert\nu_k\rvert\omega_k=\frac14\mathcal F\lvert\theta_k'\rvert\le\eta m$, so $\lvert\nu_k\,\mathrm d\tilde c_{k+}/\mathrm d\lambda\rvert\le2\eta^2m\,\lvert\tilde c_{k-}\rvert$, and $2\eta m\lambda=\frac\tau{2m}\mathcal F\lambda$.

Hence, **exactly**, for all $\mathcal F\ge0$, $\lambda\ge0$, $p$ and $k$:

$$
\lvert c_{k-}(p,\lambda)\rvert\le\frac{k\epsilon}4+\eta\,\bigl(2+\Lambda\,\mathcal F\lambda\bigr),\qquad\Lambda:=1+\frac\tau{2m}+\frac{3\tau^2}{m^2}.
\tag{9.25}
$$

In the regime S4, $\eta=O(\epsilon)$ and $\mathcal F\lambda$ is fixed, so (9.25) gives $\sup_{[0,\lambda]}\lvert\tilde c_{k-}\rvert=O(\epsilon)$. The integral above is then $O(\epsilon^2)$. Its part with $\mathrm d\nu_k/\mathrm d\lambda$ is an oscillating integral, and a second integration by parts bounds it by $O(\eta\mathcal F/m)(1+\mathcal F\lambda)$. Its part with $\nu_k\,\mathrm d\tilde c_{k+}/\mathrm d\lambda$ is at most $2\eta^2m\lambda\,\sup\lvert\tilde c_{k-}\rvert=O(\epsilon)\cdot O(\epsilon)$. Inserting the result into the first equation of (9.3), the term $-i\nu_k(p_\lambda)e^{-2i\Phi_k}$ gives the non-oscillating $-2i\nu_k^2\omega_k$, and the other terms oscillate and integrate to $O(\epsilon^2)$. Hence

$$
\tilde c_{k-}(\lambda)=\sin\delta_k(p)+i\nu_k(p)-i\nu_k(p_\lambda)\,e^{-2i\Phi_k}+O(\epsilon^2),\qquad
\tilde c_{k+}(\lambda)=1-i\,\Xi_k+O(\epsilon^2),\quad\Xi_k:=\int_0^\lambda2\nu_k^2\omega_k\,\mathrm d\lambda' .
\tag{9.5}
$$

Here $\Xi_k\le2\eta^2m\lambda=O(\epsilon)$ is a phase, and $\Xi_k-\Xi_{k'}=O(N\epsilon\,\Xi_k)=O(\epsilon^2)$.

**Answer to 1(a).** The label is $p(\lambda)=p_0+\mathcal F\lambda$ (exact, every $\mathcal F$). The condition on $\mathcal F$ is (9.4): the branch angle changes slowly compared with the gap. Its range is given by (9.25): the amplitude of the negative branch stays small as long as $\eta\,(2+\Lambda\mathcal F\lambda)\ll1$. In the regime S4 this holds for small $\epsilon$, and by (9.5) the neglected weight of the negative branch of level $k$ is $\lvert c_{k-}\rvert^2\le\bigl(\frac{k\epsilon}4+2\eta\bigr)^2+O(\epsilon^3)$, of second order. The part $\frac{k\epsilon}4$ is present without a gradient (it is the mismatch $\delta_k$ of (5.24)); the gradient adds at most $2\eta=\mathcal F\tau/2m^2$ to the amplitude.

### Step 3. Displacement (item 1(b))

By W1 and W2, $\bar x_{{\rm lab},1}=\langle\psi_1\vert\hat x\vert\psi_1\rangle$ and $\mathrm d\bar x_{{\rm lab},1}/\mathrm d\lambda=\langle i[A,\hat x]\rangle$, because $B$ and $G$ commute with $\hat x$. On $\lvert p\rangle\otimes\beta\otimes\lvert f_k\rangle$, $i[A,\hat x]$ acts as $\partial_pC_k=-\tau\cos p\,\sigma_1$, with $\langle\beta_{k,\pm}\vert\partial_pC_k\vert\beta_{k,\pm}\rangle=\pm w_k$ and $\langle\beta_{k,-}\vert\partial_pC_k\vert\beta_{k,+}\rangle=-\tau\cos p\cos\theta_k$. With (9.1), **exactly**,

$$
\frac{\mathrm d\bar x_{{\rm lab},1}}{\mathrm d\lambda}=\int\frac{\mathrm dp}{2\pi}\lvert g_1\rvert^2\,\frac1N\sum_k\Bigl[\bigl(1-2\lvert c_{k-}\rvert^2\bigr)\,w_k(p_\lambda)-2\operatorname{Re}\bigl(c_{k+}^{*}c_{k-}\bigr)\,\tau\cos p_\lambda\cos\theta_k(p_\lambda)\Bigr],
\tag{9.6}
$$

which is (5.24) at $\mathcal F=0$. By (9.5), $2\operatorname{Re}(c_{k+}^{*}c_{k-})=2\sin\delta_k(p)\cos2\Phi_k-2\nu_k(p)\sin2\Phi_k+O(\epsilon^2)$: the co-rotating admixture $-i\nu_k(p_\lambda)$ is imaginary and drops out, so to first order the gradient adds no non-oscillating term to the velocity, only a second oscillating one. The **oscillating part** of the velocity is the second term of (9.6) with $2\operatorname{Re}(c_{k+}^{*}c_{k-})$ replaced by these two first-order terms; $\Delta\bar x^{\rm osc}_{{\rm lab},1}(\lambda)$ is its integral over $[0,\lambda]$.

- *Non-oscillating part.* $\int_0^\lambda w_k(p_{\lambda'})\,\mathrm d\lambda'=[\omega_k(p_\lambda)-\omega_k(p)]/\mathcal F$. Taylor's formula in $\mu_k$ about $\bar\mu$, with $\sum_kj=0$ and $\lvert\partial_\mu^2w\rvert\le3\tau^2/2m^3$, gives $\frac1N\sum_kw_k=\bar w+O\bigl(\frac{\tau^2}m(N\epsilon)^2\bigr)$, uniformly in $p$; the factor $1-2\lvert c_{k-}\rvert^2$ is $1+O(\epsilon^2)$; S3 replaces $p$ by $p_0$.
- *Oscillating part.* Put $a_k:=\tau\cos p_\lambda\cos\theta_k/2\omega_k$, $\lvert a_k\rvert\le\tau/2m$. With $2\omega_k\cos2\Phi_k=\frac{\mathrm d}{\mathrm d\lambda}\sin2\Phi_k$ and $2\omega_k\sin2\Phi_k=-\frac{\mathrm d}{\mathrm d\lambda}\cos2\Phi_k$, one integration by parts gives the boundary term $-\bigl[a_k\bigl(2\sin\delta_k\sin2\Phi_k+2\nu_k(p)\cos2\Phi_k\bigr)\bigr]_0^\lambda$ and an integral that contains $\mathrm da_k/\mathrm d\lambda=O(\mathcal F\tau/m)$. That integral is again oscillating: a second integration by parts bounds it by $O(\epsilon)\cdot O(\mathcal F\tau/m^2)(1+\mathcal F\lambda)=O(\epsilon^2)$.
- *Rest of the cross term.* By (9.5) it is $\tau\,O(\epsilon^2)$ in the velocity, uniformly on $[0,\lambda]$, hence $\tau\lambda\,O(\epsilon^2)$ in the displacement. It may contain non-oscillating terms of second order and belongs to $\mathcal R_x$ ($\tau/m$ is fixed, S4).

$$
\Delta\bar x_1(\lambda)=\kappa\sigma_X\,\Delta\bar x_{{\rm lab},1}(\lambda)\,\hat e,\qquad
\Delta\bar x_{{\rm lab},1}(\lambda)=\frac{\bar\omega(p_0+\mathcal F\lambda)-\bar\omega(p_0)}{\mathcal F}+\mathcal R_x,\qquad
\mathcal R_x=\frac{\tau^2\lambda}m\,O\bigl(\epsilon^2+\delta p^2\bigr),
\tag{9.7}
$$

with $\bar\omega(p)=\sqrt{\bar\mu^2+\tau^2\sin^2p}$ and $\bar\mu=m\bigl(1+\frac{(N-1)\epsilon}2\bigr)$. The leading term is $\int_0^\lambda\bar w(p(\lambda'))\,\mathrm d\lambda'$, of size up to $\frac{\tau^2\lambda}{2\bar\mu}$; "relative" refers to this scale, since the leading term itself vanishes at the meetings of Step 6. The whole first order in $\epsilon$ sits in $\bar\mu$. The oscillating part of the velocity contributes

$$
\bigl\lvert\Delta\bar x^{\rm osc}_{{\rm lab},1}(\lambda)\bigr\rvert\le\frac\tau m\Bigl(\frac{(N-1)\epsilon}4+\frac{\mathcal F\tau}{2m^2}\Bigr)+O(\epsilon^2),
\tag{9.8}
$$

a bounded term of order $\epsilon$ in label units. Relative to $\tau^2\lambda/m=O(\epsilon^{-1})$ it is of order $\epsilon^2$: it contributes nothing to (9.7) at first order and is part of $\mathcal R_x$. The velocity of item 2, computed from (9.7), is

$$
V_1(\lambda)=\kappa\sigma_X\,\bar w\bigl(p(\lambda)\bigr)\,\hat e,\qquad
b(\lambda):=\frac{\lVert V_1(\lambda)\rVert}{v_{\max}}=\frac{\tau\,\lvert\sin p(\lambda)\cos p(\lambda)\rvert}{\bar\omega\bigl(p(\lambda)\bigr)} .
\tag{9.9}
$$

### Step 4. Clock reading and clock advance (items 1(c), 1(d))

With $\langle a\vert f_k\rangle=N^{-1/2}e^{2\pi ika/N}$, (9.1) and the orthogonality of plane waves give, **exactly** (W1),

$$
q_1(a;\lambda)=\int\frac{\mathrm dp}{2\pi}\lvert g_1\rvert^2\,\Bigl\lVert\frac1N\sum_{k}e^{2\pi ika/N}\beta_k(p,\lambda)\Bigr\rVert^2 .
\tag{9.10}
$$

1. *Spinors.* Let $\bar\beta_\pm$ be (5.6) with $\mu_k\to\bar\mu$ (angle $\bar\theta$) and $\zeta_k:=(\bar\theta-\theta_k)/2$, $\lvert\zeta_k\rvert\le\lvert j\rvert\epsilon/4$. The $\bar\beta_-$ component of $\beta_k$ is $-c_{k+}\sin\zeta_k+c_{k-}\cos\zeta_k=O(\epsilon)$; being orthogonal to $\bar\beta_+$, it adds $O(\epsilon^2)$ to the squared norm. The $\bar\beta_+$ component is $c_{k+}+O(\epsilon^2)$.
2. *Phases.* By (9.5), $c_{k+}=e^{-i\Phi_k-i\Xi_k}(1+O(\epsilon^2))$ and $\Xi_k$ is common to all $k$ up to $O(\epsilon^2)$. The squared norm is therefore $\bigl\lvert\frac1N\sum_ke^{2\pi ika/N-i\Phi_k}\bigr\rvert^2+O(\epsilon^2)$.
3. *Expansion in $j$.* Integrating (5.20) along $p_{\lambda'}$ and using $m\epsilon=2\pi/N\lambda_0$: $\Phi_k=\bar\Phi+\frac{2\pi j}N\vartheta(p,\lambda)+\varphi_j+O(\epsilon^2)$ with $\vartheta(p,\lambda):=\frac1{\lambda_0}\int_0^\lambda\Gamma(p_{\lambda'})\,\mathrm d\lambda'$ and $\varphi_j:=\frac12(m\epsilon j)^2\int_0^\lambda\bar c\,\mathrm d\lambda'=O(\epsilon)$, even in $j$. With $u:=2\pi(a-\vartheta)/N$, the sums $S_n:=\sum_j\varphi_j^n\,e^{iuj}$ are real ($j\to-j$), so $\lvert S_0-iS_1\rvert^2=S_0^2+S_1^2$: an even phase of first order changes the modulus only at second order. By (5.1) the squared norm is $P_N(a;\vartheta(p,\lambda))+O(\epsilon^2)$.
4. *Spread.* $P_N$ is a trigonometric polynomial in $\vartheta$ and $\partial_p\vartheta$, $\partial_p^2\vartheta$ are bounded by $\frac\lambda{\lambda_0}\sup\lvert\Gamma'\rvert$, $\frac\lambda{\lambda_0}\sup\lvert\Gamma''\rvert$; S3 replaces $p$ by $p_0$ with a remainder $O(\delta p^2)$.

$$
q_1(a;\lambda)=P_N\bigl(a;\vartheta_1(\lambda)\bigr)+\mathcal R_1,\qquad\mathcal R_1=O(\epsilon^2)+O(\delta p^2)+O(\kappa),
\tag{9.11}
$$

$$
\vartheta_1(\lambda):=\frac1{\lambda_0}\int_0^\lambda\Gamma\bigl(p_0+\mathcal F\lambda'\bigr)\,\mathrm d\lambda'=\frac{\mathcal I(p_0+\mathcal F\lambda)-\mathcal I(p_0)}{\mathcal F\lambda_0},\qquad
\mathcal I(\varphi):=\int_0^\varphi\frac{\bar\mu\,\mathrm dp}{\sqrt{\bar\mu^2+\tau^2\sin^2p}} .
\tag{9.12}
$$

$\mathcal I(\varphi)=F(\varphi\mid-\tau^2/\bar\mu^2)$ is the incomplete elliptic integral of the first kind; it is odd, and $\mathcal I(\varphi+\pi)=\mathcal I(\varphi)+2\mathcal K$ with $\mathcal K:=\mathcal I(\pi/2)$. The term $O(\kappa)$ is the bound of S1 at finite $L$; it is absent in the order of limits of S2.

**Answer to 1(c).** Yes: $q_1$ has the form of (5.21) with $\Gamma(p_0)\lambda/\lambda_0$ replaced by the clock advance $\vartheta_1(\lambda)$ of (9.12).

**Answer to 1(d).** No. From (9.12),

$$
\lambda_0\,\frac{\mathrm d\vartheta_1}{\mathrm d\lambda}=\Gamma\bigl(p(\lambda)\bigr)=\frac{\bar\mu}{\sqrt{\bar\mu^2+\tau^2\sin^2p(\lambda)}} ,
\tag{9.13}
$$

the function $\Gamma$ of (5.21) at the label $p(\lambda)$; $\mathcal F$ enters only through $p(\lambda)$. Explicit dependence on $\mathcal F$ lies in $\mathcal R_1$. One such term is identified (order estimate): the part of $\Xi_k$ linear in $j$ shifts the rate by $\partial_\mu(2\nu^2\omega)\vert_{\bar\mu}=O(\eta^2)$, of second order.

### Step 5. Rate along the path (item 2)

Write $\Gamma$ for (9.13), $p=p(\lambda)$ and $s:=\sin^2p$. From (9.9), $b^2=\tau^2s\,(1-s)/\bar\omega^2$, and $\tau^2s/\bar\omega^2=1-\Gamma^2$. Both forms below are **exact** identities between the expressions (9.9) and (9.13), so they hold to the orders of items 1(b) and 1(c), for all labels:

$$
\Bigl(\lambda_0\frac{\mathrm d\vartheta_1}{\mathrm d\lambda}\Bigr)^2\Bigl(1+\frac{\tau^2}{\bar\mu^2}\sin^4p(\lambda)\Bigr)=1-\frac{\lVert V_1\rVert^2}{v_{\max}^2},
\qquad\text{equivalently}\qquad
\frac{\lVert V_1\rVert^2}{v_{\max}^2}=\bigl(1-\Gamma^2\bigr)\Bigl[1-\frac{\bar\mu^2}{\tau^2}\,\frac{1-\Gamma^2}{\Gamma^2}\Bigr].
\tag{9.14}
$$

The first form is (5.29) with $R=1/\Gamma$ at the label $p(\lambda)$. The second contains no label: the speed is a function of the rate. The converse needs a branch. $b^2=\tau^2s(1-s)/(\bar\mu^2+\tau^2s)$ vanishes at $s=0$ and $s=1$, and its derivative has the sign of $\bar\mu^2-2\bar\mu^2s-\tau^2s^2$, which changes once, at $s_*:=\bar\mu/(\bar\mu+E)<\frac12$. There $b=b_*:=(E-\bar\mu)/\tau<1$ and $\Gamma=\Gamma_*:=\sqrt{\bar\mu/E}$.
- **Lower branch**, $0\le s\le s_*$: $b$ rises strictly from $0$ to $b_*$ while $\Gamma$ falls from $1$ to $\Gamma_*$.
- **Upper branch**, $s_*\le s\le1$: $b$ falls strictly from $b_*$ to $0$ while $\Gamma$ falls from $\Gamma_*$ to $\bar\mu/E$.

Solving $\tau^2s^2-\tau^2(1-b^2)s+b^2\bar\mu^2=0$ for $s$ gives the inversion, **exactly**:

$$
\Bigl(\lambda_0\frac{\mathrm d\vartheta_1}{\mathrm d\lambda}\Bigr)^2=\frac{\bar\mu^2}{\bar\mu^2+\tau^2s_\mp(b)},\qquad
s_\mp(b):=\frac{1-b^2\mp\sqrt{(1-b^2)^2-4b^2\bar\mu^2/\tau^2}}2,\qquad0\le b\le b_*,
\tag{9.24}
$$

with $s_-$ on the lower branch and $s_+$ on the upper branch. On each branch the rate is a function of the speed. Across both it is not: every speed $b<b_*$ belongs to two rates, and the branch, i.e. the label, must be given. For example $b=0$ belongs to the rate $1$ (label $0$) and to the rate $\bar\mu/E$ (label $\frac\pi2$). The lower branch is the one of small labels. For small labels, since $1-\frac x2\le(1+x)^{-1/2}\le1$,

$$
\lambda_0\frac{\mathrm d\vartheta_1}{\mathrm d\lambda}=\sqrt{1-\frac{\lVert V_1(\lambda)\rVert^2}{v_{\max}^2}}\;\bigl(1+\rho(\lambda)\bigr),\qquad-\frac{\tau^2}{2\bar\mu^2}\sin^4p(\lambda)\le\rho(\lambda)\le0 .
\tag{9.15}
$$

The bound on $\rho$ is exact, on both branches. The deviation of the rate from $1$ is $\frac{\tau^2}{2\bar\mu^2}p^2+O(p^4)$, so $\rho$ is smaller by a factor of order $p^2$. At fixed $\tau/\bar\mu$ the leading order is equally $1-\frac12b^2$; the square root is the more accurate form, and the only one that stays valid when $\tau p/\bar\mu$ is not small (it then needs only $\tau^2p^4/\bar\mu^2\ll1$). Integrating (9.15), with $\bar\rho$ a weighted mean of $\rho$:

$$
\vartheta_1(\lambda)=\frac{1+\bar\rho}{\lambda_0}\int_0^\lambda\sqrt{1-\frac{\lVert V_1(\lambda')\rVert^2}{v_{\max}^2}}\;\mathrm d\lambda',\qquad
-\frac{\tau^2}{2\bar\mu^2}\max_{0\le\lambda'\le\lambda}\sin^4p(\lambda')\le\bar\rho\le0 .
\tag{9.16}
$$

To leading order for small labels on the whole path ($\lvert p_0\rvert\ll1$ and $\lvert p_0+\mathcal F\lambda\rvert\ll1$), $\bar\rho\to0$ and $\vartheta_1$ is a functional of the path $V_1$ alone.

### Step 6. Joint reading and meetings (items 3(a), 3(b))

*3(a).* At leading order in $\kappa$ (W3), $C=(C_{\rm D}+G)\otimes\mathbb 1\otimes\mathbb 1_c+\mathbb 1\otimes C_{\rm D}\otimes\mathbb 1_c$ is a sum of commuting terms on different factors. The product start evolves into $\psi_1(\lambda)\otimes\psi_2(\lambda)\otimes\chi$, and the class of $(a_1,a_2)$ has the projector $\Pi_{a_1}\otimes\Pi_{a_2}\otimes\mathbb 1_c$. The types are pairwise different, so (1.11) imposes nothing. Body $2$ is the case $\mathcal F=0$, $p_0=0$ of Steps 1–4, i.e. (5.21) with $\Gamma(0)=1$:

$$
q_{12}(a_1,a_2;\lambda)=q_1(a_1;\lambda)\,q_2(a_2;\lambda)+O(\kappa),\qquad
q_2(a;\lambda)=P_N\bigl(a;\vartheta_2(\lambda)\bigr)+\mathcal R_1,\qquad\vartheta_2(\lambda)=\frac\lambda{\lambda_0} .
\tag{9.17}
$$

The factorization is exact at $\kappa=0$.

*3(b).* For body $2$, Step 3 with $\mathcal F=0$ and $p_0=0$ gives $\Delta\bar x_{{\rm lab},2}=\bar w(0)\lambda+\mathcal R_x=\mathcal R_x$, in agreement with (5.25): no displacement to the order of (9.7). The starts have equal mean labels and $t_\pi=0$, so by W2 the bodies meet iff the leading term of (9.7) vanishes, $\sin^2(p_0+\mathcal F\lambda)=\sin^2p_0$:

$$
\mathcal F\lambda\in\{\,n\pi:\ n\ge1\,\}\ \cup\ \{\,2\lvert p_0\rvert+n\pi:\ n\ge0\,\}\setminus\{0\},\qquad
\lambda_R=\frac{2\lvert p_0\rvert}{\mathcal F}\ \ (p_0<0),\qquad\lambda_R=\frac\pi{\mathcal F}\ \ (p_0=0).
\tag{9.18}
$$

For $p_0<0$, body $1$ leaves towards decreasing labels ($\bar w(p_0)<0$), stops at $\lambda_R/2$ where $p=0$, at the displacement $-[\bar\omega(p_0)-\bar\mu]/\mathcal F$, and returns with label $\lvert p_0\rvert$; on $[0,\lambda_R]$, $\lvert p(\lambda)\rvert\le\lvert p_0\rvert$, so the path stays on the lower branch of (9.24) iff $\sin^2p_0\le s_*$. The label $p(\lambda)$ of (9.1), taken mod $2\pi$, has period $2\pi/\mathcal F$ in $\lambda$. The functions $\bar\omega$, $\bar w$ and $\Gamma$ have period $\pi$ in the label, so the retained displacement (9.7), the velocity (9.9) and the rate (9.13) have period $\pi/\mathcal F$. The meetings with $n\ge1$ lie after the label has passed $\frac\pi2$ and repeat those of the first period. For $p_0=0$ the two families coincide. Because of $\mathcal R_x$, $\lambda_R$ for $p_0<0$ is determined up to the relative shift $\frac{\bar\omega(p_0)}{m\lvert\sin p_0\cos p_0\rvert}\,O(\epsilon^2+\delta p^2)$, which is not uniform for $p_0\to0^-$.

### Step 7. Advances at the meetings (items 3(c)–3(e))

*3(c).* Insert (9.18) into (9.12) and (9.17), with $\mathcal I$ odd. For $p_0<0$:

$$
\vartheta_2(\lambda_R)=\frac{2\lvert p_0\rvert}{\mathcal F\lambda_0},\qquad
\vartheta_1(\lambda_R)=\frac{2\,\mathcal I(\lvert p_0\rvert)}{\mathcal F\lambda_0},\qquad
\frac{\vartheta_1(\lambda_R)}{\vartheta_2(\lambda_R)}=\frac{\mathcal I(\lvert p_0\rvert)}{\lvert p_0\rvert}=\frac1{\lvert p_0\rvert}\int_0^{\lvert p_0\rvert}\frac{\bar\mu\,\mathrm dp}{\sqrt{\bar\mu^2+\tau^2\sin^2p}} .
\tag{9.19}
$$

For $p_0=0$: $\vartheta_2(\lambda_R)=\pi/\mathcal F\lambda_0$, $\vartheta_1(\lambda_R)=2\mathcal K/\mathcal F\lambda_0$, ratio $2\mathcal K/\pi$. The ratio does not depend on $\mathcal F$; it depends on $p_0$ and on $\tau/\bar\mu$, with $\bar\mu=m\bigl(1+\frac{(N-1)\epsilon}2\bigr)$.

*3(d).* $\Gamma(p)\le1$ with equality only at $\sin p=0$, and $p(\lambda')$ is not constant because $\mathcal F>0$. By (9.12) and (9.17), for all $p_0\in(-\frac\pi2,0]$, $\mathcal F>0$, $\tau>0$:

$$
\vartheta_1(\lambda)<\vartheta_2(\lambda)\quad\text{for every }\lambda>0,\qquad
\frac{2\mathcal K}\pi\le\frac{\vartheta_1}{\vartheta_2}\Big\vert_{\text{meeting}}<1 .
\tag{9.20}
$$

The clock of body $2$, the body without gradient, has advanced more at $\lambda_R$, and the advances are never equal at $\lambda>0$. The ratio tends to $1$ only for $p_0\to0^-$, where $\lambda_R\to0$, or for $\tau/\bar\mu\to0$. The same holds at every later meeting: there the ratio is $2\mathcal K/\pi$ (at $\mathcal F\lambda=n\pi$) or $\frac{2\mathcal I(\lvert p_0\rvert)+2n\mathcal K}{2\lvert p_0\rvert+n\pi}$. The lower bound follows because $\Gamma$ decreases on $[0,\frac\pi2]$, so its running mean $\mathcal I(\varphi)/\varphi$ decreases to $2\mathcal K/\pi$. At $p_0=0$ the first meeting comes after one full period $\pi/\mathcal F$ of the retained displacement (half a period of the label), and the ratio jumps to $2\mathcal K/\pi$. These are statements about the advances (9.12); the readings distinguish the two advances only where the distributions (9.11) and (9.17) differ by more than $\mathcal R_1$.

*3(e).* With $\Gamma=1-\frac{\tau^2}{2\bar\mu^2}p^2+O(p^4)$ and, from (9.9), $b^2=\frac{\tau^2}{\bar\mu^2}p^2+O(p^4)$, and $\mathrm d\lambda=\mathrm dp/\mathcal F$:

$$
\frac{\vartheta_1(\lambda_R)}{\vartheta_2(\lambda_R)}=1-\frac{\tau^2p_0^2}{6\bar\mu^2}+O(p_0^4)=1-\frac12\Bigl\langle\frac{\lVert V_1\rVert^2}{v_{\max}^2}\Bigr\rangle+O(p_0^4),\qquad
\Bigl\langle\frac{\lVert V_1\rVert^2}{v_{\max}^2}\Bigr\rangle:=\frac1{\lambda_R}\int_0^{\lambda_R}b^2\,\mathrm d\lambda=\frac{\tau^2p_0^2}{3\bar\mu^2}+O(p_0^4).
\tag{9.21}
$$

Each $O(p_0^4)$ is bounded by a multiple of $\frac{\tau^2}{\bar\mu^2}\bigl(1+\frac{\tau^2}{\bar\mu^2}\bigr)p_0^4$; in the second form it equals $-\bigl(\frac{\tau^2}{10\bar\mu^2}+\frac{\tau^4}{40\bar\mu^4}\bigr)p_0^4$ to this order.

### Step 8. Certain readings at the first meeting (item 3(f))

By (5.1), $P_N(a;\vartheta)=\delta_{a,\,\vartheta\bmod N}$ if $\vartheta\in\mathbb Z$, and $P_N(a;\vartheta)>0$ for every $a$ otherwise. Both readings at $\lambda_R$ are therefore certain, up to $\mathcal R_1$, iff both advances are integers. Since $0<\vartheta_1<\vartheta_2$ by (9.20), for $p_0<0$:

$$
\mathcal F\lambda_0=\frac{2\lvert p_0\rvert}{n_2},\qquad\frac{\mathcal I(\lvert p_0\rvert)}{\lvert p_0\rvert}=\frac{n_1}{n_2},\qquad n_1,n_2\in\mathbb N,\ 1\le n_1<n_2
\quad\Longrightarrow\quad
q_{12}(a_1,a_2;\lambda_R)=\delta_{a_1,\,n_1\bmod N}\,\delta_{a_2,\,n_2\bmod N}+O(\mathcal R_1).
\tag{9.22}
$$

Clock $2$ shows $n_2\bmod N$ and clock $1$ shows $n_1\bmod N$; they differ iff $N\nmid(n_2-n_1)$. For $p_0=0$ the conditions are $\mathcal F\lambda_0=\pi/n_2$ and $2\mathcal K/\pi=n_1/n_2$, a condition on $\tau/\bar\mu$. Since $\mathcal I(\varphi)/\varphi$ decreases strictly from $1$ to $2\mathcal K/\pi$ on $(0,\frac\pi2]$, every pair with $2\mathcal K/\pi<n_1/n_2<1$ fixes exactly one $p_0\in(-\frac\pi2,0)$ and then $\mathcal F\lambda_0$. Because $P_N(n;n+\delta)=1-O(\delta^2)$, the conditions need to hold only up to $O(\epsilon)$. The constant in $\mathcal R_1$ is the one of S4 at $\lambda/\lambda_0=n_2$ and $\mathcal F\lambda_0=2\lvert p_0\rvert/n_2$.

## Result

Exact within S1, S2: (9.1), (9.23), (9.25), the identities (9.14) and (9.24), the bounds of (9.15) and (9.16), and the factorization in (9.17) at $\kappa=0$. Order estimates in the sense of S4: (9.7), (9.8), (9.11) and the weight bound under (9.4). The statements (9.13), (9.18)–(9.20), (9.22) are exact properties of the first-order expressions (9.7) and (9.12); (9.21) is their expansion in $p_0$.

- **Limit** (9.23): errors of the state, of the reading statistics and of the mean position label on a chain of length $2L+1$; they vanish for $L\to\infty$ at fixed start and $\lambda$, after $\kappa\to0$.
- **1(a)** (9.1): $p(\lambda)=p_0+\mathcal F\lambda$, exact. (9.4): condition $\eta=\mathcal F\tau/4m^2\ll1$. (9.25): $\lvert c_{k-}\rvert\le k\epsilon/4+\eta(2+\Lambda\mathcal F\lambda)$ for all parameters; in the regime S4 the neglected weight is at most $(k\epsilon/4+2\eta)^2+O(\epsilon^3)$.
- **1(b)** (9.7): $\Delta\bar x_1=\kappa\sigma_X\hat e\,[\bar\omega(p_0+\mathcal F\lambda)-\bar\omega(p_0)]/\mathcal F$, relative remainder $O(\epsilon^2+\delta p^2)$. (9.8): the oscillating part of the velocity contributes a bounded term of relative order $\epsilon^2$. (9.9): $V_1$ and $b=\lVert V_1\rVert/v_{\max}$.
- **1(c)** (9.11), (9.12): $q_1=P_N(a;\vartheta_1(\lambda))+\mathcal R_1$ with $\vartheta_1=[\mathcal I(p_0+\mathcal F\lambda)-\mathcal I(p_0)]/\mathcal F\lambda_0$; yes, of the form (5.21).
- **1(d)** (9.13): $\lambda_0\,\mathrm d\vartheta_1/\mathrm d\lambda=\Gamma(p(\lambda))$; no other dependence on $\mathcal F$ to first order.
- **2** (9.14): relation without expansion in the label. (9.24): its inversion on the two branches; the rate is a function of the speed only on one branch. (9.15): $\lambda_0\,\mathrm d\vartheta_1/\mathrm d\lambda=\sqrt{1-\lVert V_1\rVert^2/v_{\max}^2}\,(1+\rho)$. (9.16): $\vartheta_1$ as a functional of the path.
- **3(a)** (9.17): $q_{12}=q_1q_2$, $\vartheta_2=\lambda/\lambda_0$. **3(b)** (9.18): all meetings and $\lambda_R$. **3(c)** (9.19): advances and ratio at $\lambda_R$. **3(d)** (9.20): $\vartheta_1<\vartheta_2$ at every $\lambda>0$, hence at every meeting; never equal. **3(e)** (9.21): ratio $=1-\frac12\langle\lVert V_1\rVert^2/v_{\max}^2\rangle+O(p_0^4)$. **3(f)** (9.22): integer advances; readings $n_1\bmod N$ and $n_2\bmod N$ with $n_1<n_2$.

## Consistency checks

1. **Dimensions.** $x$, $p$, $a$ are pure numbers; $\tau,m,\mathcal F,\omega_k,\bar\mu$ and $\kappa\sigma_X$ have the dimension $[\lambda]^{-1}$. Then $\mathcal F\lambda$, $\eta$, $\Lambda$, $\nu_k$, $\Xi_k$, $\vartheta_i$, $b$, $s_\mp$, $M_2$, $\varepsilon_L$ and $\Delta\bar x_{\rm lab}$ in (9.7) are pure numbers, and $V_1$ and $v_{\max}=\tau\kappa\sigma_X$ both have $[\lambda]^{-2}$.
2. **$\mathcal F\to0$ at fixed $\lambda$.** $p(\lambda)\to p_0$; $\nu_k\to0$ and $\Phi_k\to\omega_k\lambda$, so (9.3) gives $\tilde c_{k+}=\cos\delta_k$, $\tilde c_{k-}=\sin\delta_k$, (9.25) becomes $\lvert\sin\delta_k\rvert\le k\epsilon/4$, and (9.6) becomes (5.24) exactly; (9.7) $\to\bar w(p_0)\lambda$, which is (5.25); (9.12) $\to\Gamma(p_0)\lambda/\lambda_0$, which is (5.21); (9.14) becomes (5.29).
3. **$\tau\to0$.** Then $A=0$, $[B,G]=0$ and $\beta_{0,+}=(1,0)^{\rm T}$, so each clock label evolves by $e^{-i(m+D)\lambda}\lvert0\rangle$ and $q_i=P_N(a;\lambda/\lambda_0)$ exactly, for every $\mathcal F$. Accordingly $\Gamma\equiv1$, $\vartheta_1=\vartheta_2$, (9.7) $\to0$, the ratio (9.19) $\to1$, the deviation in (9.21) vanishes as $\tau^2$, and in (9.24) $b_*\to0$.

## Open issues

- The remainders $\mathcal R_x$, $\mathcal R_1$ are order estimates at fixed $\mathcal F\lambda$; their constants were not computed. The exact bound (9.25) grows linearly in $\mathcal F\lambda$, and nothing is claimed for $\mathcal F\lambda\to\infty$ (many periods), where transfer to the negative branch may accumulate.
- The explicit dependence of the rate on $\mathcal F$ at second order ($O(\eta^2)$) is identified in Step 4 but not derived in full.
- A meeting is an equality of the mean positions only. $\lambda_R$ is determined up to the shift stated in Step 6, which is not uniform for $p_0\to0^-$; the first meeting changes discontinuously between $p_0\to0^-$ and $p_0=0$.
- The limits are not interchanged: $\kappa\to0$ is not uniform in $L$, and $L\to\infty$ is not uniform in the start (through $M_2$) or in $\lambda$. Corrections in $\kappa$, including any influence of one body on the other through the medium, are out of scope.
- At fixed $\tau/\bar\mu$ the small-label result (9.15) does not distinguish the square root from $1-\frac12b^2$ at leading order; the exact relation is (9.14), with the inversion (9.24).

## Methods used

- Translation of the plane-wave label by a uniform gradient (interaction picture with respect to $G$)
- Adiabatic following in a two-branch system; integration by parts of oscillating integrals
- Rate of change of a mean value from a commutator
- Expansion of the level frequencies about the mean level; symmetry of an even phase
- Taylor expansion about the mean of a concentrated weight
- Incomplete elliptic integral of the first kind; monotonicity of running means
- Dyson series and Poisson-tail bound for the boundary error; Chebyshev's inequality and second-moment growth for the long-chain limit

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