04-ilang-time / 09-proper-time
09proper timeverified
Determines the time shown by a clock carried by a body under a uniform cost gradient, and compares two carried clocks between two meetings of their bodies.
# Carried clock under a uniform gradient and two clocks between two meetings
- **Subproject:** 04-ilang-time
- **Package:** 09-proper-time
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-10
## 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.** $p_\lambda:=p+\mathcal F\lambda$; $\theta_k':=\partial_p\theta_k=-\mu_k\tau\cos p/\omega_k^2$; $\Phi_k(p,\lambda):=\int_0^\lambda\omega_k(p_{\lambda'})\,\mathrm d\lambda'$.
**Assumptions and their range.**
- **S1. Weak records.** Leading order in $\kappa$ at fixed $\lambda$ and fixed chain length. It 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 $X\to\kappa X$ (Steps 3, 6); (W3) the joint statistics of the two bodies factorize (Step 6). Corrections vanish with $\kappa$; (5.21) quotes them as $O(\kappa)$.
- **S2. Long chains.** For $x\in\{-L,\dots,L\}$ (open ends) and a start supported in $\lvert x-x_0\rvert\le\ell_0$, the limits are taken in the order: leading order in $\kappa$ (S1), then $L\to\infty$ at fixed $\lambda$, start and parameters, then the expansions below. The limit $L\to\infty$ exists: $B$ and $G$ are diagonal in $x$ and $A$ moves $x$ by one step with $\lVert A\rVert\le\tau$, so the Dyson series in $A$ bounds the amplitude beyond distance $d$ by $\sum_{n\ge d}(\tau\lambda)^n/n!\le(e\tau\lambda/d)^d$, for every $\mathcal F$; with $d=L-\lvert x_0\rvert-\ell_0$ this vanishes. On $x\in\mathbb Z$, $\langle p\vert p'\rangle=2\pi\delta(p-p')$.
- **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 $\lvert0\rangle=N^{-1/2}\sum_k\lvert f_k\rangle$ and $\int\frac{\mathrm dp}{2\pi}\lvert g_i\rvert^2=1$. The weight $\lvert g_i\rvert^2/2\pi$ has mean $p_0\in(-\frac\pi2,0]$ for body $1$ and $0$ for body $2$, and variance $\delta p^2\ll1$. For smooth $f$, $\int\frac{\mathrm dp}{2\pi}\lvert g_i\rvert^2f=f(\text{mean})+r$ with $\lvert r\rvert\le\frac12\delta p^2\sup\lvert f''\rvert$.
- **S4. Orders.** $O(\epsilon^n)$ and $O(\delta p^2)$ denote bounds $K\epsilon^n$, $K\delta p^2$ with $K$ depending only on $N$, $\tau/m$, $\lambda/\lambda_0$ and $\mathcal F\lambda_0$; they are uniform in $p$, $p_0$, $k$ and $a$. 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 $\tilde c_{k-}(\lambda)=\sin\delta_k(p)-\bigl[i\nu_ke^{-2i\Phi_k}\tilde c_{k+}\bigr]_0^\lambda+i\int_0^\lambda e^{-2i\Phi_k}\frac{\mathrm d}{\mathrm d\lambda'}(\nu_k\tilde c_{k+})\,\mathrm d\lambda'$. In the last integral, $\mathrm d\nu_k/\mathrm d\lambda=O(\eta\mathcal F)$ multiplies an oscillating factor (a second integration by parts gives $O(\eta\mathcal F/m)(1+\mathcal F\lambda)$), and $\lvert\nu_k\,\mathrm d\tilde c_{k+}/\mathrm d\lambda\rvert\le2\eta^2\omega_k\lvert\tilde c_{k-}\rvert$ integrates to $O(\eta^2m\lambda)\sup\lvert\tilde c_{k-}\rvert$. By S4, $\eta=O(\epsilon)$ and $\eta^2m\lambda=O(\epsilon)$, and $\lvert\delta_k\rvert\le\frac{d_k}2\sup\lvert\partial_\mu\theta\rvert\le\frac{k\epsilon}4$ (from $\lvert\partial_\mu\theta\rvert=\tau\lvert\sin p\rvert/\omega^2\le1/2\mu$). 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. 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 $\Xi_k\le2\eta^2\omega_k\lambda=O(\epsilon)$ 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). Under (9.4) the start stays in the branch of positive eigenvalue: the neglected weight of the negative branch of level $k$ is $\lvert c_{k-}\rvert^2\le\bigl(\frac{k\epsilon}4+2\eta\bigr)^2+O(\epsilon^3)$. 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. With $\mathcal F\lambda_0$ fixed, (9.4) holds automatically for small $\epsilon$ and the neglected weight is of second order.
### 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.
- *Non-oscillating part.* $\int_0^\lambda w_k(p_{\lambda'})\,\mathrm d\lambda'=[\omega_k(p_\lambda)-\omega_k(p)]/\mathcal F$. By (5.20) and $\sum_kj=0$, $\frac1N\sum_kw_k=\bar w+O\bigl(\frac{\tau^2}m(N\epsilon)^2\bigr)$; the factor $1-2\lvert c_{k-}\rvert^2$ is $1+O(\epsilon^2)$; S3 replaces $p$ by $p_0$.
- *Oscillating part.* With $2\omega_k\cos2\Phi_k=\frac{\mathrm d}{\mathrm d\lambda}\sin2\Phi_k$ and $a_k:=\tau\cos p_\lambda\cos\theta_k/2\omega_k$, $\lvert a_k\rvert\le\tau/2m$, one integration by parts gives $-\bigl[a_k\bigl(2\sin\delta_k\sin2\Phi_k+2\nu_k(p)\cos2\Phi_k\bigr)\bigr]_0^\lambda+O(\epsilon^2)$.
$$
\Delta\bar x_1(\lambda)=\kappa\sigma_X\,\Delta\bar x_{{\rm lab},1}(\lambda)\,\hat e,\qquad
\Delta\bar x_{{\rm lab},1}(\lambda)=\frac{\bar\omega(p_0+\mathcal F\lambda)-\bar\omega(p_0)}{\mathcal F}+\mathcal R_x,\qquad
\mathcal R_x=\frac{\tau^2\lambda}m\,O\bigl(\epsilon^2+\delta p^2\bigr),
\tag{9.7}
$$
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 $\frac{\tau^2\lambda}{2\bar\mu}$; 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 $\mathcal K:=\mathcal I(\pi/2)$.
**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) and $p=p(\lambda)$. From (9.9), $b^2=\tau^2\sin^2p\,(1-\sin^2p)/\bar\omega^2$, and $\tau^2\sin^2p/\bar\omega^2=1-\Gamma^2$. 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. Conversely the rate is a function of the speed only where $b$ is monotonic in $\lvert\sin p\rvert$, i.e. for $\sin^2p\le\bar\mu/(\bar\mu+\sqrt{\bar\mu^2+\tau^2})$, where $b$ rises to its largest value $b_*=(\sqrt{\bar\mu^2+\tau^2}-\bar\mu)/\tau<1$. 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. 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 $2$, (5.25) gives $\Delta\bar x_{{\rm lab},2}=\bar w(0)\lambda+\mathcal R_x=\mathcal R_x$: 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 $\lvert p_0\rvert$. The meetings with $n\ge1$ lie after the label has passed $\pi/2$; the label motion has period $\pi/\mathcal F$ in $\lambda$. 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 is the full period and the ratio jumps to $2\mathcal K/\pi$. These are statements about the advances (9.12); for the distributions they hold where $\vartheta_2-\vartheta_1$ exceeds the remainders of (9.11).
*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 $O(\epsilon)$.
## Result
Exact within S1, S2: (9.1), the identities (9.14), the bounds of (9.15) and (9.16), and the factorization in (9.17) at $\kappa=0$. Order estimates in the sense of S4: (9.4), (9.7), (9.8), (9.11), (9.21). The statements (9.13), (9.18)–(9.20), (9.22) are exact properties of the first-order expressions (9.7) and (9.12).
- **1(a)** (9.1): $p(\lambda)=p_0+\mathcal F\lambda$, exact. (9.4): condition $\eta=\mathcal F\tau/4m^2\ll1$; neglected negative-branch weight 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 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. (9.15): $\lambda_0\,\mathrm d\vartheta_1/\mathrm d\lambda=\sqrt{1-\lVert V_1\rVert^2/v_{\max}^2}\,(1+\rho)$. (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 $\mathcal F\lambda$, $\eta$, $\nu_k$, $\Xi_k$, $\vartheta_i$, $b$ 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 $\tilde c_{k+}=\cos\delta_k$, $\tilde c_{k-}=\sin\delta_k$ 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) $\to1$ and the deviation in (9.21) vanishes as $\tau^2$.
## Open issues
- The remainders $\mathcal R_x$, $\mathcal R_1$ are order estimates at fixed $\mathcal F\lambda$; their constants were not computed. Nothing is claimed for $\mathcal F\lambda\to\infty$ (many periods of the label), 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 $p_0=0$.
- Corrections in $\kappa$, including any influence of one body on the other through the medium, are out of scope; the results assume the order of limits of S2.
- 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).
## 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