Version 1 · earlier version; the current one is v3 · External review, round 1: major errors

# Clock rate and acceleration from rest of a body with position-dependent factors

- **Subproject:** 05-ilang-free-fall
- **Package:** 03-clock-and-fall
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-10

## Setup and assumptions

- **System.** Body $b$ (only instance of its type) and medium $c$; $C=C_b\otimes\mathbb 1_c+\sum_xP_x\otimes K_x$ with $P_x=\lvert x\rangle\langle x\rvert$, $K_x=\kappa xX$; start $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $\lambda\ge0$. The class $\mathcal C$, the linear profiles and the starts are assumptions of this package (M8).
- **Effective body contract.** By (8.19) of 04-ilang-time/08-maximal-speed, $C_{\rm eff}=C_b+\sum_x\langle K_x\rangle P_x$. It is again a member of $\mathcal C$; its plain part contains $\kappa x\langle X\rangle$. All profiles below are those of $C_{\rm eff}$. With $\hat x:=\sum_xxP_x$,
$$C=C_{\rm eff}\otimes\mathbb 1_c+\kappa\,\hat x\otimes\bigl(X-\langle X\rangle\bigr).\tag{3.1}$$
- **M6.** A replacement $C_b\to C_b+\sum_x\alpha_xP_x$, $K_x\to K_x-\alpha_x\mathbb 1_c$ leaves both terms of (3.1) unchanged; the family (8.1) with a second body changes $C_{\rm eff}$ by a multiple of $\mathbb 1$ only (8.20). A quantity is therefore well defined if it is a function of $C_{\rm eff}$ modulo $\mathbb 1$. On the clock level $k$ (the contract commutes with $\lvert f_k\rangle\langle f_k\rvert$, so $k$ is conserved) the on-site block at $x$ is $M_{x,k}\sigma_3+V_x\mathbb 1_2$, $M_{x,k}:=m_x+\eta_xd_k$, and the hopping block from $x$ to $x+1$ is $T_x=-\tfrac{i\tau}{2}\zeta_x\sigma_1$. The well-defined quantities used are
$$E^{\pm}_{x,k}:=V_x\pm M_{x,k}\ (\text{modulo a common constant}),\quad \Delta_{x,k}:=2M_{x,k},\quad t_x:=\lVert T_x\rVert=\tfrac{\tau\zeta_x}{2},\quad \delta E^{\pm}_{x,k}:=E^{\pm}_{x+1,k}-E^{\pm}_{x,k},\quad\delta t_x:=t_x-t_{x-1}.\tag{3.2}$$
The profiles are functions of these: $m_x=\tfrac12\Delta_{x,0}$, $\eta_xd_k=\tfrac12(\Delta_{x,k}-\Delta_{x,0})$, $V_{x+1}-V_x=\tfrac12(\delta E^+_{x,k}+\delta E^-_{x,k})$.
- **Chain.** Positions $\lvert x\rvert\le L$, bonds $x\to x+1$ for $-L\le x<L$, open ends; in sums, $\zeta_{-L-1}=\zeta_L:=0$. "$x\in\mathbb Z$" is the limit $L\to\infty$ at fixed start and fixed $\lambda$ (Step 1).
- **Notation.** $R_x:=\lvert x+1\rangle\langle x\rvert$, $R:=\sum_x\zeta_xR_x$; $\tilde\phi_X(p):=\sum_xe^{-ipx}\phi_X(x)$, so $\hat x=i\partial_p$ on the infinite chain. For linear profiles $M_{x,k}=M_k+G_kx$ with $M_k=m+d_k$ and $G_k:=m\,g_m+d_k\,g_\eta$.
- **Inputs used** (as quoted in the question): (5.21), (5.22); (6.12), (6.13), (6.24), (6.14), (6.15), (6.20); (8.19), (8.20); (9.12); (17.15) of 03-ilang-space/17-motion. Standard facts: Duhamel formula, Taylor's theorem, first-order perturbation theory in a term of the contract.
- **Orders.** Every statement is marked *exact* or *order estimate*; $c$ denotes a pure number that may differ from line to line.

## Derivation

### Step 1. Weak records and the long chain

Decompose $\chi$ over the eigenvalues $\xi$ of $X$. In the sector $\xi$ the body evolves with $C_{\rm eff}+\epsilon_\xi\hat x$, $\epsilon_\xi=\kappa(\xi-\langle X\rangle)$, with weights $w_\xi$, $\sum_\xi w_\xi\epsilon_\xi=0$, $\sum_\xi w_\xi\epsilon_\xi^2=\kappa^2\sigma_X^2$. For a body operator $O$, $f(\epsilon):=\langle\phi\vert e^{i(C_{\rm eff}+\epsilon\hat x)\lambda}Oe^{-i(C_{\rm eff}+\epsilon\hat x)\lambda}\vert\phi\rangle$ has $\lvert f''\rvert\le(2\lambda\lVert\hat x\rVert)^2\lVert O\rVert$ (Duhamel, twice). Taylor's theorem and the vanishing first moment give, **exactly**, for every member of $\mathcal C$, every start and every $\lambda$,
$$\Bigl\lvert\langle O\otimes\mathbb 1_c\rangle_{\Psi(\lambda)}-\langle\phi\vert e^{iC_{\rm eff}\lambda}Oe^{-iC_{\rm eff}\lambda}\vert\phi\rangle\Bigr\rvert\le2\,\kappa^2\sigma_X^2\,L^2\lambda^2\,\lVert O\rVert .\tag{3.3}$$
In the weak-record regime ($\kappa\to0$ at fixed $\lambda$, chain, start and fixed $C_{\rm eff}$) the place weights and the clock statistics are those of $e^{-iC_{\rm eff}\lambda}\phi$; what is neglected is bounded by (3.3). From here on $\kappa=0$ unless stated.

*Finite propagation.* In the Dyson series in $A_\zeta$ (interaction picture of the on-site part, which is diagonal in $x$) each factor moves $x$ by one and $\lVert A_\zeta\rVert\le\tau\zeta_{\max}$. Hence, with $\theta:=\tau\zeta_{\max}\lambda$,
$$\bigl\lVert\text{part of }e^{-iC_{\rm eff}\lambda}\phi\text{ farther than }r\text{ from the support of }\phi\bigr\rVert\le e^{\theta}\,\frac{\theta^{\,r}}{r!}\qquad(\text{exact}).\tag{3.4}$$
Two chains that agree within the distance $r$ of the support give expectations of $O$ that differ by at most $4\lVert O\rVert e^\theta\theta^r/r!$. The limit $L\to\infty$ at fixed start and fixed $\lambda$ therefore exists whenever $\sup_x\zeta_x<\infty$; for unbounded $\zeta_x$ the results below hold on every finite chain and the limit is not claimed.

### Step 2. The operators (item 1)

Only $A_\zeta=-\tfrac{i\tau}{2}(R-R^\dagger)\otimes\sigma_1$ fails to commute with $\hat x$ (the record term of (3.1) and the on-site terms are diagonal in $x$). With $[R_x,\hat x]=-R_x$:
$$\frac{\mathrm d\bar x_{\rm lab}}{\mathrm d\lambda}=\langle\hat v\otimes\mathbb 1_c\rangle,\qquad \hat v:=i[C_{\rm eff},\hat x]=-\frac\tau2\sum_x\zeta_x\bigl(R_x+R_x^\dagger\bigr)\otimes\sigma_1\otimes\mathbb 1_N .\tag{3.5}$$
Exact for every $\kappa$. Next $\hat a:=i[C_{\rm eff},\hat v]$ has three parts: $i[A_\zeta,\hat v]=-\tfrac{\tau^2}{2}[R,R^\dagger]$ with $RR^\dagger=\sum_x\zeta_{x-1}^2P_x$, $R^\dagger R=\sum_x\zeta_x^2P_x$; the $\sigma_3$ part, by $\sigma_3\sigma_1=i\sigma_2$, gives an anticommutator; the plain part uses $[P_{x'},R_x]=(\delta_{x',x+1}-\delta_{x',x})R_x$. With $\mathsf M_x:=m_x\mathbb 1_N+\eta_xD$,
$$\hat a=\frac{\tau^2}{2}\sum_x\bigl(\zeta_x^2-\zeta_{x-1}^2\bigr)P_x+\frac\tau2\sum_x\zeta_x\Bigl[(R_x+R_x^\dagger)\otimes\sigma_2\otimes(\mathsf M_x+\mathsf M_{x+1})+(V_{x+1}-V_x)\,(-iR_x+iR_x^\dagger)\otimes\sigma_1\otimes\mathbb 1_N\Bigr].\tag{3.6}$$
Since $i[\hat x,\hat v]=A_\zeta$, the second derivative is, **exactly** for every $\kappa$, every member and every start,
$$\frac{\mathrm d^2\bar x_{\rm lab}}{\mathrm d\lambda^2}=\bigl\langle\hat a\otimes\mathbb 1_c+\kappa\,A_\zeta\otimes(X-\langle X\rangle)\bigr\rangle_{\Psi(\lambda)} .\tag{3.7}$$
The second term vanishes at $\lambda=0$ (product start) and is bounded by $2\kappa^2\sigma_X^2\,\tau\zeta_{\max}L\lambda$ (sector argument of Step 1); it is neglected in the weak-record regime. For uniform profiles and $V_x=-\mathcal Fx$, (3.5)–(3.6) are (6.12).

*Starts $S_k$, $S'_k$.* In $\lvert1\rangle$ and in $\lvert2\rangle$, $\langle\sigma_1\rangle=\langle\sigma_2\rangle=0$. Hence, exactly, for both starts and every $\kappa$,
$$\frac{\mathrm d\bar x_{\rm lab}}{\mathrm d\lambda}(0)=0,\qquad\frac{\mathrm d^2\bar x_{\rm lab}}{\mathrm d\lambda^2}(0)=\frac{\tau^2}{2}\sum_x\bigl(\zeta_x^2-\zeta_{x-1}^2\bigr)\phi_X(x)^2=2\sum_x\bigl(t_x^2-t_{x-1}^2\bigr)\phi_X(x)^2 ,\tag{3.8}$$
which is $\tau^2g_\zeta\bigl(1+g_\zeta\bar x_{\rm lab}(0)\bigr)$ for linear profiles (support inside the chain). It depends on the bond profile only, and not on $k$, $m$, the branch or the clock.

### Step 3. Linear profiles: decomposition

On level $k$, with $A_1=-\tau\sin\hat p\,\sigma_1$ and $(x+\tfrac12)R_x=\tfrac12\{\hat x,R_x\}$, the level block is, exactly,
$$C^{(k)}=C_0+C_1,\qquad C_0=-\tau\sin\hat p\,\sigma_1+M_k\sigma_3,\qquad C_1=\tfrac12\{\hat x,B(\hat p)\},\qquad B(p):=-g_\zeta\tau\sin p\,\sigma_1+G_k\sigma_3-\mathcal F .\tag{3.9}$$
*Recentring (exact).* With $y=x-x_0$ the contract has the same form about $x_0$, with $(\tau,M_k,g_\zeta)\to(\tau^0,M^0_k,g^0_\zeta):=\bigl(\tau(1+g_\zeta x_0),\,M_{x_0,k},\,g_\zeta/(1+g_\zeta x_0)\bigr)$, $G_k$ and $\mathcal F$ unchanged, plus $V_{x_0}\mathbb 1$. Local and original parameters differ at second order in the gradients.

From (3.6), $\hat a=\hat a_0+\hat a_1+\hat a_2$ with $\hat a_0=2\tau M_k\cos\hat p\,\sigma_2$ and
$$\hat a_1=\tau^2g_\zeta+\tfrac12\{\hat x,F_a(\hat p)\}+\mathcal F\tau\sin\hat p\,\sigma_1,\qquad F_a(p):=2\tau(M_kg_\zeta+G_k)\cos p\,\sigma_2 .$$

### Step 4. Acceleration to first order in the gradients (item 2(a))

Infinite chain, start $S_k(x_0)$. Let $U_t(p):=e^{-iC_0(p)t}$, $\chi_t(p):=U_t\lvert1\rangle=\bigl(\cos\omega t-i\tfrac{M_k}{\omega}\sin\omega t\bigr)\lvert1\rangle+i\tfrac{\tau\sin p}{\omega}\sin\omega t\,\lvert2\rangle$, $\omega=\omega_k(p)$. The zeroth-order state is $\psi_0(\lambda)=\tilde\phi_X\chi_\lambda$, and $\psi_1(\lambda)=-i\int_0^\lambda\mathrm ds\,U_{\lambda-s}C_1\psi_0(s)$ (first-order perturbation theory). For spinor functions $u,v$ and $\xi:=-\partial_p\arg\tilde\phi_X$, an integration by parts gives the identity
$$\bigl\langle\tilde\phi_Xu\bigm\vert\tfrac12\{\hat x,F(\hat p)\}\bigm\vert\tilde\phi_Xv\bigr\rangle=\int\frac{\mathrm dp}{2\pi}\lvert\tilde\phi_X\rvert^2\Bigl\{\xi\,u^\dagger Fv+\tfrac i2\bigl[u^\dagger F\,\partial_pv-(\partial_pu^\dagger)Fv\bigr]\Bigr\}.$$
*Parity.* $C_0(-p)=\sigma_3C_0(p)\sigma_3$, $B(-p)=\sigma_3B(p)\sigma_3$, $\chi_t(-p)=\sigma_3\chi_t(p)$, $\sigma_3\sigma_2\sigma_3=-\sigma_2$, and $\phi_X$ real makes $\lvert\tilde\phi_X\rvert^2$ and $\xi$ even. Consequences, all exact: the zeroth-order acceleration $\langle\psi_0\vert\hat a_0\vert\psi_0\rangle$ vanishes for all $\lambda$; every term with $\xi$ (which carries $x_0$ and any asymmetry of $w$) has an odd coefficient and integrates to zero. With $u_s^\dagger:=\chi_\lambda^\dagger\,\hat a_0(p)\,U_{\lambda-s}$, the first-order coefficient $a^{(1)}=\langle\psi_0\vert\hat a_1\vert\psi_0\rangle+2\operatorname{Re}\langle\psi_0\vert\hat a_0\vert\psi_1\rangle$ is, **exactly** (first Taylor coefficient, infinite chain, every real $\phi_X$, every $x_0$),
$$a^{(1)}(\lambda)=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert\tilde\phi_X(p)\rvert^2\alpha(p,\lambda),\quad
\alpha:=\tau^2g_\zeta+\mathcal F\tau\sin p\,\chi_\lambda^\dagger\sigma_1\chi_\lambda-\operatorname{Im}\bigl[\chi_\lambda^\dagger F_a\partial_p\chi_\lambda\bigr]+\int_0^\lambda\!\mathrm ds\operatorname{Re}\bigl[u_s^\dagger B\,\partial_p\chi_s-(\partial_pu_s^\dagger)B\chi_s\bigr],\tag{3.10}$$
with $\alpha$ smooth and even in $p$ and independent of $x_0$.

*Value at $p=0$.* There $\chi_t=e^{-iM_kt}\lvert1\rangle$, $\partial_p\chi_t=\tfrac{i\tau}{M_k}\sin(M_kt)\lvert2\rangle$, $\partial_pU_t=\tfrac{i\tau}{M_k}\sin(M_kt)\sigma_1$, $\hat a_0=2\tau M_k\sigma_2$, $\partial_p\hat a_0=0$, $B=G_k\sigma_3-\mathcal F$. Then $-\operatorname{Im}[\chi_\lambda^\dagger F_a\partial_p\chi_\lambda]=-\tfrac{2\tau^2}{M_k}(M_kg_\zeta+G_k)\sin^2(M_k\lambda)$, and
$$\begin{aligned}
\operatorname{Re}\bigl[u_s^\dagger B\partial_p\chi_s\bigr]&=-2\tau^2(G_k+\mathcal F)\cos\bigl(M_k(2\lambda-s)\bigr)\sin(M_ks),\\
\operatorname{Re}\bigl[(\partial_pu_s^\dagger)B\chi_s\bigr]&=\tau^2(G_k-\mathcal F)\bigl[\sin(2M_k\lambda)+\sin\bigl(2M_k(\lambda-s)\bigr)\bigr],
\end{aligned}$$
whose difference integrates to $-2\tau^2G_k\lambda\sin(2M_k\lambda)+\tfrac{2\tau^2}{M_k}\mathcal F\sin^2(M_k\lambda)$. Therefore
$$\boxed{\;a^{(1)}_\infty(\lambda):=\alpha(0,\lambda)=\tau^2g_\zeta\cos(2M_k\lambda)+\frac{2\tau^2}{M_k}\bigl(\mathcal F-G_k\bigr)\sin^2(M_k\lambda)-2\tau^2G_k\,\lambda\,\sin(2M_k\lambda)\;}\tag{3.11}$$
This is the limit of a wide start. The third term grows with $\lambda$: on the level $k$ the two branch components move in opposite directions in $p$ under $G_k\hat x\sigma_3$, so the oscillation between the branches has an amplitude proportional to $\lambda$.

*What the limits require.* With $\delta_W^2:=\sum_x(\phi_X(x+1)-\phi_X(x))^2=O(W^{-2})$, $p^2\le\tfrac{\pi^2}{4}\,4\sin^2\tfrac p2$ and evenness of $\alpha$,
$$\bigl\lvert a^{(1)}(\lambda)-a^{(1)}_\infty(\lambda)\bigr\rvert\le\frac{\pi^2}{8}\,\delta_W^2\,\sup_p\lvert\partial_p^2\alpha\rvert\ \ (\text{exact}),\qquad \sup_p\lvert\partial_p^2\alpha\rvert\le c\,\tau\,b_1\,(1+M_k\lambda)(1+\tau\lambda)^3,\tag{3.12}$$
with $b_1:=(\tau+M_k)\lvert g_\zeta\rvert+\lvert G_k\rvert+\lvert\mathcal F\rvert$; the second bound holds for all $\lambda\ge0$ up to the number $c$ (each $\partial_p$ of $U_t$ costs at most a factor $c(1+\tau t)$, by Duhamel). Requirements:
- $L$: on the chain $\lvert x\rvert\le L$ with $\lvert x_0\rvert+W<L$ the first-order coefficient differs from (3.10) by at most $c\,\tau M_k(1+b_1L\lambda)\,e^{\tau\lambda}(\tau\lambda)^r/r!$, $r=L-\lvert x_0\rvert-W$ (order estimate, from (3.4)); the limit is $r\to\infty$ at fixed $\lambda$, and the correction is negligible once $r\gg e\,\tau\lambda$.
- $x_0$: nothing on the infinite chain ((3.10) does not depend on $x_0$).
- $W$: $W\to\infty$ at fixed $\lambda$; by (3.12) the error is small on $[0,\lambda_{\max}]$ if $W^2\gg(1+M_k\lambda_{\max})(1+\tau\lambda_{\max})^3$ relative to $\tau b_1$ (sufficient).
- $\lambda$: fixed; none of the limits is uniform in $\lambda$.

In the background (17.15) the acceleration is $\kappa\,a^{(1)}\,e$.

### Step 5. Window averages (item 2(b))

Integrating (3.11) with $\mathrm d\bar x_{\rm lab}/\mathrm d\lambda(0)=0$ (exact consequences of (3.11)):
$$v^{(1)}_\infty(\lambda)=\frac{\tau^2}{M_k}(\mathcal F-G_k)\,\lambda+\frac{\tau^2G_k}{M_k}\,\lambda\cos(2M_k\lambda)+\frac{\tau^2}{2M_k^2}\,(M_kg_\zeta-\mathcal F)\sin(2M_k\lambda).\tag{3.13}$$
For the window $[\lambda_1,T]$, $T=\lambda_1+\Lambda$, with $\rho$ of (6.24) and $\varsigma:=\bigl(T\cos(2M_kT)-\lambda_1\cos(2M_k\lambda_1)\bigr)/\Lambda$, the average $\overline a=(v(T)-v(\lambda_1))/\Lambda$ is, exactly for every window,
$$\overline{a^{(1)}_\infty}=\frac{\tau^2}{M_k}\Bigl[(\mathcal F-G_k)+G_k\,\varsigma+(M_kg_\zeta-\mathcal F)\,\rho\Bigr],\qquad\lvert\rho\rvert\le\frac1{M_k\Lambda},\qquad\varsigma=\cos(2M_kT)+\frac{\lambda_1}{\Lambda}\bigl[\cos(2M_kT)-\cos(2M_k\lambda_1)\bigr].\tag{3.14}$$
*Coefficients for a long window* ($\Lambda\to\infty$ at fixed $\lambda_1$, i.e. $\Lambda\gg1/(2M_k)$):
- of $\mathcal F$: $\tfrac{\tau^2}{M_k}(1-\rho)\to\tfrac{\tau^2}{M_k}$ (limit exists, error $\le\tau^2/(M_k^2\Lambda)$);
- of $g_\zeta$: $\tau^2\rho\to0$;
- of $g_m$ and $g_\eta$: $-\tfrac{\tau^2m}{M_k}(1-\varsigma)$ and $-\tfrac{\tau^2d_k}{M_k}(1-\varsigma)$. **These have no limit:** $1-\varsigma=2\sin^2(M_kT)+O(\lambda_1/\Lambda)$ takes every value in $[0,2]$ however long the window is. Example: over $[0,n\pi/M_k]$ the $G_k$ part of $\overline a$ is exactly $0$; over $[\tfrac{\pi}{2M_k},\tfrac{\pi}{2M_k}+\tfrac{n\pi}{M_k}]$ it is $-2\tau^2G_k/M_k$.

What does converge is the displacement. Integrating (3.13),
$$\bar x^{(1)}_{\rm lab}(\lambda)-\bar x_{\rm lab}(0)=\tfrac12\,a_{\rm sec}\,\lambda^2+\frac{\tau^2G_k}{2M_k^2}\,\lambda\sin(2M_k\lambda)+\frac{\tau^2}{2M_k^3}\bigl(M_kg_\zeta-\mathcal F-G_k\bigr)\sin^2(M_k\lambda),\qquad\boxed{\;a_{\rm sec}:=\frac{\tau^2}{M_k}\bigl(\mathcal F-G_k\bigr)\;}\tag{3.15}$$
The **secular acceleration** $a_{\rm sec}$ is the limit $\Lambda\to\infty$ of the doubly averaged acceleration $\tfrac{2}{\Lambda^2}\int_0^\Lambda(\Lambda-\lambda)a^{(1)}_\infty\,\mathrm d\lambda=2[\bar x^{(1)}_{\rm lab}(\Lambda)-\bar x_{\rm lab}(0)]/\Lambda^2$, with a relative error $O(1/(M_k\Lambda))$ (exact from (3.15)); it is also the mean of (3.14) over the window end $T$ over one period $\pi/M_k$, up to $O(1/(M_k\Lambda))$. The plain window average equals $a_{\rm sec}$ plus $\tfrac{\tau^2}{M_k}[G_k\varsigma+(M_kg_\zeta-\mathcal F)\rho]$.

*Range of validity of the first-order treatment.* (3.10), (3.11), (3.13)–(3.15) are exact statements about the first Taylor coefficient. For the remainder, take a chain that reaches at most $\ell\ge1$ sites from $x_0$, parameters at $x_0$ (Step 3), and $b_1\ell\le M_k$. Taylor's theorem in a common scale of the gradients, with $\lVert C_1\rVert\le b_1\ell$, $\lVert\hat a\rVert\le c\tau M_k$, $\lVert\hat a_1\rVert\le c\tau b_1\ell$, $\lVert\hat a_2\rVert\le c\tau(b_1\ell)^2/M_k$ and Duhamel's formula, gives for all $\lambda\ge0$
$$\bigl\lvert a(\lambda)-a^{(0)}_\ell(\lambda)-a^{(1)}_\ell(\lambda)\bigr\rvert\le c\,\tau M_k\,\bigl[b_1\,\ell\,(\lambda+M_k^{-1})\bigr]^2 ,\tag{3.16}$$
rigorous up to the number $c$; $a^{(0)}_\ell,a^{(1)}_\ell$ are the coefficients on that chain, which differ from $0$ and (3.10) by the edge terms of Step 4. The first-order treatment is valid where (3.16) is small compared with (3.11), for which $b_1\ell^2(1+M_k\lambda)\ll\tau$ suffices. This is a sufficient condition (order estimate); it is not shown to be necessary. For variant (d) the quoted (6.14) gives the range $\mathcal F_{\max}T\ll\min(1,M_k/\tau)$.

### Step 6. Well-defined form, variants, other branch (items 2(c)–(e))

*(c).* With (3.2) at $x_0$: $\tau^2/M_k=8t^2/\Delta_k$, $G_k=\tfrac12\delta\Delta_k$, $\mathcal F-G_k=-\delta E^+_k$, $g_\zeta=\delta t/t$, so
$$a_{\rm sec}=-\frac{8t^2}{\Delta_k}\,\delta E^+_k,\qquad\overline{a^{(1)}_\infty}=\frac{8t^2}{\Delta_k}\Bigl[-\delta E^+_k+\tfrac12\,\delta\Delta_k\,\varsigma+\Bigl(\tfrac{\Delta_k}{2}\tfrac{\delta t}{t}+\tfrac12(\delta E^+_k+\delta E^-_k)\Bigr)\rho\Bigr],\tag{3.17}$$
where $\rho,\varsigma$ depend on the contract through $\Delta_k=2M_k$ only. Only gaps, norms of hopping blocks and differences of on-site eigenvalues between neighbouring positions occur; no single cost occurs. The secular acceleration of the start $S_k$ is fixed by the neighbour difference of the *upper* on-site eigenvalue of level $k$.

*(e), other branch.* Conjugation with $\sigma_1$ leaves $A_\zeta$, $\hat v$, $V_x$ unchanged, maps $\lvert2\rangle\to\lvert1\rangle$ and $(M_k,G_k)\to(-M_k,-G_k)$; the $p=0$ data of Step 4 hold for either sign of $M_k$. For $S'_k(x_0)$:
$$a'^{(1)}_\infty=\tau^2g_\zeta\cos(2M_k\lambda)-\frac{2\tau^2}{M_k}(\mathcal F+G_k)\sin^2(M_k\lambda)-2\tau^2G_k\lambda\sin(2M_k\lambda),\qquad a'_{\rm sec}=-\frac{\tau^2}{M_k}(\mathcal F+G_k)=+\frac{8t^2}{\Delta_k}\,\delta E^-_k,\tag{3.18}$$
and $\overline{a'^{(1)}_\infty}=a'_{\rm sec}+\tfrac{\tau^2}{M_k}[G_k\varsigma+(M_kg_\zeta+\mathcal F)\rho]$.

*(d), variants.* With $M_k=m+d_k$, $d_k=2\pi k/(N\lambda_0)$:
$$a_{\rm sec}:\quad\text{(a)}\ -g\tau^2,\qquad\text{(b1)}\ -g\tau^2\frac{m}{m+d_k},\qquad\text{(b2)}\ -g\tau^2\frac{d_k}{m+d_k},\qquad\text{(c)}\ -g\tau^2,\qquad\text{(d)}\ \frac{\tau^2\mathcal F}{m+d_k}.\tag{3.19}$$

| Variant | $a_{\rm sec}$, start $S_k$ | depends on $m$ | on $k$ | on $N$, $\lambda_0$ | on $\tau$ | $a'_{\rm sec}$, start $S'_k$ |
| --- | --- | --- | --- | --- | --- | --- |
| (a) | $-g\tau^2$ | no | no | no | yes | $-g\tau^2$ |
| (b1) | $-g\tau^2\,m/(m+d_k)$ | yes for $k\ge1$ | yes | yes for $k\ge1$ | yes | $-g\tau^2\,m/(m+d_k)$ |
| (b2) | $-g\tau^2\,d_k/(m+d_k)$ | yes for $k\ge1$ | yes | yes for $k\ge1$ | yes | $-g\tau^2\,d_k/(m+d_k)$ |
| (c) | $-g\tau^2$ | no | no | no | yes | $-g\tau^2$ |
| (d) | $\tau^2\mathcal F/(m+d_k)$ | yes | yes | yes for $k\ge1$ | yes | $-\tau^2\mathcal F/(m+d_k)$ |

The plain window averages are $-g\tau^2(1-\varsigma)$ for (a), $-g\tau^2(1-\varsigma-\rho)$ for (c), $(1-\varsigma)$ times the entry for (b1), (b2), and $(1-\rho)$ times the entry for (d). Through $\varsigma$ and $\rho$ they depend on $M_k$, hence on $m$, $k$, $N$, $\lambda_0$, in every variant; for (d) this dependence vanishes as $1/(M_k\Lambda)$, for the others it does not vanish. In (a), (b1), (b2), (c) both branches have the same secular acceleration; in (d) they have opposite ones. The bond gradient alone gives $a_{\rm sec}=0$.

### Step 7. Clock rate, uniform profiles (item 3(a))

For uniform profiles $C_b=\zeta_\ast A_1+\sigma_3\otimes(m_\ast+\eta_\ast D)+V_\ast\mathbb 1$ is, identically, the Model D contract with $(\tau,m,\lambda_0)\to(\tau\zeta_\ast,\,m_\ast,\,\lambda_0/\eta_\ast)$ plus $V_\ast\mathbb 1$; the vectors $\lvert f_k\rangle$ are unchanged and $V_\ast\mathbb 1$ changes only the common phase (1.2). Hence (5.21) and (5.22) hold with these substitutions. With $\bar d=\pi(N-1)/(N\lambda_0)$, $\bar\mu_\ast:=m_\ast+\eta_\ast\bar d$, $\epsilon_\ast:=2\pi\eta_\ast/(N\lambda_0m_\ast)$:
$$q(a;\lambda)=P_N\Bigl(a;\Gamma_\ast(p_0)\frac{\eta_\ast\lambda}{\lambda_0}\Bigr)+\mathcal R_\ast,\qquad\Gamma_\ast(p_0)=\frac{\bar\mu_\ast}{\sqrt{\bar\mu_\ast^2+\tau^2\zeta_\ast^2\sin^2p_0}},\qquad\mathcal R_\ast=O\bigl((N\epsilon_\ast)^2(1+\eta_\ast\lambda/\lambda_0)^2\bigr)+O(\delta p^2)+O(\kappa),\tag{3.20}$$
$$\lambda_\ast(p_0)=\frac{\lambda_0}{\eta_\ast\,\Gamma_\ast(p_0)},\qquad\lambda_\ast(0)=\frac{\lambda_0}{\eta_\ast}.\tag{3.21}$$
The tick at $p_0=0$ depends on $\eta_\ast$ only; it does not depend on $m_\ast$, $\zeta_\ast$ or $V_\ast$.

### Step 8. Clock rate, linear profiles (item 3(b))

Start $S_{\rm clock}(x_0)$, $\lvert0\rangle=N^{-1/2}\sum_k\lvert f_k\rangle$. With $\psi_k(\lambda):=e^{-iC^{(k)}\lambda}(\phi_X\otimes\lvert1\rangle)$, exactly at $\kappa=0$,
$$q(a;\lambda)=\frac1{N^2}\sum_{k,k'}e^{2\pi i(k-k')a/N}\,\langle\psi_{k'}(\lambda)\vert\psi_k(\lambda)\rangle .$$
*Zeroth order in the gradients at fixed profiles at $x_0$:* $C^{(k)}\to-\tau^0\sin\hat p\,\sigma_1+M_{x_0,k}\sigma_3+V_{x_0}$ (Step 3). From $\chi_t$ of Step 4, $\chi_{k',\lambda}^\dagger\chi_{k,\lambda}=e^{-i(\omega_k-\omega_{k'})\lambda}+O(\tau^{0\,2}p^2/m_{x_0}^2)$ and $\lvert(\omega_k-\omega_{k'})-(M_{x_0,k}-M_{x_0,k'})\rvert\le\lvert M_{x_0,k}-M_{x_0,k'}\rvert\,\tau^{0\,2}p^2/(2m_{x_0}^2)$, while $M_{x_0,k}-M_{x_0,k'}=\eta_{x_0}\tfrac{2\pi(k-k')}{N\lambda_0}$ is exactly linear in $k$. Summing the geometric series,
$$q(a;\lambda)=P_N\Bigl(a;\frac{\eta_{x_0}\lambda}{\lambda_0}\Bigr)+\mathcal R_{\rm cl},\qquad\mathcal R_{\rm cl}=O\Bigl(\frac{\tau^{0\,2}}{m_{x_0}^2}\,\delta_W^2\,\bigl(1+\eta_{x_0}\lambda/\lambda_0\bigr)\Bigr)+(\text{gradient terms below})+O\bigl(\kappa^2\sigma_X^2L^2\lambda^2\bigr).\tag{3.22}$$
At rest $\Gamma=1$ for every level, so the first-order form in the level spacing has no correction of higher order in the level spacing except with the factor $\delta_W^2=O(W^{-2})$. **Definition:** the advance is $\vartheta_{x_0}(\lambda)=\eta_{x_0}\lambda/\lambda_0$, and
$$\text{rate at }x_0:=\frac{\mathrm d\vartheta_{x_0}}{\mathrm d\lambda}=\frac{\eta_{x_0}}{\lambda_0}=\frac{N}{2\pi}\bigl(E^+_{x_0,k+1}-E^+_{x_0,k}\bigr),\qquad\text{tick }\frac{\lambda_0}{\eta_{x_0}} .\tag{3.23}$$
It is a difference of on-site eigenvalues (M6), the spacing of adjacent clock levels at $x_0$.

*First corrections.*
- *Reason for the order.* Let $\Pi$ be the reflection $y\to-y$ about $x_0$ times $\sigma_3$. Then $\Pi C_0\Pi=C_0$ and $\Pi C_1\Pi=-C_1$ (parameters at $x_0$), the clock reading commutes with $\Pi$, and for $w$ even the start is $\Pi$-invariant. Hence, exactly (infinite chain, $w$ even), $q$ is an even function of $(g_m,g_\eta,g_\zeta,\mathcal F)$ jointly at fixed profiles at $x_0$: no first-order term.
- *$w$ not even.* By the identity of Step 4 with $u=\chi_{k'}$, $v=\chi_k$, the first-order change of $\langle\psi_{k'}\vert\psi_k\rangle$ at $p=0$ is the factor $-i\lambda\,g_\eta(d_k-d_{k'})\,\bar y_0$, $\bar y_0:=\bar x_{\rm lab}(0)-x_0=O(W)$: the rate is $\eta$ at the mean position of the start. This is first order in $g_\eta W$ and is a relabelling of the place; the other first-order terms carry $\delta_W^2$.
- *Width.* At $\tau=0$ the contract is diagonal in $x$ and, exactly, $q(a;\lambda)=\sum_x\phi_X(x)^2P_N(a;\eta_x\lambda/\lambda_0)$, a mixture of clocks with different rates. Expanding about the mean position, the width first changes $q$ at order $(g_\eta\,s_W\,\lambda/\lambda_0)^2$, $s_W^2:=\sum_x(x-\bar x_{\rm lab}(0))^2\phi_X(x)^2\le W^2$: second power of $g_\eta W$, and through the clock gradient only.
- *Beginning motion.* By the evenness above, it first changes $q$ at second order in the gradients times $\lambda$. For variant (d), (9.12) at $p_0=0$ gives $\vartheta=\tfrac{\lambda}{\lambda_0}\bigl[1-\tfrac{\tau^2\mathcal F^2\lambda^2}{6\bar\mu^2}+O((\mathcal F\lambda)^4)\bigr]$, order $(\tau\mathcal F\lambda/\bar\mu)^2$. For the other variants the second-order coefficient is not computed here.

### Step 9. Two places and comparison (items 3(c), 4)

*Procedure.* The rates at $x_1$ and $x_2$ are taken from two runs of the same description (same contract list, starts $S_{\rm clock}(x_1)$ and $S_{\rm clock}(x_2)$), with the clock reading applied at the same value of $\lambda$ in both. The comparison relies on A8: readings at equal $\lambda$ are compared through the order of $\lambda$, which is not a time (M1). The ratio of rates is the ratio of the advances at equal $\lambda$, the inverse ratio of the ticks. By (3.23),
$$\frac{\text{rate at }x_1}{\text{rate at }x_2}=\frac{\eta_{x_1}}{\eta_{x_2}}=\frac{1+g_\eta x_1}{1+g_\eta x_2}:\qquad\text{(a), (b2), (c)}:\ \frac{1+gx_1}{1+gx_2};\qquad\text{(b1), (d)}:\ 1 .\tag{3.24}$$
It holds to the order of (3.22). It does not depend on the unit of $\lambda$ (a rescaling of $\lambda$ rescales both rates), nor on $m$, $N$, $\lambda_0$ in any variant.

*Comparison (item 4), first order in the gradients, wide start:*

| Variant | $a_{\rm sec}$ from rest the same for all $k$ and all $m$ | clock rate depends on the place |
| --- | --- | --- |
| (a) | yes, $-g\tau^2$ | yes |
| (b1) | no | no |
| (b2) | no | yes |
| (c) | yes, $-g\tau^2$ | yes |
| (d) | no | no |

$$\text{Both properties: the variants (a) and (c), and only these among the five.}\tag{3.25}$$
The statement for (a) and (c) is for the secular acceleration (3.15). The plain window average (3.14) and the function (3.11) depend on $k$ and $m$ in (a) and (c) as well, through $\cos(2M_kT)$ and $\sin(2M_k\lambda)$, with an amplitude that does not decrease.

## Result

- **Item 1.** Operators (3.5), (3.6) and the exact second derivative (3.7); values at $\lambda=0$ (3.8), equal for $S_k$ and $S'_k$.
- **Item 2.** First-order acceleration (3.10) and its wide-start limit (3.11), with the requirements (3.12); velocity (3.13); window average (3.14), whose coefficients of $g_m$, $g_\eta$ have no long-window limit; secular acceleration and displacement (3.15); remainder (3.16); well-defined form (3.17); other branch (3.18); variants (3.19) and table.
- **Item 3.** Uniform profiles (3.20), tick (3.21); linear profiles (3.22), rate (3.23), orders of the first corrections (Step 8); ratio (3.24).
- **Item 4.** Table of Step 9 and (3.25).
- **Controls (M9).** A fall independent of $k$ and $m$ fails in (b1), (b2), (d); a place-dependent rate fails in (b1), (d); equal fall of the two branches fails in (d) (start $S'_k$).

## Consistency checks

1. **Dimensions and zero gradients.** With $x$ a pure label, $\tau$, $M_k$, $G_k$, $\mathcal F$ have the unit $1/\lambda$ and $g_\bullet$ none; (3.8), (3.11), (3.15) have the unit $1/\lambda^2$, (3.23) $1/\lambda$. For zero gradients (3.6) is $\hat a_0$ of (6.12), (3.11) vanishes, (3.22) is $P_N(a;\lambda/\lambda_0)$ and (3.24) is $1$. Passed.
2. **Variant (d).** (3.5)–(3.6) reduce to (6.12); (3.11) gives $\tfrac{2\tau^2}{M_k}\mathcal F\sin^2(M_k\lambda)$, the label form of (6.13); (3.14) gives $\tfrac{\tau^2}{M_k}\mathcal F(1-\rho)$, in agreement with (6.24), (6.15) and $\overline{\ddot{\bar m}}=\mathcal F/I^{\rm lab}_k$ of (6.20). Passed.
3. **$\tau\to0$ and order $\tau^2$.** All accelerations are $\propto\tau^2$ and vanish; the clock formula becomes the exact mixture of Step 8. An independent expansion to order $\tau^2$ in the interaction picture of the on-site term (hopping block phases $e^{i(2M_k+G_k(2x+1))\lambda\sigma_3}$) reproduces the three terms of (3.13), including $-\tfrac{2\tau^2G_k}{M_k}\lambda\sin^2(M_k\lambda)$ for the $G_k$ part. Passed.

## Open issues

- At strict first order (the order of limits of the question) the plain window average of the acceleration has no long-window limit when $G_k\neq0$; this is proven. What happens to the growing oscillation of (3.11) beyond first order (in particular when $\lvert G_k\rvert W\lambda$ is not small, where the branch oscillation has different phases across the start) is not derived here; it belongs to the package on long falls.
- The range (3.16) is a sufficient condition; the sharp range of the first-order treatment is not determined.
- The second-order coefficient of the beginning motion in $q$ is given only for variant (d).
- For unbounded bond profiles the limit of a long chain is not shown.
- The two rates are compared across two runs through A8; a comparison inside one description is not treated.

## Methods used

- Heisenberg equations of motion, commutators on a chain
- Effective body contract and sector decomposition of the records
- First-order perturbation theory (Duhamel), Taylor remainder bounds
- Plane-wave representation, parity and reflection symmetry
- Finite-propagation (Dyson series) bound
- Window averaging of oscillating terms
- Spectrum of a ring clock (geometric sum)

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