Version 2 · earlier version; the current one is v3 · External review, round 2: minor issues

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 03-clock-and-fall
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-10

## Changes from previous version

- **Step 5 (I2).** New: the first-order velocity of the named start at finite $W$ is decomposed exactly (3.26); its window average has a long-window limit $\bar a_W$ for every finite $W$ (3.27); the two orders of the limits (long window, wide start) are compared (3.28). The average with the wide start taken first, (3.13)–(3.14), is unchanged.
- **Steps 6 and 9 (I1).** Every answer to items 2(c), 2(d), 2(e) and 4 is now stated for the acceleration of item 2(a) and its window average of item 2(b): new (3.29)–(3.31) and new tables. (3.25) keeps its tag with a changed statement. The secular acceleration, (3.15) and (3.17)–(3.19), keeps its tags as an additional result; it is now proven to be the limit $W\to\infty$ of $\bar a_W$ (3.28).
- **(3.16) (I3).** Statement changed: an exact bound with an explicit number, valid on the open chain. The edge estimates of Step 4 are corrected ($\tau M_k\to\tau(\tau+M_k)$).
- **(3.22) (I4).** Statement changed: an explicit absolute remainder at finite $\tau$ for every real $w$; new (3.33) for even $w$.
- **(I5).** The error of the double average is stated as an absolute bound (3.32).
- Unchanged in statement: (3.1)–(3.15), (3.20), (3.21), (3.23), (3.24).

## Response to verification

- **I1: Accepted and fixed in Steps 6 and 9.** The tables, (3.29)–(3.31), (3.25) and the Result section answer for the acceleration (3.10)/(3.11) and its window average. $a_{\rm sec}$ is labelled an additional result; its relation to the named quantity is (3.28).
- **I2: Accepted and fixed in Step 5.** Finite $W$: (3.26), (3.27). Wide start first: (3.14). The limits do not commute when $G_k\neq0$ (3.28). The distinction is carried into (3.29)–(3.31) and (3.25).
- **I3: Accepted and fixed in Step 5, "Range of validity".** (3.16) is rederived with norm bounds that hold on the open chain ($\lVert\hat a_0\rVert\le2\tau(\tau+M_k)$, ends included).
- **I4: Accepted and fixed in Step 8.** Finite-$\tau$ bounds (3.22), (3.33); symmetric and asymmetric starts are separated; what is and is not proven about the powers is listed.
- **I5: Accepted and fixed in Step 5,** (3.32).

## 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.** By (8.1), the replacements that leave $C$ unchanged add real numbers $\alpha_h$ to the diagonal entries of $C_b$ at the places $h=(x,s,a)$ and subtract $\alpha_h\mathbb 1_c$ from the record of $h$; both terms of (3.1) are unchanged by them, for every dependence of $\alpha_h$ on the place. With a second body, (8.1) also allows $Z$, which 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; $\langle f\rangle_W:=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert\tilde\phi_X(p)\rvert^2f(p)$, with $\langle1\rangle_W=1$; $\delta_W^2:=\sum_x(\phi_X(x+1)-\phi_X(x))^2=\langle4\sin^2\tfrac p2\rangle_W=O(W^{-2})$. For linear profiles $M_{x,k}=M_k+G_kx$ with $M_k=m+d_k$ and $G_k:=m\,g_m+d_k\,g_\eta$; $\omega_k(p)=\sqrt{M_k^2+\tau^2\sin^2p}$ as in (5.6); $b_1:=(\tau+M_k)\lvert g_\zeta\rvert+\lvert G_k\rvert+\lvert\mathcal F\rvert$.
- **Inputs used** (as quoted in the question): (5.6), (5.21), (5.22); (6.12), (6.13), (6.24), (6.14), (6.15), (6.20); (8.1), (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, the stationary-phase estimate for a phase with nondegenerate stationary points.
- **Orders.** Every statement is marked *exact* or *order estimate*; $c$ denotes a pure number that may differ from line to line. An order estimate is a bound that is rigorous up to the value of $c$.

## 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), and $\lVert\hat x\rVert=L$. Taylor's theorem and the vanishing first moment give, **exactly**, on the chain $\lvert x\rvert\le L$, 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 $\lVert A_\zeta\rVert\le\tau\zeta_{\max}$ on every chain, open ends included (a hopping operator with amplitudes of modulus $\le h$ has norm $\le2h$). 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 evolved states that differ in norm by at most $2e^\theta\theta^r/r!$, and 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 $\mathsf M_x:=m_x\mathbb 1_N+\eta_xD$,
$$\hat a=\frac{\tau^2}{2}\sum_x\bigl(\zeta_x^2-\zeta_{x-1}^2\bigr)P_x+\frac\tau2\sum_x\zeta_x\Bigl[(R_x+R_x^\dagger)\otimes\sigma_2\otimes(\mathsf M_x+\mathsf M_{x+1})+(V_{x+1}-V_x)\,(-iR_x+iR_x^\dagger)\otimes\sigma_1\otimes\mathbb 1_N\Bigr].\tag{3.6}$$
On the open chain the first sum contains the end terms $\tfrac{\tau^2}{2}(\zeta_{-L}^2P_{-L}-\zeta_{L-1}^2P_L)$. 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 in modulus by $2\kappa^2\sigma_X^2\,\tau\zeta_{\max}L\lambda$ (sector argument of Step 1, exact); 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 $(\tau,M_k,g_\zeta)\to(\tau^0,M^0_k,g^0_\zeta):=\bigl(\tau(1+g_\zeta x_0),\,M_{x_0,k},\,g_\zeta/(1+g_\zeta x_0)\bigr)$, $G_k$, $\mathcal F$ and $\tau^0g^0_\zeta=\tau g_\zeta$ unchanged, plus $V_{x_0}\mathbb 1$. Local and original parameters differ at second order in the gradients.

From (3.5)–(3.6), $\hat v=\hat v_0+\hat v_1$ with $\hat v_0=C_0'(\hat p)=-\tau\cos\hat p\,\sigma_1$ and $\hat v_1=\tfrac12\{\hat x,B'(\hat p)\}$, and $\hat a=\hat a_0+\hat a_1+\hat a_2$ 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
$$\bigl\langle\tilde\phi_Xu\bigm\vert\tfrac12\{\hat x,F(\hat p)\}\bigm\vert\tilde\phi_Xv\bigr\rangle=\Bigl\langle\xi\,u^\dagger Fv+\tfrac i2\bigl[u^\dagger F\,\partial_pv-(\partial_pu^\dagger)Fv\bigr]\Bigr\rangle_W.$$
*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 $x_0$),
$$a^{(1)}(\lambda)=\bigl\langle\alpha(p,\lambda)\bigr\rangle_W,\quad
\alpha:=\tau^2g_\zeta+\mathcal F\tau\sin p\,\chi_\lambda^\dagger\sigma_1\chi_\lambda-\operatorname{Im}\bigl[\chi_\lambda^\dagger F_a\partial_p\chi_\lambda\bigr]+\int_0^\lambda\!\mathrm ds\operatorname{Re}\bigl[u_s^\dagger B\,\partial_p\chi_s-(\partial_pu_s^\dagger)B\chi_s\bigr],\tag{3.10}$$
with $\alpha$ smooth and even in $p$ and independent of $x_0$. This is the acceleration of item 2(a) for the named start at finite $W$; it is written $a^{(1)}_W$ where the width matters.

*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 at fixed $\lambda$. 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 $p^2\le\tfrac{\pi^2}{4}\,4\sin^2\tfrac p2$ and the evenness of $\alpha$,
$$\bigl\lvert a^{(1)}_W(\lambda)-a^{(1)}_\infty(\lambda)\bigr\rvert\le\frac{\pi^2}{8}\,\delta_W^2\,\sup_p\lvert\partial_p^2\alpha\rvert\ \ (\text{exact}),\qquad \sup_p\lvert\partial_p^2\alpha\rvert\le c\,\tau\,b_1\,(1+M_k\lambda)(1+\tau\lambda)^3\ \ (\text{order estimate}),\tag{3.12}$$
for all $\lambda\ge0$ (each $\partial_p$ of $U_t$ costs at most a factor $c(1+\tau t)$, by Duhamel; no norm of an operator on the chain enters). Requirements:
- $L$: on the open chain $\lvert x\rvert\le L$ with $r:=L-\lvert x_0\rvert-W>0$, the zeroth-order coefficient is not zero but at most $c\,\tau(\tau+M_k)\,e^{\tau\lambda}(\tau\lambda)^r/r!$ in modulus, and the first-order coefficient differs from (3.10) by at most $c\,\tau\,b_1L\,\bigl(1+(\tau+M_k)\lambda\bigr)\,e^{\tau\lambda}(\tau\lambda)^r/r!$ (order estimates, from (3.4) with $\lVert\hat a_0\rVert\le2\tau(\tau+M_k)$ and $\lVert\hat a_1\rVert\le4\tau b_1L$, both valid with the end terms of (3.6); see Step 5). The limit is $r\to\infty$ at fixed $\lambda$; the corrections are 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 absolute error is at most $c\,\delta_W^2\,\tau b_1(1+M_k\lambda_{\max})(1+\tau\lambda_{\max})^3$ on $[0,\lambda_{\max}]$ (sufficient, order estimate).
- $\lambda$: fixed. None of the limits is uniform in $\lambda$; Step 5 shows that the wide-start form fails for $\lambda\gtrsim\lambda_W:=M_kW^2/\tau^2$.

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

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

For the window $[\lambda_1,T]$, $T=\lambda_1+\Lambda$, the average of an acceleration is $\overline a=(v(T)-v(\lambda_1))/\Lambda$ (exact). The first-order velocity of the named start is $v^{(1)}_W(\lambda)=\langle\nu(p,\lambda)\rangle_W$ with $\nu:=\int_0^\lambda\alpha(p,s)\,\mathrm ds$ ((3.10) and $v(0)=0$ by (3.8)).

**5.1 Finite $W$: structure of $\nu$.** Level $k$, infinite chain, $\omega=\omega_k(p)$. Let $P_\pm(p)$ project on the eigenvectors $\beta_{k,\pm}$ of $C_0(p)$ (5.6), $\Sigma:=P_+-P_-$, so $C_0=\omega\Sigma$. For a matrix function $O(p)$ write $O_{\rm d}:=\sum_\sigma P_\sigma OP_\sigma$, $O_{\rm o}:=O-O_{\rm d}$, $O^I(s):=e^{iC_0s}Oe^{-iC_0s}=O_{\rm d}+(P_+OP_-e^{2i\omega s}+\text{h.c.})$. Facts, from (5.6) (the eigenvectors are real, $\beta_\pm'=\pm\tfrac{\theta'}2\beta_\mp$, $\cos\theta=M_k/\omega$, $\sin\theta=-\tau\sin p/\omega$):
- $\Sigma'=\theta'\bigl(\lvert\beta_-\rangle\langle\beta_+\rvert+\lvert\beta_+\rangle\langle\beta_-\rvert\bigr)$ is off-diagonal, and $V:=\hat v_0^I(\lambda)=\omega'\Sigma+\omega(\Sigma')^I(\lambda)$;
- $\hat x^I(s)=\hat x+X(s)$, $X(s)=\int_0^s\hat v_0^I=s\,\omega'\Sigma+X_{\rm o}(s)$, where $X_{\rm o}$ is bounded and off-diagonal, with the $s$-independent part $\tfrac i2\Sigma\Sigma'$;
- $b_\sigma:=\langle\beta_\sigma\vert B\vert\beta_\sigma\rangle=\sigma h-\mathcal F$ with $h(p):=(g_\zeta\tau^2\sin^2p+G_kM_k)/\omega$; $b_{\rm o}:=\langle\beta_+\vert B\vert\beta_-\rangle$; $b_\sigma'=(B')_{\sigma\sigma}+\sigma\theta'b_{\rm o}$; $\{\Sigma',B_{\rm o}\}=2\theta'b_{\rm o}\mathbb 1$.

First-order perturbation theory for the Heisenberg velocity gives $\hat v^{(1)}(\lambda)=\hat v_1^I(\lambda)+i\int_0^\lambda\mathrm ds\,[C_1^I(s),V]$, with $\hat v_1^I=\tfrac12\{\hat x+X(\lambda),(B')^I(\lambda)\}$ and $C_1^I(s)=\tfrac12\{\hat x+X(s),B^I(s)\}$. With $[\hat x,f(\hat p)]=if'(\hat p)$,
$$\hat v^{(1)}(\lambda)=\tfrac12\{\hat x,Y\}+\mathcal N,\qquad\mathcal N:=\tfrac12\bigl\{X(\lambda),(B')^I(\lambda)\bigr\}-\tfrac12\Bigl\{\partial_pV,\int_0^\lambda\!B^I\Bigr\}+i\int_0^\lambda\!\mathrm ds\,\bigl[\tfrac12\{X(s),B^I(s)\},V\bigr],$$
where $Y:=(B')^I(\lambda)+i\int_0^\lambda[B^I(s),V]\,\mathrm ds$ obeys $Y(-p)=-\sigma_3Y(p)\sigma_3$. In the start $\tilde\phi_X\lvert1\rangle$ the first term gives $\langle\xi\,Y_{11}\rangle_W=0$ (parity, as in Step 4), so $\nu=\mathcal N_{11}$ (exact). The terms of $\mathcal N$ that are linear in $\lambda$ and do not oscillate are:
- from the first anticommutator: $\lambda\,\omega'\Sigma(B')_{\rm d}$ (the off-diagonal part of $(B')^I$ anticommutes with $\Sigma$);
- from the second: $-\lambda\,\omega''\Sigma B_{\rm d}-\tfrac{\lambda\omega'}2\{\Sigma',B_{\rm d}\}$, and $+\tfrac{\lambda\omega'}2\{\Sigma',B_{\rm o}\}$ from the product of the term $2i\lambda\omega'\omega P_+\Sigma'P_-e^{2i\omega\lambda}+\text{h.c.}$ of $\partial_pV$ with the oscillating part $P_+BP_-(e^{2i\omega\lambda}-1)/(2i\omega)+\text{h.c.}$ of $\int_0^\lambda B^I$;
- from the commutator: $i\lambda\omega'\bigl[\tfrac12\{\tfrac i2\Sigma\Sigma',B_{\rm d}\},\Sigma\bigr]=+\tfrac{\lambda\omega'}2\{\Sigma',B_{\rm d}\}$.

Their sum is $\lambda\sum_\sigma P_\sigma a_\sigma$ with $a_\sigma=\sigma(\omega'b_\sigma'-\omega''b_\sigma)=\omega'h'-\omega''h+\sigma\mathcal F\omega''$. Every other term of $\mathcal N$ is bounded in $\lambda$, or is $\lambda^j$ ($j\le2$) times $e^{\pm2i\omega\lambda}$; the $j=2$ terms come from $s\,\omega'\Sigma$ and $2i\lambda\omega'\omega(\dots)$ and carry the factor $\mathcal F\omega'$. With $\langle1\vert P_\pm\vert1\rangle=\tfrac12(1\pm M_k/\omega)$ and $(\omega^2)''=2\tau^2\cos2p$, **exactly**,
$$\nu(p,\lambda)=a_{\rm sec}(p)\,\lambda+\nu_0(p)+\operatorname{Re}\sum_{j=0}^{2}\lambda^jA_j(p)\,e^{2i\omega_k(p)\lambda},\qquad a_{\rm sec}(p)=\frac{g_\zeta\tau^4\sin^2p-G_k\tau^2M_k\cos2p}{\omega_k^2}+\mathcal F\,\frac{\tau^2M_k\bigl(M_k^2\cos2p-\tau^2\sin^4p\bigr)}{\omega_k^4},\tag{3.26}$$
with $\nu_0$, $A_j$ smooth, $2\pi$-periodic and linear in the gradients, and $A_2=\mathcal F\,\omega_k'(p)\,\tilde A_2(p)$ with $\tilde A_2$ smooth. At $p=0$: $a_{\rm sec}(0)=\tfrac{\tau^2}{M_k}(\mathcal F-G_k)$. The amplitudes $\nu_0$, $A_0$, $A_1$ are not given in closed form for $p\neq0$; at $p=0$ they are read off from (3.13).

**5.2 Finite $W$: the long window.** The stationary points of $\omega_k$ are $p\in\{0,\pm\tfrac\pi2,\pi\}$, with $\omega_k''=\pm\tau^2/\omega_k\neq0$ there. For smooth periodic $f$, $\int f\,e^{2i\omega_k\lambda}\mathrm dp=O(\lambda^{-1/2})$ (stationary phase); for $j=2$, $\lambda^2\omega_k'e^{2i\omega_k\lambda}=\tfrac{\lambda}{2i}\partial_pe^{2i\omega_k\lambda}$ and one integration by parts reduce the term to $\lambda\,O(\lambda^{-1/2})$. Hence $v^{(1)}_W(\lambda)=\bar a_W\lambda+O_W(\lambda)$ with $\lvert O_W(\lambda)\rvert\le c_W(1+\lambda)^{1/2}$ for all $\lambda\ge0$; $c_W$ depends on the start and the parameters and is linear in the gradients. Therefore, for every finite $W$ and every $\lambda_1$, the limit exists (**exact**) and
$$\lim_{\Lambda\to\infty}\overline{a^{(1)}_W}=\bar a_W:=\langle a_{\rm sec}(p)\rangle_W=g_\zeta\,\tau^4s_W-\frac{\tau^2G_k}{M_k}\Bigl[1-(2M_k^2+\tau^2)\,s_W\Bigr]+\frac{\tau^2\mathcal F}{M_k}\,f_W,\tag{3.27}$$
$$s_W:=\Bigl\langle\frac{\sin^2p}{\omega_k^2}\Bigr\rangle_W,\quad f_W:=\Bigl\langle\frac{M_k^2(M_k^2\cos2p-\tau^2\sin^4p)}{\omega_k^4}\Bigr\rangle_W;\qquad0<s_W\le\frac{\delta_W^2}{M_k^2},\quad0\le1-f_W\le2(M_k^2+\tau^2)\,s_W\quad(\text{exact}),$$
the last from $1-f(p)=\sin^2p\,(M_k^2+\tau^2)(2M_k^2+\tau^2\sin^2p)/\omega_k^4$. At finite window, $\overline{a^{(1)}_W}=\bar a_W+[O_W(T)-O_W(\lambda_1)]/\Lambda$ (exact), and the deviation from $\bar a_W$ is at most $2c_W(1+T)^{1/2}/\Lambda$ (order estimate). *Coefficients in the long-window limit at finite $W$:* of $g_\zeta$: $\tau^4s_W$; of $g_m$: $-\tfrac{\tau^2m}{M_k}[1-(2M_k^2+\tau^2)s_W]$; of $g_\eta$: the same with $m\to d_k$; of $\mathcal F$: $\tfrac{\tau^2}{M_k}f_W$. *Rate (order estimate):* the weight $\lvert\tilde\phi_X\rvert^2$ has the width $1/W$ in $p$ and $2\omega_k\lambda\approx2M_k\lambda+\tau^2p^2\lambda/M_k$, so the oscillating part keeps its wide-start form for $\lambda\ll\lambda_W=M_kW^2/\tau^2$, and for $\lambda\gg\lambda_W$ its growing term is $c\,(\lambda_W/\lambda)^{1/2}$ times $\tfrac{\tau^2\lvert G_k\rvert}{M_k}\lambda$. The limit (3.27) is reached for $\Lambda\gg\lambda_W$.

**5.3 Wide start first.** Integrating (3.11) with $v(0)=0$ (exact consequences of (3.11)):
$$v^{(1)}_\infty(\lambda)=\frac{\tau^2}{M_k}(\mathcal F-G_k)\,\lambda+\frac{\tau^2G_k}{M_k}\,\lambda\cos(2M_k\lambda)+\frac{\tau^2}{2M_k^2}\,(M_kg_\zeta-\mathcal F)\sin(2M_k\lambda).\tag{3.13}$$
With $\rho$ of (6.24) and $\varsigma:=\bigl(T\cos(2M_kT)-\lambda_1\cos(2M_k\lambda_1)\bigr)/\Lambda$, exactly for every window,
$$\overline{a^{(1)}_\infty}=\frac{\tau^2}{M_k}\Bigl[(\mathcal F-G_k)+G_k\,\varsigma+(M_kg_\zeta-\mathcal F)\,\rho\Bigr],\qquad\lvert\rho\rvert\le\frac1{M_k\Lambda},\qquad\varsigma=\cos(2M_kT)+\frac{\lambda_1}{\Lambda}\bigl[\cos(2M_kT)-\cos(2M_k\lambda_1)\bigr].\tag{3.14}$$
By (3.12), $\overline{a^{(1)}_\infty}=\lim_{W\to\infty}\overline{a^{(1)}_W}$ at a fixed window. *Coefficients for a long window* ($\Lambda\to\infty$ at fixed $\lambda_1$):
- of $\mathcal F$: $\tfrac{\tau^2}{M_k}(1-\rho)\to\tfrac{\tau^2}{M_k}$ (absolute error $\le\tau^2/(M_k^2\Lambda)$);
- of $g_\zeta$: $\tau^2\rho\to0$ (absolute error $\le\tau^2/(M_k\Lambda)$);
- 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 the average is exactly $0$; over $[\tfrac{\pi}{2M_k},\tfrac{\pi}{2M_k}+\tfrac{n\pi}{M_k}]$ it is $-2\tau^2G_k/M_k$.

**5.4 The two orders of the limits.** From (3.27) and the exact bounds on $s_W$, $f_W$ (with $\lvert g_\zeta\rvert\tau^4/M_k^2\le\tfrac{\tau^2}{M_k}(1+\tfrac{\tau^2}{M_k^2})M_k\lvert g_\zeta\rvert$), and from 5.3:
$$\bigl\lvert\bar a_W-a_{\rm sec}\bigr\rvert\le2\,\delta_W^2\,\frac{\tau^2}{M_k}\Bigl(1+\frac{\tau^2}{M_k^2}\Bigr)b_1\ \ (\text{exact}),\qquad\lim_{W\to\infty}\lim_{\Lambda\to\infty}\overline{a^{(1)}_W}=a_{\rm sec},\qquad\lim_{\Lambda\to\infty}\lim_{W\to\infty}\overline{a^{(1)}_W}\ \text{does not exist if }G_k\neq0 .\tag{3.28}$$
The limits **do not commute** when $G_k\neq0$. They commute for the coefficients of $\mathcal F$ and $g_\zeta$ (both orders give $\tau^2/M_k$ and $0$). Here $a_{\rm sec}$ is the **secular acceleration**, an additional result, defined by the displacement obtained from (3.13):
$$\bar x^{(1)}_{\rm lab}(\lambda)-\bar x_{\rm lab}(0)=\tfrac12\,a_{\rm sec}\,\lambda^2+\frac{\tau^2G_k}{2M_k^2}\,\lambda\sin(2M_k\lambda)+\frac{\tau^2}{2M_k^3}\bigl(M_kg_\zeta-\mathcal F-G_k\bigr)\sin^2(M_k\lambda),\qquad\boxed{\;a_{\rm sec}:=\frac{\tau^2}{M_k}\bigl(\mathcal F-G_k\bigr)\;}\tag{3.15}$$
It is not the window average of item 2(b); its relation to it is (3.28). It is also the long-window limit of the doubly averaged wide-start acceleration $\tfrac{2}{\Lambda^2}\int_0^\Lambda(\Lambda-\lambda)a^{(1)}_\infty\,\mathrm d\lambda$, with the absolute error (exact, from (3.15))
$$\Bigl\lvert\frac{2\bigl[\bar x^{(1)}_{\rm lab}(\Lambda)-\bar x_{\rm lab}(0)\bigr]}{\Lambda^2}-a_{\rm sec}\Bigr\rvert\le\frac{\tau^2\lvert G_k\rvert}{M_k^2\,\Lambda}+\frac{\tau^2\lvert M_kg_\zeta-\mathcal F-G_k\rvert}{M_k^3\,\Lambda^2}.\tag{3.32}$$
A relative error $\le\varrho^{-1}\bigl[(M_k\Lambda)^{-1}+(M_k\Lambda)^{-2}\bigr]$ follows only under the condition $\lvert\mathcal F-G_k\rvert\ge\varrho\,\max(\lvert G_k\rvert,\lvert M_kg_\zeta-\mathcal F-G_k\rvert)$ with some $\varrho>0$.

**5.5 Range of validity of the first-order treatment.** (3.10), (3.11), (3.13)–(3.15), (3.26)–(3.28) are exact statements about the first Taylor coefficient on the infinite chain. *Remainder (exact).* Take any chain, open ends included, all of whose positions lie within $\ell$ of the expansion point (the position $0$ with the parameters $\tau$, $M_k$, or $x_0$ with the local parameters of Step 3), and scale the gradients by a common $\varepsilon\in[0,1]$: $C=C_0+\varepsilon C_1$, $\hat v=\hat v_0+\varepsilon\hat v_1$, $\hat a=\hat a_0+\varepsilon\hat a_1+\varepsilon^2\hat a_2$ with $\hat a_0=i[C_0,\hat v_0]$, $\hat a_1=i[C_0,\hat v_1]+i[C_1,\hat v_0]$, $\hat a_2=i[C_1,\hat v_1]$. On this chain, with $\Omega_k:=\tau+M_k$: $\lVert C_0\rVert\le\Omega_k$, $\lVert\hat v_0\rVert\le\tau$, $\lVert C_1\rVert\le\ell\,(\lvert G_k\rvert+\lvert\mathcal F\rvert+\tau\lvert g_\zeta\rvert)\le\ell b_1$, $\lVert\hat v_1\rVert\le\tau\lvert g_\zeta\rvert\ell$, hence
$$\lVert\hat a_0\rVert\le2\tau\Omega_k,\qquad\lVert\hat a_1\rVert\le4\tau\,b_1\ell,\qquad\lVert\hat a_2\rVert\le2\tau\,(b_1\ell)^2/\Omega_k .$$
These follow from the commutators, so the end terms of (3.6) are included ($\lVert\hat a_0\rVert\ge\tau^2/2$ on the open chain is consistent with $2\tau\Omega_k$). For $f(\varepsilon)=\langle\phi\vert U_\varepsilon^\dagger\hat aU_\varepsilon\vert\phi\rangle$, Duhamel's formula gives $\lVert\partial_\varepsilon^jU_\varepsilon\rVert\le(\lambda\lVert C_1\rVert)^j$, so $\lvert f''\rvert\le4(\lambda b_1\ell)^2\lVert\hat a\rVert+4\lambda b_1\ell\,\lVert\partial_\varepsilon\hat a\rVert+2\lVert\hat a_2\rVert$. If $b_1\ell\le\Omega_k$, then $\lVert\hat a\rVert\le8\tau\Omega_k$ and $\lVert\partial_\varepsilon\hat a\rVert\le8\tau b_1\ell$, and Taylor's theorem gives, for every start on the chain, all $\lambda\ge0$ and $\kappa=0$,
$$\bigl\lvert a(\lambda)-a^{(0)}_\ell(\lambda)-a^{(1)}_\ell(\lambda)\bigr\rvert\le16\,\tau\,(\tau+M_k)\,\Bigl[b_1\,\ell\,\bigl(\lambda+(\tau+M_k)^{-1}\bigr)\Bigr]^2\qquad\bigl(b_1\ell\le\tau+M_k\bigr),\tag{3.16}$$
where $a^{(0)}_\ell$, $a^{(1)}_\ell$ are the Taylor coefficients on that chain. *Order estimates.* On the long chain, apply (3.16) to the sub-chain within $\ell=W+r$ of $x_0$; by (3.4) this costs at most $c\,\tau(\tau+M_k+\lvert\mathcal F\rvert)\,e^{\theta}\theta^r/r!$ in $a(\lambda)$, and $a^{(0)}_\ell$, $a^{(1)}_\ell$ differ from $0$ and (3.10) by the edge terms of Step 4. The first-order treatment therefore has an absolute error below a fraction $\epsilon_{\rm tol}$ of the scale $\tau^2b_1/M_k$ of (3.11) if $16\,b_1\ell^2\bigl(1+(\tau+M_k)\lambda\bigr)^2\le\epsilon_{\rm tol}\,\tau$ and $r\gg e\,\tau\lambda$. This is a sufficient condition; it is not shown to be necessary. For the long window of 5.2 it has to hold up to $\lambda=T\gg\lambda_W$, which it does for small enough gradients at fixed $W$ and $T$ (the order of limits of the question); otherwise only the regime $T\ll\lambda_W$ of (3.14) lies inside the range. 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 $x_0$ (to first order in the gradients every position of the chain gives the same result): $\tau=2t$, $M_k=\tfrac12\Delta_k$, $G_k=\tfrac12\delta\Delta_k=\tfrac12(\delta E^+_k-\delta E^-_k)$, $\mathcal F=-\tfrac12(\delta E^+_k+\delta E^-_k)$, $g_\zeta=\delta t/t$, $\omega_k^2=\tfrac14\Delta_k^2+4t^2\sin^2p$. The averaged acceleration of item 2(b) for the start $S_k$, at finite $W$ in the long-window limit, is
$$\bar a_W=16\,t^3\,\delta t\;s_W-\frac{4t^2}{\Delta_k}\,\delta\Delta_k\Bigl[1-\bigl(\tfrac12\Delta_k^2+4t^2\bigr)s_W\Bigr]-\frac{4t^2}{\Delta_k}\bigl(\delta E^+_k+\delta E^-_k\bigr)f_W ,\tag{3.29}$$
where $s_W$, $f_W$ are functions of $\Delta_k$, $t$ and the place amplitudes $\phi_X$ of the start only. With the wide start first (3.14), and in the limit $W\to\infty$ of (3.29) (additional result),
$$\overline{a^{(1)}_\infty}=\frac{8t^2}{\Delta_k}\Bigl[-\delta E^+_k+\tfrac12\,\delta\Delta_k\,\varsigma+\Bigl(\tfrac{\Delta_k}{2}\tfrac{\delta t}{t}+\tfrac12(\delta E^+_k+\delta E^-_k)\Bigr)\rho\Bigr],\qquad a_{\rm sec}=\lim_{W\to\infty}\bar a_W=-\frac{8t^2}{\Delta_k}\,\delta E^+_k ,\tag{3.17}$$
where $\rho,\varsigma$ depend on the contract through $\Delta_k$ only. Only gaps, norms of hopping blocks and differences of on-site eigenvalues between neighbouring positions occur; no single cost occurs, so all three expressions are invariant under (8.1).

*(e), other branch.* Conjugation with $\sigma_1$ leaves $A_\zeta$, $\hat v$, $V_x$, $g_\zeta$, $\mathcal F$ unchanged and maps $\lvert2\rangle\to\lvert1\rangle$, $(M_k,G_k)\to(-M_k,-G_k)$; Steps 4 and 5 hold for either sign of $M_k$, and $s_W$, $f_W$ are even in $M_k$. Under this map only the $\mathcal F$ term of (3.26) changes sign. For $S'_k(x_0)$ at finite $W$, long window:
$$\bar a'_W=g_\zeta\,\tau^4s_W-\frac{\tau^2G_k}{M_k}\Bigl[1-(2M_k^2+\tau^2)\,s_W\Bigr]-\frac{\tau^2\mathcal F}{M_k}\,f_W ,\tag{3.30}$$
with the bound and the non-commutation of (3.28) unchanged ($a_{\rm sec}\to a'_{\rm sec}$). With the wide start first, and the limit $W\to\infty$ of (3.30) (additional result):
$$a'^{(1)}_\infty=\tau^2g_\zeta\cos(2M_k\lambda)-\frac{2\tau^2}{M_k}(\mathcal F+G_k)\sin^2(M_k\lambda)-2\tau^2G_k\lambda\sin(2M_k\lambda),\qquad a'_{\rm sec}=-\frac{\tau^2}{M_k}(\mathcal F+G_k)=+\frac{8t^2}{\Delta_k}\,\delta E^-_k,\tag{3.18}$$
and $\overline{a'^{(1)}_\infty}=a'_{\rm sec}+\tfrac{\tau^2}{M_k}[G_k\varsigma+(M_kg_\zeta+\mathcal F)\rho]$.

*(d), variants.* Let $u_W:=(2M_k^2+\tau^2)s_W$ and $u^{\rm c}_W:=2(M_k^2+\tau^2)s_W$; exactly, $0<u_W\le(2+\tfrac{\tau^2}{M_k^2})\delta_W^2$ and $0<u^{\rm c}_W\le2(1+\tfrac{\tau^2}{M_k^2})\delta_W^2$. From (3.27), for the start $S_k$ at finite $W$ in the long-window limit,
$$\bar a_W:\quad\text{(a)}\ -g\tau^2(1-u_W),\quad\text{(b1)}\ -g\tau^2\tfrac{m}{M_k}(1-u_W),\quad\text{(b2)}\ -g\tau^2\tfrac{d_k}{M_k}(1-u_W),\quad\text{(c)}\ -g\tau^2(1-u^{\rm c}_W),\quad\text{(d)}\ \tfrac{\tau^2\mathcal F}{M_k}f_W ,\tag{3.31}$$
and their limits $W\to\infty$ (additional result, $M_k=m+d_k$, $d_k=2\pi k/(N\lambda_0)$):
$$a_{\rm sec}:\quad\text{(a)}\ -g\tau^2,\qquad\text{(b1)}\ -g\tau^2\frac{m}{m+d_k},\qquad\text{(b2)}\ -g\tau^2\frac{d_k}{m+d_k},\qquad\text{(c)}\ -g\tau^2,\qquad\text{(d)}\ \frac{\tau^2\mathcal F}{m+d_k}.\tag{3.19}$$

**Table 6.1: the window average of item 2(b), start $S_k$ and start $S'_k$.**

| Variant | finite $W$, $\Lambda\to\infty$: $\bar a_W$ ($S_k$) | $\bar a'_W$ ($S'_k$) | wide start first, window $[\lambda_1,T]$ ($S_k$) | the same for $S'_k$ | $\Lambda\to\infty$ after $W\to\infty$ |
| --- | --- | --- | --- | --- | --- |
| (a) | $-g\tau^2(1-u_W)$ | equal to $\bar a_W$ | $-g\tau^2(1-\varsigma)$ | equal | no limit |
| (b1) | $-g\tau^2\frac{m}{M_k}(1-u_W)$ | equal to $\bar a_W$ | $-g\tau^2\frac{m}{M_k}(1-\varsigma)$ | equal | no limit |
| (b2) | $-g\tau^2\frac{d_k}{M_k}(1-u_W)$ | equal to $\bar a_W$ | $-g\tau^2\frac{d_k}{M_k}(1-\varsigma)$ | equal | no limit for $k\ge1$; $0$ for $k=0$ |
| (c) | $-g\tau^2(1-u^{\rm c}_W)$ | equal to $\bar a_W$ | $-g\tau^2(1-\varsigma-\rho)$ | equal | no limit |
| (d) | $\frac{\tau^2\mathcal F}{M_k}f_W$ | $-\frac{\tau^2\mathcal F}{M_k}f_W$ | $\frac{\tau^2\mathcal F}{M_k}(1-\rho)$ | $-\frac{\tau^2\mathcal F}{M_k}(1-\rho)$ | $\pm\frac{\tau^2\mathcal F}{M_k}$ |

**Table 6.2: dependence of the window average on the body (start $S_k$; the answers for $S'_k$ are the same).** "At $\delta_W^2$" means: only through $u_W$, $u^{\rm c}_W$ or $f_W$, with a variation bounded by the exact bounds above and vanishing as $W\to\infty$.

| Variant | $\bar a_W$ depends on $m$ | on $k$ | on $N$, $\lambda_0$ | on $\tau$ | wide start first: depends on $m$, $k$ (and on $N$, $\lambda_0$ for $k\ge1$) |
| --- | --- | --- | --- | --- | --- |
| (a) | yes, at $\delta_W^2$ | yes, at $\delta_W^2$ | yes for $k\ge1$, at $\delta_W^2$ | yes | yes, at order one, through $\varsigma$ |
| (b1) | yes (order one for $k\ge1$; at $\delta_W^2$ for $k=0$) | yes | yes for $k\ge1$ | yes | yes, at order one |
| (b2) | yes for $k\ge1$ (it is $0$ for $k=0$) | yes | yes for $k\ge1$ | yes | yes, at order one |
| (c) | yes, at $\delta_W^2$ | yes, at $\delta_W^2$ | yes for $k\ge1$, at $\delta_W^2$ | yes | yes, at order one, through $\varsigma$ and $\rho$ |
| (d) | yes | yes | yes for $k\ge1$ | yes | yes; through $\rho$ the extra dependence vanishes as $1/(M_k\Lambda)$ |

That the dependence at $\delta_W^2$ is really present: $\partial u^{\rm c}_W/\partial(M_k^2)=-2\tau^2\langle\sin^2p\cos^2p/\omega_k^4\rangle_W<0$ for every start (exact), and $\partial u_W/\partial(M_k^2)=-\tau^2\langle\sin^2p\cos2p/\omega_k^4\rangle_W$, which is negative for wide starts ($\langle\sin^2p\rangle_W>2\langle\sin^4p\rangle_W$ at leading order in $1/W$; order estimate). In the iterated limit $W\to\infty$ after $\Lambda\to\infty$ the entries of (a) and (c) become $-g\tau^2$ for every $m$, $k$, $N$, $\lambda_0$; they still depend on $\tau$. The bond gradient alone gives $\bar a_W=g\tau^4s_W\to0$. In (a), (b1), (b2), (c) the two branches have the same window average in every column; in (d) they have opposite ones.

### 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 $\kappa=0$,
$$q(a;\lambda)=\frac1{N^2}\sum_{k,k'}e^{2\pi i(k-k')a/N}\,\langle\psi_{k'}(\lambda)\vert\psi_k(\lambda)\rangle .$$
Let $C_{\rm loc}$ be the uniform member of $\mathcal C$ with the profiles at $x_0$ ($\tau^0$, $m_{x_0}$, $\eta_{x_0}$, $V_{x_0}$ of Step 3) and $q_{\rm loc}$ its distribution for the same start. All parameters in (i) are the local ones.

*(i) Zeroth order in the gradients (exact, infinite chain).* From Step 4, $\chi_{k,\lambda}=e^{-i\omega_k\lambda}\lvert1\rangle+r_k$ with $r_k=i\sin(\omega_k\lambda)\bigl[(1-\tfrac{M_k}{\omega_k})\lvert1\rangle+\tfrac{\tau\sin p}{\omega_k}\lvert2\rangle\bigr]$, so $\lvert\langle1\vert r_k\rangle\rvert\le\tfrac{\tau^2\sin^2p}{2M_k^2}$ and $\lVert r_k\rVert^2\le\tfrac{\tau^2\sin^2p}{M_k^2}$; and $\lvert(\omega_k-\omega_{k'})-(M_k-M_{k'})\rvert\le\lvert M_k-M_{k'}\rvert\tfrac{\tau^2\sin^2p}{2m^2}$ with $\lvert M_k-M_{k'}\rvert=\eta\tfrac{2\pi\lvert k-k'\rvert}{N\lambda_0}<\tfrac{2\pi\eta}{\lambda_0}$. Hence $\lvert\chi_{k',\lambda}^\dagger\chi_{k,\lambda}-e^{-i(M_k-M_{k'})\lambda}\rvert\le\tfrac{\tau^2\sin^2p}{m^2}(2+\pi\eta\lambda/\lambda_0)$. Since $M_k-M_{k'}$ is exactly linear in $k-k'$, the geometric sum gives $P_N$, and with $\sin^2p\le4\sin^2\tfrac p2$: $\lvert q_{\rm loc}-P_N(a;\eta_{x_0}\lambda/\lambda_0)\rvert\le\delta_W^2(\tau^0/m_{x_0})^2(2+\pi\eta_{x_0}\lambda/\lambda_0)$.

*(ii) Gradients (exact, every chain that contains the start, every real $w$).* On the full clock space $C=C_{\rm loc}+C_1$ with $C_1=\sum_yy\,P_y\otimes\bigl[\sigma_3\otimes(mg_m+g_\eta D)-\mathcal F\bigr]$ plus the bond part $\tfrac{\tau g_\zeta}{2}\sum_y(y+\tfrac12)(R_y\otimes(-i\sigma_1)+\text{h.c.})$, $y=x-x_0$. For every vector $\psi$, $\lVert C_1\psi\rVert\le b_{\rm cl}\,(\lVert\hat y\psi\rVert+\tfrac12)$ with
$$b_{\rm cl}:=m\lvert g_m\rvert+d_{N-1}\lvert g_\eta\rvert+\lvert\mathcal F\rvert+\tau\lvert g_\zeta\rvert\qquad(d_{N-1}=\lVert D\rVert),$$
and under $C_{\rm loc}$, $\lVert\hat y\psi(s)\rVert\le W+\tau^0s$, because $\lVert[C_{\rm loc},\hat y]\rVert=\lVert\hat v\rVert\le\tau^0$ on every chain, open ends included. Duhamel's formula gives $\lVert\psi(\lambda)-\psi_{\rm loc}(\lambda)\rVert\le b_{\rm cl}\lambda(W+\tfrac12+\tfrac12\tau^0\lambda)$ and $\lvert q-q_{\rm loc}\rvert\le2\lVert\psi-\psi_{\rm loc}\rVert$.

*(iii) Ends and records.* By (3.4), $q_{\rm loc}$ on the chain and on the infinite chain differ by at most $4e^{\theta_0}\theta_0^r/r!$, $\theta_0=\tau^0\lambda$, $r$ the distance from the support of the start to the nearer end; the records add at most (3.3). Together, **exactly**, on the chain $\lvert x\rvert\le L$, for every real $w$:
$$q(a;\lambda)=P_N\Bigl(a;\frac{\eta_{x_0}\lambda}{\lambda_0}\Bigr)+\mathcal R_{\rm cl},\qquad\lvert\mathcal R_{\rm cl}\rvert\le\delta_W^2\frac{\tau^{0\,2}}{m_{x_0}^2}\Bigl(2+\frac{\pi\eta_{x_0}\lambda}{\lambda_0}\Bigr)+2\,b_{\rm cl}\,\lambda\Bigl(W+\tfrac12+\tfrac12\tau^0\lambda\Bigr)+4e^{\theta_0}\frac{\theta_0^{\,r}}{r!}+2\kappa^2\sigma_X^2L^2\lambda^2 .\tag{3.22}$$
In the limit of item 3(b) ($g_\bullet W\to0$, $g_\bullet\tau\lambda\to0$, $\mathcal F\lambda\to0$ at fixed $\lambda$, $W$ and profiles at $x_0$) the second term vanishes. 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: what is proven.*
- *Every real $w$ (asymmetric starts included).* By (3.22), the width changes $q$ by at most $2b_{\rm cl}\lambda(W+\tfrac12)$ and the beginning motion by at most $b_{\rm cl}\tau^0\lambda^2$: first power of the gradients; first power of $W$ and of $\lambda$ for the width, power $\lambda^2$ for the motion. A first-power term is present in general: at $\tau=0$ the contract is diagonal in $x$ and, exactly, $q=\sum_x\phi_X(x)^2P_N(a;\eta_x\lambda/\lambda_0)$, whose expansion about $x_0$ starts with $\partial_\vartheta P_N\cdot g_\eta\,\bar y_0\,\lambda/\lambda_0$, $\bar y_0:=\bar x_{\rm lab}(0)-x_0=O(W)$. It is the rate at the mean position of the start, and it vanishes if the rate is taken at $\bar x_{\rm lab}(0)$. By continuity in $\tau$ it is present for small $\tau>0$; for general $\tau>0$ only the bound is proven.
- *Even $w$, chain symmetric about $x_0$.* Let $\Pi$ be the reflection $y\to-y$ times $\sigma_3$. Then $\Pi C_{\rm loc}\Pi=C_{\rm loc}$, $\Pi C_1\Pi=-C_1$, the clock reading commutes with $\Pi$, and the start is $\Pi$-invariant. Hence $q$ is an even function of $(g_m,g_\eta,g_\zeta,\mathcal F)$ jointly at fixed profiles at $x_0$ (exact): no term of odd total degree. With the second-order Duhamel formula and $\lVert\hat y^{\,n}\psi(s)\rVert\le c\,(W+1+\tau_{\max}s)^n$, $n\le2$, $\tau_{\max}:=\tau\max_x\zeta_x$,
$$\lvert q-q_{\rm loc}\rvert\le c\,\Bigl[b_{\rm cl}\,\lambda\,\bigl(W+1+\tau_{\max}\lambda\bigr)\Bigr]^2\qquad(\text{order estimate; even }w,\ \text{symmetric chain}).\tag{3.33}$$
The width then first enters at the second power of the gradients times $W\lambda$, the beginning motion at the second power of the gradients times $\tau\lambda^2$. On a chain that is not symmetric about $x_0$ the end term of (3.22) is added.
- *Second-order coefficients.* Width: at $\tau=0$ the exact mixture gives $q=P_N(a;\vartheta)+\tfrac12\partial_\vartheta^2P_N\,(g_\eta s_W\lambda/\lambda_0)^2+O\bigl((g_\eta W\lambda/\lambda_0)^3\bigr)$ about the mean position, $s_W^2:=\sum_x(x-\bar x_{\rm lab}(0))^2\phi_X(x)^2\le W^2$; the coefficient is not zero, and it comes from the clock gradient only. Motion: for variant (d), (9.12) at $p_0=0$ gives $\vartheta=\tfrac{\lambda}{\lambda_0}\bigl[1-\tfrac{\tau^2\mathcal F^2\lambda^2}{6\bar\mu^2}+O((\mathcal F\lambda)^4)\bigr]$ for the start of 09-proper-time, which differs from $S_{\rm clock}$ by $O(\delta_W\tau/m)$ in norm. **Not proven:** that the second-order coefficients are nonzero for $g_m$, $g_\zeta$, for the motion term of $g_\eta$, and for the width at general $\tau>0$; there the powers above are upper bounds, not shown to be attained.

### 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 accuracy of (3.22) for each of the two runs. 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.* "The acceleration from rest" is the acceleration of item 2(a), (3.10)/(3.11), and its window average of item 2(b).

| Variant | (3.10)/(3.11) the same for all $k$, $m$ (at $\lambda>0$) | $\bar a_W$ (finite $W$, $\Lambda\to\infty$) the same | $\lim_{W\to\infty}\bar a_W$ the same | window average with the wide start first the same | clock rate depends on the place |
| --- | --- | --- | --- | --- | --- |
| (a) | no | no; $\lvert\bar a_W+g\tau^2\rvert\le\lvert g\rvert\tau^2(2+\tfrac{\tau^2}{M_k^2})\delta_W^2$ | yes, $-g\tau^2$ | no (order one, no long-window limit) | yes |
| (b1) | no | no | no | no | no |
| (b2) | no | no | no | no | yes |
| (c) | no | no; $\lvert\bar a_W+g\tau^2\rvert\le2\lvert g\rvert\tau^2(1+\tfrac{\tau^2}{M_k^2})\delta_W^2$ | yes, $-g\tau^2$ | no (order one, no long-window limit) | yes |
| (d) | no | no | no | no (the limit $\tfrac{\tau^2\mathcal F}{M_k}$ exists and depends on $k$, $m$) | no |

In the first column, (3.11) is a non-constant function of $M_k\lambda$ in every variant (for (a): $-2g\tau^2[\sin^2(M_k\lambda)+M_k\lambda\sin(2M_k\lambda)]$); only at $\lambda=0$ is the acceleration the same for all $k$ and $m$, by (3.8). Therefore:
$$\begin{aligned}&\text{For the acceleration of item 2(a), and for its window average at finite }W\text{ or with the wide start taken first, no variant has an acceleration from rest}\\&\text{that is the same for all }k\text{ and }m.\ \text{In the iterated limit }\Lambda\to\infty\text{, then }W\to\infty\text{, the variants (a) and (c), and only these among the five,}\\&\text{have both properties (window average }-g\tau^2\text{; place-dependent clock rate); at finite }W\text{ they have the first up to the bounds of the table.}\end{aligned}\tag{3.25}$$

## 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(a).** First-order acceleration at finite $W$ (3.10), its wide-start limit (3.11), requirements (3.12) and Step 4.
- **Item 2(b).** Finite $W$: structure (3.26); the long-window limit $\bar a_W$ exists and is (3.27), with its coefficients. Wide start first: (3.13), (3.14); the coefficients of $g_m$, $g_\eta$ have no long-window limit. The two limits do not commute for $G_k\neq0$ (3.28). Remainder of the first-order treatment (3.16), exact on the open chain; the range is a sufficient condition.
- **Item 2(c).** (3.29) for $\bar a_W$; (3.17) for the wide-start-first average.
- **Item 2(d), 2(e).** (3.31), (3.30), Tables 6.1 and 6.2.
- **Item 3.** Uniform profiles (3.20), tick (3.21); linear profiles (3.22) with an absolute remainder, rate (3.23), first corrections (3.33) and Step 8; ratio (3.24).
- **Item 4.** Table of Step 9 and (3.25).
- **Additional result (not an answer to an item).** The secular acceleration $a_{\rm sec}$ (3.15), with (3.17)–(3.19) and the absolute error (3.32); it equals $\lim_{W\to\infty}\lim_{\Lambda\to\infty}$ of the window average (3.28).
- **Controls (M9).** A window average independent of $k$ and $m$ fails in (b1), (b2), (d) in every limit, and in (a), (c) at finite $W$ and with the wide start first; 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, $\tau$, $M_k$, $G_k$, $\mathcal F$, $b_1$ have the unit $1/\lambda$ and $g_\bullet$ none; $s_W$ has the unit $\lambda^2$, and $u_W$, $f_W$ none; (3.8), (3.11), (3.15), (3.16), (3.26), (3.27) have the unit $1/\lambda^2$, (3.23) $1/\lambda$, and (3.22), (3.33) none. For zero gradients (3.6) is $\hat a_0$ of (6.12), (3.11) and (3.27) vanish, (3.22) is $P_N(a;\lambda/\lambda_0)$ up to its first term, 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), and both orders of the limits give $\tau^2\mathcal F/M_k$, as (3.28) states. In (3.26) the branch values are $a_\sigma=\sigma\mathcal F\omega_k''$, which is $\tfrac{\mathrm d}{\mathrm d\lambda}\sigma\omega_k'(p+\mathcal F\lambda)$ for the exact shift of the plane-wave label in (9.1). Passed.
3. **Independent derivation of (3.26), and $\tau\to0$.** The branch function $E_\sigma(p,x)=\sigma\sqrt{M_{x,k}^2+\tau^2\zeta(x)^2\sin^2p}+V_x$ with $\dot x=\partial_pE_\sigma$, $\dot p=-\partial_xE_\sigma$ gives, at first order in the gradients, $\ddot x=-\partial_p^2E_\sigma\,\partial_xE_\sigma+\partial_x\partial_pE_\sigma\,\partial_pE_\sigma$, which reproduces the three coefficients of $a_\sigma$ (bond gradient: $g_\zeta\tau^4\sin^2p/\omega_k^2$; $G_k$: $-G_kM_k\tau^2\cos2p/\omega_k^2$ on both branches; $\mathcal F$: $\sigma\mathcal F\omega_k''$); at $p=0$ it is (3.15). For $\tau\to0$ all accelerations vanish as $\tau^2$ and the clock formula becomes the exact mixture of Step 8. Passed.

## Open issues

- The amplitudes $\nu_0$, $A_0$, $A_1$ of (3.26) are not given in closed form for $p\neq0$, and the rate at which the window average reaches (3.27) is an order estimate ($\Lambda\gg\lambda_W$); the constant $c_W$ is not determined.
- (3.27) and (3.14) are statements about the first Taylor coefficient. The range (3.16) is a sufficient condition; the sharp range is not determined, and the behaviour of the growing oscillation beyond first order (when $\lvert G_k\rvert W\lambda$ is not small) is not derived here; it belongs to the package on long falls.
- For variant (a) the dependence of $\bar a_W$ on $M_k$ is shown to be present for wide starts only (order estimate); for variant (c) it is exact for every start.
- Clock corrections: the second-order coefficients are computed only at $\tau=0$ (width) and for variant (d) (motion); for the other cases the powers are upper bounds.
- 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 with explicit operator norms on the open chain
- Plane-wave representation, band projectors, parity and reflection symmetry
- Separation of secular and oscillating terms; stationary-phase estimate
- Finite-propagation (Dyson series) bound
- Window averaging of oscillating terms; iterated limits
- Spectrum of a ring clock (geometric sum)

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