Version 5 · current

# Limits of the level-independent fall: level superpositions, records, the operator under the factor

- **Subproject:** 05-ilang-free-fall
- **Package:** 07-limits
- **Version:** v5
- **Mode:** internal regeneration
- **Date:** 2026-10-11

## Changes from previous version

Internal regeneration after `review.v4.md` (I1 major, I2 and I3 minor). Only the passages that the three issues name, and the places that repeat them, are changed; everything else is v4 word for word. Tags (7.1)–(7.46) are kept; new displays carry (7.47)–(7.51). No tag is withdrawn. The section "Response to verification" of v4 is not carried over.

**Restated displays that keep their tag.**
- **(7.44):** restated (I1). v4: one case A, defined by "$\mathcal D_k\to\infty$ or a long window", with (7.15) and (7.16) for both alternatives. Now: case A is the window average at fixed $g$; the instantaneous value after dephasing is a separate case A′; every cell states what holds at the order $\kappa^2$ and at the order $\kappa^3$. Cases B and "between" are unchanged.
- **(7.15):** formula unchanged. Status now stated for each order of limits: sharp-label coefficient of the window average (case A, bound (7.33)) and of the limit $\lambda\to\infty$ of the instantaneous acceleration at fixed $g$ (case A′, (7.47), with the rates (7.49)).
- **(7.16):** formula unchanged; status changed (I1). It is the derivative of the secular velocity term and the leading part of the window average under the window condition (7.48). It is **not** the instantaneous acceleration after dephasing, (7.50).
- **(7.46):** inequality unchanged; status: sufficient, not necessary (I2).
- **(7.22):** row "Records".

**New displays.** (7.47): exact identity for the instantaneous acceleration of order $\kappa^2$, with a bound and its limit. (7.48): leading term of $R_3$ and the window condition at the order $\kappa^3$. (7.49): size of the oscillating part at the order $\kappa^2$, for a label weight that is free of stationary points and for one that contains a stationary point. (7.50): instantaneous acceleration of order $\kappa^3$ at rest after dephasing. (7.51): the necessary and sufficient condition for a given accuracy between two levels (I2).

**Statements withdrawn (no tag):** case A as "$\mathcal D_k\to\infty$ or a long window" with (7.16) for both; the sentence "Without a window … $a_2[g](\lambda)/\lambda\to2s_2$" in its v4 form (range "away from the stationary points" without a scale; replaced by (7.47), (7.49)); "that it is the fastest-growing part rests on the estimate of $R_3$" as a statement about the acceleration; "(7.46) is also necessary up to factors of order one …"; "'weak records' has to mean (7.46)"; "no further reduction to moments exists" (I3).

**Changed passages.**
1. Step 5, part 6: the first sentence (I3).
2. Step 8, part 1: two sentences and (7.47) added at the end. Part 2: the table (7.44); the paragraph "Case A" (the text between (7.33) and (7.15), with (7.48), and the sentence after (7.16)); a new paragraph "Case A′" with (7.49), (7.50). Part 3: the items "Case A" and "Case A′", and the last item. Part 4: the sentence before (7.17); the text after (7.46), with (7.51); the first list item of the range; the last paragraph (I1, I2).
3. Step 10: row "Records". Result: the bullets "(7.41), (7.44), (7.47)", "(7.15), (7.16), (7.33), (7.48)", "(7.49), (7.50)" (new), "(7.17), (7.18), (7.34), (7.46), (7.51)". Consistency check 3: one sentence added for (7.47), (7.50). Open issues: the third item restated, two items added. Methods used: one keyword added.

Unchanged word for word: Setup; Steps 1–4, 6, 7, 9; Step 5 apart from that sentence; in Step 8 the displays (7.41), (7.33), (7.15), (7.16), (7.42), (7.43), (7.45), (7.17), (7.18), (7.34), (7.46) and the paragraph "Case B".

## Setup and assumptions

- **Setting (question.md, assumptions of this package, M8, M9).** One body $b$ of the class $\mathcal C_{\rm lev}$ with linear level profiles ($M_k,G_k,\mathcal F_k,g_{\zeta,k},V_k,\tau_k$), a medium $c$ with $K_x=\kappa xX$, no field object; $\lambda\ge0$; infinite chain as the limit of long chains. Starts: branch starts $B_{k,\sigma}[g]$ and level-superposed starts $\Phi_\sigma[\alpha,g]$, times $\chi$.
- **Effective blocks and the record term.** The level blocks $C^{(k)}$ are those of the effective body contract (8.19) of 04-ilang-time/08-maximal-speed: $\mathcal F_k=\mathcal F^{\rm body}_k-\kappa\langle X\rangle$. The total contract is therefore $C=\sum_k\bigl(C^{(k)}\otimes\mathbb 1_c+\kappa\hat x\otimes(X-\langle X\rangle\mathbb 1_c)\bigr)\otimes\lvert f_k\rangle\langle f_k\rvert$; the mean record term is counted once, in $C^{(k)}$. In (4.1), $p^{[\xi]}$ belongs to the plain gradients $\mathcal F^{\rm body}_k-\kappa\xi=\mathcal F_k-\varepsilon_\xi$.
- **Inputs, as quoted.** (5.5), (5.6), (5.11) of 04-ilang-time/05-moving-clock; (8.1), (8.19), (8.20) of 04-ilang-time/08-maximal-speed; (17.15) of 03-ilang-space/17-motion; (3.2), (3.23), (3.36) of 03-clock-and-fall; (4.1), (4.2), (4.5), (4.7), (4.10), (4.11), (4.16), (4.17), (4.19)–(4.25) of 04-branch-fall.
- **Notation.** $\sigma=\pm1$; $w_\xi:=\lVert Q_\xi\chi\rVert^2$, $\varepsilon_\xi:=\kappa(\xi-\langle X\rangle)$, so $\sum_\xi w_\xi\varepsilon_\xi=0$, $\sum_\xi w_\xi\varepsilon_\xi^2=\kappa^2\sigma_X^2$, $\sum_\xi w_\xi\varepsilon_\xi^3=\kappa^3\mu_3$, $\mu_3:=\langle(X-\langle X\rangle)^3\rangle$. Inside one level: $\omega,\theta$ of (5.6) of 04-ilang-time/05-moving-clock with $\tau\to\tau_k$, $\mu_k\to M_k$; a prime is $\partial_p$; $\hat x=i\partial_p$ on plane-wave amplitudes. $\nu_k:=\omega_k''(0)=\tau_k^2/M_k$, $t_k:=\tau_k^2/M_k^2$, $p_c:=\min(1,M_k/\tau_k)$ (for a pair of levels the smaller value). $\operatorname{Var}_\alpha(y):=\sum_k\lvert\alpha_k\rvert^2\bigl(y_k-\sum_j\lvert\alpha_j\rvert^2y_j\bigr)^2$.
- **Facts used.** $\langle\beta_{k,\sigma}\vert\sigma_3\vert\beta_{k,\sigma}\rangle=\sigma M_k/\omega_k$, $\langle\beta_{k,\sigma}\vert\sigma_1\vert\beta_{k,\sigma}\rangle=-\sigma\tau_k\sin p/\omega_k$; $\omega\omega'=\tfrac12\tau^2\sin2p$, hence $\omega''(0)=\tau^2/M$, $\omega'''=-\omega'(4+3\omega''/\omega)$, $\omega''''(0)=-\nu(4+3t)$; $\theta'=-\tau M\cos p/\omega^2$, $\lvert\theta'\rvert\le\tau/M$; at $p=0$: $\theta'=-\tau/M$, $\theta''=0$, $\theta'''=\tfrac\tau M(1+2t)$, $(\omega\theta')''=\tau(1+t)$.
- **Level offsets.** $V_k\mathbb 1$ commutes with $C^{(k)}$ and multiplies the level component $k$ by $e^{-iV_k\lambda}$. For a start in one level this is a common phase (1.2), so every result quoted from 04-branch-fall holds for any $V_k$. In a level-superposed start the offsets enter only the phases of the coherences (Step 5), never $p_x$ or $\lvert\Gamma_{kk'}\rvert$.
- **Regimes (A13).** "First order in the gradients" as in question.md. Record results: Taylor order in $\kappa$ at fixed $\lambda$, start and chain; the range on which a term dominates is stated with it. Status words: *exact* (an identity), *bound* (an exact inequality), *order estimate* (not a bound; the calculation of its scaling is given with it).

## Derivation

### Step 1. Item 1(a),(b): position statistics of a level-superposed start

1. *Exact mixture.* The total contract of the Setup is block-diagonal in the level, so $\Psi(\lambda)=\sum_k\alpha_k\,\Psi^{(k)}(\lambda)\otimes\lvert f_k\rangle$ with $\Psi^{(k)}(\lambda)$ the evolved position–branch–medium vector of $B_{k,\sigma}[g]\otimes\chi$. A reading of the position is a partition by $x$ (A4); it sums over the clock label, and $\langle f_{k'}\vert f_k\rangle=\delta_{kk'}$ removes all cross terms (witness rule (1.7) with the clock label as companion). Hence
$$
p_x(\lambda)=\sum_k\lvert\alpha_k\rvert^2\,p^{(k)}_x(\lambda),\qquad\bar x_{\rm lab}(\lambda)=\sum_k\lvert\alpha_k\rvert^2\,\bar x_k(\lambda),
\tag{7.1}
$$
with $p^{(k)}_x$, $\bar x_k$ those of $B_{k,\sigma}[g]$. **Exact** in the gradients and in $\kappa$, for every $\lambda$, chain, $g$, $\alpha$, $\chi$. It rests only on the conservation of the level and on the orthonormality of the $\lvert f_k\rangle$; it does not depend on how the mean record term is assigned. Every reading of the position alone is that of the mixture of single-level starts with weights $\lvert\alpha_k\rvert^2$; the phases of $\alpha$ and the $V_k$ do not enter.
2. *Drift acceleration.* (7.1) is linear in the single-level quantities, so (4.10) and (4.11) apply term by term, in the same order of limits (first order in the gradients of all levels with $\alpha_k\neq0$ at fixed $\lambda$, start, chain; long chain; $\Lambda\to\infty$; $\delta p\to0$):
$$
\Bigl\lvert\overline{a^{(1)}}-\sum_k\lvert\alpha_k\rvert^2\langle a_{k,\sigma}\rangle_g\Bigr\rvert\le\sum_k\lvert\alpha_k\rvert^2\frac{2\tau_k}{M_k\Lambda}\mathsf b_k\Bigl(X_g+\frac{\tau_k}{2M_k}\Bigr),\qquad a_{\rm drift}[\alpha]=\sum_k\lvert\alpha_k\rvert^2\,a_{k,\sigma}(p_0).
\tag{7.2}
$$
Taking $\alpha_k=\delta_{kk_0}$ for each $k_0$: $a_{\rm drift}[\alpha]$ is the same for all $\alpha$ **iff** $a_{k,\sigma}(p_0)$ does not depend on $k$. At rest this is line 1 ($\sigma=+$) or line 2 ($\sigma=-$) of (4.19), unchanged: a superposition of levels adds and removes no condition.
3. *What position readings distinguish.* If $a_{k,\sigma}(p_0)$ is level-independent, the drift acceleration of $\Phi_\sigma[\alpha,g]$, in the limits of (7.2), is that of one level and carries no trace of $\alpha$; at finite spread and finite window the averages $\langle a_{k,\sigma}\rangle_g$, the bounded rests of (4.7) and, for $p_0\neq0$, the velocities of Step 3 can still depend on the level. If not, (i) the mean acceleration (7.2) depends on $\lvert\alpha_k\rvert^2$, and (ii) $p_x$ is a mixture of components that separate, so the variance of the position exceeds the weighted mean of the single-level variances by the between-level variance (7.3). A body in one level has neither.

### Step 2. Item 1(c): variance of the first-order displacements, start at rest

By (4.7) the first-order velocity of level $k$ is $\lambda\langle a_{k,\sigma}\rangle_g+\rho_k(\lambda)$ with $\sup_{\lambda\ge0}\lvert\rho_k\rvert=:r_k<\infty$; integrating, $\bar x^{(1)}_k=\tfrac12\lambda^2\langle a_{k,\sigma}\rangle_g+O(\lambda)$, the rest bounded by $r_k\lambda$. With $r:=\max_kr_k$ and $\operatorname{Var}(A+B)-\operatorname{Var}(A)=2\operatorname{Cov}(A,B)+\operatorname{Var}(B)$:
$$
D(\lambda):=\operatorname{Var}_\alpha\bigl(\bar x^{(1)}\bigr)=\frac{\lambda^4}{4}\operatorname{Var}_\alpha\bigl(\langle a_{\cdot,\sigma}\rangle_g\bigr)+E(\lambda),\quad\lvert E\rvert\le r\lambda^3\sqrt{\operatorname{Var}_\alpha(\langle a_{\cdot,\sigma}\rangle_g)}+r^2\lambda^2;\qquad\lim_{\delta p\to0}\operatorname{Var}_\alpha\bigl(\langle a_{\cdot,\sigma}\rangle_g\bigr)=\operatorname{Var}_\alpha\bigl(a_{\cdot,\sigma}(0)\bigr).
\tag{7.3}
$$
Order of limits: first order in the gradients at fixed $\lambda$, start, chain; long chain; the $\lambda^4$ part dominates for $\lambda\gg r/\sqrt{\operatorname{Var}_\alpha}$ at fixed $g$. The limit $\delta p\to0$ in (7.3) is a limit of the **coefficient** of $\lambda^4$, for every sequence of starts at rest with $\delta p\to0$; it uses only the mean $0$ and the variance $\delta p^2$ of the label weight (error $\le\tfrac12\sup\lvert a''_{k,\sigma}\rvert\delta p^2$ per level, (4.11)). It is not a limit of $D$ at fixed $\lambda$: $r$ depends on $g$. $D$ is of second order in the gradients. For $\lvert\alpha_k\rvert^2=1/N$, with (4.25), $M_k=m+d_k$ and $\mathcal V_N:=\frac1N\sum_kM_k^{-2}-\bigl(\frac1N\sum_kM_k^{-1}\bigr)^2>0$ (for $N=2$: $\mathcal V_2=\tfrac14\bigl(\tfrac1{M_0}-\tfrac1{M_1}\bigr)^2$):
$$
\begin{array}{l|c|c|c|c|c}
\text{variant}&\text{(a)}&\text{(b1)}&\text{(b2)}&\text{(c)}&\text{(f)}\\\hline
\lim_{\delta p\to0}\ \lambda^4\text{-part of }D&0&\tfrac14\lambda^4\tau^4g^2m^2\,\mathcal V_N&\tfrac14\lambda^4\tau^4g^2m^2\,\mathcal V_N&0&\tfrac14\lambda^4\tau^4g^2c_{\rm f}^2\,\mathcal V_N
\end{array}
\tag{7.4}
$$
((b2): $d_k/M_k=1-m/M_k$; (f): both branches.) In (a), (c) the coefficient of $\lambda^4$ is, by (4.11), at most $\tfrac1{16}\max_k\sup_p\lvert a''_{k,\sigma}\rvert^2\,\delta p^4$ for every start at rest, and vanishes in the sharp-label limit; what remains is $E$. Control: (a) against (b1).

### Step 3. Item 1(d): velocities at $p_0\neq0$

By (4.5), exactly at zero gradients and for every smooth $g$ of any width, the level component $k$ has the constant velocity $v_k=\sigma\langle\omega_k'\rangle_g$. For a start concentrated at $p_0$ with the spread $\delta p$, Taylor's formula for $\omega_k'$ on the arc $(p_0-\pi,p_0+\pi]$ gives, using only the mean $p_0$ and the variance $\delta p^2$ of the label weight,
$$
\bigl\lvert v_k-\sigma\,\omega_k'(p_0)\bigr\rvert\le\tfrac12\sup_p\lvert\omega_k'''\rvert\,\delta p^2\ \ (\text{exact}),\qquad v_k\neq v_{k'}\ \ \text{whenever}\ \ \delta p^2<\delta p_{*}^2:=\frac{2\,\lvert\omega_k'(p_0)-\omega_{k'}'(p_0)\rvert}{\sup_p\lvert\omega_k'''\rvert+\sup_p\lvert\omega_{k'}'''\rvert}.
\tag{7.27}
$$
Hence, for every sequence of starts concentrated at $p_0$ with $\delta p\to0$,
$$
\lim_{\delta p\to0}v_k=\sigma\,\omega_k'(p_0)=\sigma\,\frac{\tau_k^2\sin p_0\cos p_0}{\sqrt{M_k^2+\tau_k^2\sin^2p_0}};\qquad\omega_k'(p_0)=\omega_{k'}'(p_0)\ (\sin2p_0\neq0)\iff\frac{\tau_k^4}{M_k^2+\tau_k^2\sin^2p_0}=\frac{\tau_{k'}^4}{M_{k'}^2+\tau_{k'}^2\sin^2p_0}.
\tag{7.5}
$$
*Can a start of this class have level components with equal velocities?*
- *Starts of finite width:* **yes**, in every member of $\mathcal C_{\rm lev}$, the class $\mathcal C$ included. Example: $\lvert g(p)\rvert^2=1+\epsilon\cos(p-p_0)$, $0<\lvert\epsilon\rvert<1$, is normalized and has the mean $p_0$ on the arc. Every $\omega_k'$ has the period $\pi$ and the mean zero, so it contains only even harmonics and $\langle\omega_k'\rangle_g=0$: all level components have the velocity zero, also for $\sin2p_0\neq0$. The spread of this start is of order one ($\delta p^2=\pi^2/3-2\epsilon$).
- *Sharp label, class $\mathcal C$* ($\tau_k=\tau$, $M_k$ strictly increasing): the sharp-label velocities $\sigma\omega_k'(p_0)$ are strictly ordered for $\sin2p_0\neq0$. So **no** in the following sense: for $p_0\notin\{0,\pm\tfrac\pi2,\pi\}$ and two levels $k\neq k'$, no start concentrated at $p_0$ with $\delta p<\delta p_*$ of (7.27) has equal velocities in these levels; the components then separate as $\lvert v_k-v_{k'}\rvert\lambda$ already at zeroth order in the gradients.
- *Sharp label, $\mathcal C_{\rm lev}$:* for two levels with different data, $(\tau_k,M_k)\neq(\tau_{k'},M_{k'})$, equality of the sharp-label velocities at one $p_0$ with $\sin2p_0\neq0$ is one condition on $(\tau_k,M_k,\tau_{k'},M_{k'})$ and needs $\tau_k\neq\tau_{k'}$ (for $\tau_k=\tau_{k'}$ the condition of (7.5) reduces to $M_k=M_{k'}$). For two levels with equal data the velocities are equal for every $g$, and equality at all $p_0$ holds iff the data are equal (compare the coefficients of $\sin^2p_0$).

### Step 4. Item 1(e): rate of the mean label

1. *Exact transport of the label weight.* On the infinite chain, (4.2) gives for the amplitudes $c(p,\lambda)\in\mathbb C^2$ of a level component $i\partial_\lambda c=C_0(p)c+\tfrac i2\bigl(\partial_p(Bc)+B\partial_pc\bigr)$. $C_0$ and $B$ are Hermitian, so $\partial_\lambda(c^\dagger c)=\partial_p(c^\dagger Bc)$. Multiplying by $p$ and integrating over the arc $(p_0-\pi,p_0+\pi]$ by parts, for a normalized component:
$$
\frac{\mathrm d\bar p_k}{\mathrm d\lambda}=-\int\frac{\mathrm dp}{2\pi}\,c^\dagger B_k\,c+\bigl(c^\dagger B_kc\bigr)(p_0+\pi)\qquad(\text{exact in the gradients}).
\tag{7.6}
$$
The last term is the weight crossing the cut, bounded by $\lVert B_k\rVert\,\lVert c(p_0+\pi)\rVert^2$.
2. *First order.* Insert the zeroth-order amplitudes $c=g\,e^{-i(V_k+\sigma\omega_k)\lambda}\beta_{k,\sigma}$ (the correction to $c$ contributes at second order). The integrand is $\lambda$-independent, so no window is needed. With the facts above and $B_k=-g_{\zeta,k}\tau_k\sin p\,\sigma_1+G_k\sigma_3-\mathcal F_k$, for every smooth $g$:
$$
\frac{\mathrm d\bar p_k}{\mathrm d\lambda}\Big\vert_{1}=\langle\pi_{k,\sigma}\rangle_g-\lvert g(p_0+\pi)\rvert^2\pi_{k,\sigma}(p_0+\pi),\quad\pi_{k,\sigma}(p):=\mathcal F_k-\sigma\frac{G_kM_k+g_{\zeta,k}\tau_k^2\sin^2p}{\omega_k(p)};\qquad\lim_{\substack{\delta p\to0\\ \lvert g(\pi)\rvert^2\to0}}=\pi_{k,\sigma}(0)=\mathcal F_k-\sigma G_k\ \ (p_0=0).
\tag{7.7}
$$
The weight at the cut does not vanish with the spread: the variance controls the integrated weight, not the value $\lvert g(p_0+\pi)\rvert^2$, which can be any number $\ge0$. For a start at rest, $\pi_{k,\sigma}(\pi)=\pi_{k,\sigma}(0)$ and Taylor's formula for $\pi_{k,\sigma}$ give, using only the mean $0$ and the variance $\delta p^2$:
$$
\Bigl\lvert\frac{\mathrm d\bar p_k}{\mathrm d\lambda}\Big\vert_{1}-\bigl(1-\lvert g(\pi)\rvert^2\bigr)\bigl(\mathcal F_k-\sigma G_k\bigr)\Bigr\rvert\le\tfrac12\sup_p\lvert\pi_{k,\sigma}''\rvert\,\delta p^2\qquad(\text{exact, every start at rest}).
\tag{7.28}
$$
Order of limits: first order in the gradients at fixed $\lambda$ and start; long chain; then a sequence of starts at rest with $\delta p\to0$ and $\lvert g(\pi)\rvert^2\to c_{\rm cut}\in[0,\infty)$. The limiting rate is $(1-c_{\rm cut})(\mathcal F_k-\sigma G_k)$; it is $\mathcal F_k-\sigma G_k$ **only for $c_{\rm cut}=0$**. The factor $1-c_{\rm cut}$ is the same for all levels, because $\Phi_\sigma[\alpha,g]$ has the same $g$ in every level. With (4.16), $a_{k,\sigma}(0)=\sigma\,\nu_k\,\pi_{k,\sigma}(0)$ for every start; the drift acceleration (4.11) has no cut term.
3. *Equivalences.* For sequences of starts at rest with $\delta p\to0$ and $c_{\rm cut}\neq1$: the limiting rate is the same for all levels **iff** $\mathcal F_k-\sigma G_k$ does not depend on $k$. Together with $a_{k,\sigma}(0)=a\neq0$ for all $k$ this holds **iff** in addition $\tau_k^2/M_k$ does not depend on $k$. In the class $\mathcal C$ the limiting rate is level-independent iff $g_\eta=0$, and then $a_{k,\sigma}(0)=\tau^2(\sigma\mathcal F-mg_m)/M_k$ is level-independent only if it is zero: **no**, in $\mathcal C$ the two cannot hold together with $a\neq0$ (variant (a): $\pi_{k,\sigma}(0)=-\sigma gM_k$). In $\mathcal C_{\rm lev}$ they can, at the price $\tau_k^2\propto M_k$. For sequences with $c_{\rm cut}=1$ the limiting rate of the arc mean is zero in every level, whatever the gradients, and then holds trivially together with any level-independent $a$.

### Step 5. Item 1(f): clock readings and coherences

1. *Formula.* The reading of the clock label is the partition by $a$; with $\langle a\vert f_k\rangle=N^{-1/2}e^{2\pi ika/N}$ and $\Psi=\sum_k\psi_k\otimes\lvert f_k\rangle$:
$$
P(a;\lambda)=\frac1N\sum_{k,k'}\alpha_{k'}^{*}\alpha_k\,\Gamma_{kk'}(\lambda)\,e^{2\pi i(k-k')a/N},\qquad\Gamma_{kk}=1,\quad\Gamma_{k'k}=\Gamma_{kk'}^{*},\quad\lvert\Gamma_{kk'}\rvert\le1.
\tag{7.8}
$$
This is exact if $\psi_k$ includes the medium, and then $\Gamma_{kk'}=\sum_\xi w_\xi\Gamma^{[\xi]}_{kk'}$ as in (4.1) (plain gradients $\mathcal F_k-\varepsilon_\xi$ in every level). At leading order in $\kappa$ ($\kappa\to0$ at fixed $\lambda$, chain, start) $\Gamma_{kk'}$ is that of the effective body contract, used below.
2. *Zero gradients.* From (5.6) of 04-ilang-time/05-moving-clock, $\langle\beta_{k',\sigma}\vert\beta_{k,\sigma}\rangle=\cos\tfrac{\theta_k-\theta_{k'}}2=:F$, so
$$
\Gamma^{(0)}_{kk'}(\lambda)=e^{-i(V_k-V_{k'})\lambda}\Bigl\langle\cos\tfrac{\theta_k-\theta_{k'}}2\,e^{-i\sigma(\omega_k-\omega_{k'})\lambda}\Bigr\rangle_g;\qquad g(-p)=g(p)\ \Longrightarrow\ \Gamma_{kk'}[\text{gradients}]=\Gamma_{kk'}[-\text{gradients}]\ \ \text{exactly}.
\tag{7.9}
$$
Proof of the second statement: let $\mathsf P$ be the reflection $x\to-x$ times $\sigma_3$. By (4.2), $\mathsf PC_0\mathsf P^{-1}=C_0$ and $\mathsf PC_1\mathsf P^{-1}=-C_1$ in every level; $\sigma_3\beta_{k,\sigma}(-p)=\sigma\beta_{k,\sigma}(p)$ gives $\mathsf PB_{k,\sigma}[g]=\sigma B_{k,\sigma}[g^-]$ with $g^-(p):=g(-p)$, the same sign in both levels; $\mathsf P$ is unitary. For every smooth $g$, every $p_0$ and $\lambda$, exactly:
$$
\Gamma_{kk'}\bigl[g;\,\text{gradients}\bigr]=\Gamma_{kk'}\bigl[g^-;\,-\text{gradients}\bigr].
\tag{7.23}
$$
Hence for every start of **definite parity**, $g^-=\pm g$ (both are starts at rest), $\lvert\Gamma_{kk'}\rvert$ is exactly even in the gradients.
3. *Exact expansion of $\lvert\Gamma_{kk'}\rvert$ to second order.* Class: every smooth $g$, every $p_0$, every $\lambda\ge0$; effective body contract; Taylor expansion in the gradients of the two levels at fixed $\lambda$ and start, infinite chain. Let $\varphi_k:=B_{k,\sigma}[g]$ (position and branch labels), $U_{0k}(\lambda):=e^{-iC_0^{(k)}\lambda}$, $D_k:=U_{0k}^\dagger e^{-iC^{(k)}\lambda}$ with $C^{(k)}$ without $V_k$, $Z:=U_{0k'}^\dagger U_{0k}$, and
$$
\Omega_k:=\int_0^\lambda U_{0k}(s)^\dagger C_1^{(k)}U_{0k}(s)\,\mathrm ds,\qquad\Theta_1:=\Omega_k-Z^\dagger\Omega_{k'}Z,\qquad e^{-i(\Theta_1+\Theta_2+\dots)}:=\bigl(Z^\dagger D_{k'}^\dagger Z\bigr)D_k ,
$$
with $\Theta_n$ self-adjoint of order $n$ (the right side is unitary; $D_k=\exp[-i\Omega_k-\tfrac12\int_0^\lambda\!\mathrm ds\int_0^s\!\mathrm ds'\,[\tilde C_1(s),\tilde C_1(s')]+\dots]$, and $\Theta_2$ follows by the Baker–Campbell–Hausdorff formula). Then $\Gamma_{kk'}=e^{-i(V_k-V_{k'})\lambda}\langle Z^\dagger\varphi_{k'}\vert e^{-i\Theta}\vert\varphi_k\rangle$. Split $Z^\dagger\varphi_{k'}=\gamma_0^{*}\varphi_k+\chi_\perp$ with $\gamma_0:=\langle\varphi_{k'}\vert Z\vert\varphi_k\rangle=e^{i(V_k-V_{k'})\lambda}\Gamma^{(0)}_{kk'}$, $\chi_\perp\perp\varphi_k$, $\lVert\chi_\perp\rVert^2=1-\lvert\Gamma^{(0)}_{kk'}\rvert^2$. With $\langle\cdot\rangle:=\langle\varphi_k\vert\cdot\vert\varphi_k\rangle$, $\delta\Theta_1:=\Theta_1-\langle\Theta_1\rangle$, and $\lvert1+z_1+z_2\rvert=1+\operatorname{Re}z_1+\operatorname{Re}z_2+\tfrac12(\operatorname{Im}z_1)^2+O(3)$, the terms with $\langle\Theta_1\rangle$ and $\langle\Theta_2\rangle$ cancel (they are phases), and where $\Gamma^{(0)}_{kk'}(\lambda)\neq0$:
$$
\begin{aligned}
\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}&=1+\operatorname{Re}\epsilon_1+Q_2+O(\text{grad}^3),\qquad Q_2:=-\tfrac12\mathcal V+\tfrac12(\operatorname{Im}\epsilon_1)^2+\operatorname{Re}\epsilon_2,\qquad\mathcal V:=\bigl\lVert\delta\Theta_1\varphi_k\bigr\rVert^2,\\
\epsilon_1&:=-\frac{i}{\gamma_0}\langle\chi_\perp\vert\Theta_1\varphi_k\rangle,\qquad\epsilon_2:=-\frac{1}{\gamma_0}\bigl\langle\chi_\perp\big\vert\bigl(i\Theta_2+\tfrac12\delta\Theta_1^2\bigr)\varphi_k\bigr\rangle,\qquad\lvert\epsilon_1\rvert\le\eta_0\,\mathcal V^{1/2},\quad\eta_0:=\frac{\sqrt{1-\lvert\Gamma^{(0)}_{kk'}\rvert^2}}{\lvert\Gamma^{(0)}_{kk'}\rvert}.
\end{aligned}
\tag{7.35}
$$
(7.35) is exact; the bound is the Cauchy–Schwarz inequality with $\chi_\perp\perp\varphi_k$; likewise $\lvert\epsilon_2\rvert\le\eta_0\bigl(\operatorname{Var}(\Theta_2)^{1/2}+\tfrac12\lVert\delta\Theta_1^2\varphi_k\rVert\bigr)$. $\operatorname{Re}\epsilon_1=\operatorname{Re}(\Gamma^{(1)}_{kk'}/\Gamma^{(0)}_{kk'})$ is the first-order term, given in closed form in part 4. The first-order term is bounded by $\eta_0\mathcal V^{1/2}$: it exists only because $\lvert\Gamma^{(0)}_{kk'}\rvert<1$. **The order of the first deviation**, for a pair $k\neq k'$ and a given $\lambda$, is therefore, exactly:
$$
\begin{array}{l|l|l}
\text{case}&\text{defining condition}&\text{first deviation of }\lvert\Gamma_{kk'}\rvert\\\hline
\text{(O0)}&(\tau,M,G,\mathcal F,g_\zeta)\text{ equal in the two levels}&\text{none, at any order: }\lvert\Gamma_{kk'}\rvert=1\\
\text{(OZ)}&\Gamma^{(0)}_{kk'}(\lambda)=0&\lvert\Gamma_{kk'}\rvert=\lvert\Gamma^{(1)}_{kk'}\rvert+O(\text{grad}^2)\text{: degree one, not linear}\\
\text{(O1)}&\Gamma^{(0)}_{kk'}\neq0,\ \operatorname{Re}\epsilon_1\neq0&\text{first order, the term }\operatorname{Re}\epsilon_1\text{ of (7.24)}\\
\text{(O2)}&\Gamma^{(0)}_{kk'}\neq0,\ \operatorname{Re}\epsilon_1=0,\ Q_2\neq0&\text{second order, the term }Q_2\text{ of (7.35)}\\
\text{(O3)}&\Gamma^{(0)}_{kk'}\neq0,\ \operatorname{Re}\epsilon_1=Q_2=0&\text{third order or higher (out of scope)}
\end{array}
\tag{7.36}
$$
(O0): the two level blocks differ by a constant. Sufficient for $\operatorname{Re}\epsilon_1=0$ at all $\lambda$: definite parity (7.23), or equal data $(\tau_k,M_k)=(\tau_{k'},M_{k'})$ (then $\eta_0=0$). Parts 5 and 6 determine in which of (O1)–(O3) a start at rest lies and give the leading term.
4. *First-order term for every $g$ (exact).* Notation: $\rho:=\lvert g\rvert^2$; $j:=\operatorname{Im}(g^{*}g')$ ($\langle\hat x\rangle_g=-\int\frac{\mathrm dp}{2\pi}j$); $\Delta\omega:=\omega_k-\omega_{k'}$; $\pi_k:=\pi_{k,\sigma}(p)$ of (7.7); $\lVert f\rVert^2:=\int\frac{\mathrm dp}{2\pi}\lvert f\rvert^2$. First-order perturbation theory in $C_1$ of (4.2), with $C_1\psi=\tfrac i2(B_k'\psi+2B_k\psi')$, $\beta_{k,\sigma}'=\tfrac\sigma2\theta_k'\beta_{k,-\sigma}$, $\langle\beta_{k,\sigma}\vert B_k\vert\beta_{k,\sigma}\rangle=-\pi_k$, gives $-i\Omega_k\varphi_k=a_k\beta_{k,\sigma}+a_{k,\perp}\beta_{k,-\sigma}$ with $a_k=-\lambda\bigl(\pi_kg'+\tfrac12\pi_k'g\bigr)+\tfrac{i\sigma}2\lambda^2\omega_k'\pi_k\,g$ (the transport of Step 4; it conserves the norm) and the inter-branch amplitude
$$
\begin{aligned}
a_{k,\perp}&=\tfrac12\bigl(b_{0,k}\,g+2\,b_{\perp,k}\,g'\bigr)I_{0,k}-i\sigma\,\omega_k'\,b_{\perp,k}\,g\,I_{1,k},\qquad I_{n,k}:=\int_0^\lambda s^n e^{-2i\sigma\omega_ks}\,\mathrm ds,\quad I_{1,k}=\tfrac{i\sigma}{2\omega_k}\bigl(\lambda e^{-2i\sigma\omega_k\lambda}-I_{0,k}\bigr),\\
b_{\perp,k}&:=\langle\beta_{k,-\sigma}\vert B_k\vert\beta_{k,\sigma}\rangle=\frac{\tau_k\sin p}{\omega_k}\bigl(G_k-M_kg_{\zeta,k}\bigr),\qquad b_{0,k}:=-g_{\zeta,k}\tau_k\cos p\,\frac{M_k}{\omega_k}-\sigma\,\theta_k'\,\pi_{k,-\sigma},\\
\lVert a_{k,\perp}\rVert&\le A_{k,\perp}(\lambda):=\frac{\tau_k\mathsf b_k}{2M_k^2}\Bigl(2+\frac{\tau_k}{M_k}+2\lVert\sin p\;g'\rVert\Bigr)+\frac{\tau_k^3\mathsf b_k}{2M_k^3}\Bigl(\lambda+\frac1{M_k}\Bigr)\langle\sin^4p\rangle_g^{1/2}\qquad(\text{bound}).
\end{aligned}
\tag{7.29}
$$
(The bound uses $\lvert I_{0,k}\rvert\le1/\omega_k$, $\lvert I_{1,k}\rvert\le\lambda/(2\omega_k)+1/(2\omega_k^2)$, $\omega_k\ge M_k$, $\lvert b_{\perp,k}\rvert\le\tfrac{\tau_k}{M_k}\mathsf b_k\lvert\sin p\rvert$, $\lvert\omega_k'b_{\perp,k}\rvert\le\tfrac{\tau_k^3}{M_k^2}\mathsf b_k\sin^2p$, $\lvert b_{0,k}\rvert\le\tfrac{\tau_k}{M_k}\mathsf b_k(2+\tfrac{\tau_k}{M_k})$.) Inserting in $\Gamma_{kk'}=\int\frac{\mathrm dp}{2\pi}c_{k'}^\dagger c_k$ and integrating the term with $\rho'$ by parts on the circle:
$$
\Gamma^{(1)}_{kk'}=e^{-i(V_k-V_{k'})\lambda}\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,e^{-i\sigma\Delta\omega\lambda}\Bigl\{\rho\Bigl[\tfrac\lambda2(\pi_k+\pi_{k'})F'+\tfrac{i\sigma}2\lambda^2\bigl(\omega_{k'}'\pi_k-\omega_k'\pi_{k'}\bigr)F\Bigr]-i\lambda(\pi_k-\pi_{k'})F\,j\Bigr\}+\Gamma^{(1),\perp}_{kk'},\qquad\Bigl(\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\Bigr)_{1}=\operatorname{Re}\frac{\Gamma^{(1)}_{kk'}}{\Gamma^{(0)}_{kk'}}\ \ \text{where }\Gamma^{(0)}_{kk'}(\lambda)\neq0.
\tag{7.24}
$$
With $\langle\beta_{k',\sigma}\vert\beta_{k,-\sigma}\rangle=-\sigma\sin\tfrac{\theta_k-\theta_{k'}}2=-\langle\beta_{k',-\sigma}\vert\beta_{k,\sigma}\rangle$ the inter-branch part is
$$
\Gamma^{(1),\perp}_{kk'}=-\sigma\,e^{-i(V_k-V_{k'})\lambda}\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,\sin\tfrac{\theta_k-\theta_{k'}}2\Bigl[g^{*}a_{k,\perp}\,e^{i\sigma(\omega_k+\omega_{k'})\lambda}-a_{k',\perp}^{*}\,g\,e^{-i\sigma(\omega_k+\omega_{k'})\lambda}\Bigr],\qquad\bigl\lvert\Gamma^{(1),\perp}_{kk'}\bigr\rvert\le\tfrac12\Bigl(\frac{\tau_k}{M_k}+\frac{\tau_{k'}}{M_{k'}}\Bigr)\langle\sin^2p\rangle_g^{1/2}\bigl(A_{k,\perp}+A_{k',\perp}\bigr).
\tag{7.30}
$$
(7.24) with (7.29), (7.30) is the complete first-order term, exact for every smooth $g$, $p_0$ and $\lambda\ge0$. A sufficient condition for $\Gamma^{(0)}_{kk'}\neq0$, for a start at rest, from $F\cos y\ge F-(1-\cos y)$, $1-\cos y\le\lvert y\rvert$, $1-F\le\tfrac18(\theta_k-\theta_{k'})^2$ and Taylor's formula for $\Delta\omega$ (mean $0$ and variance $\delta p^2$ only):
$$
1-\bigl\lvert\Gamma^{(0)}_{kk'}(\lambda)\bigr\rvert\le\epsilon_0(\lambda):=\Bigl[\tfrac12\lambda\sup_p\lvert\omega_k''-\omega_{k'}''\rvert+\tfrac18\Bigl(\frac{\tau_k}{M_k}+\frac{\tau_{k'}}{M_{k'}}\Bigr)^2\Bigr]\delta p^2\qquad(\text{bound, every start at rest}),
\tag{7.31}
$$
so $\Gamma^{(0)}_{kk'}\neq0$ wherever $\epsilon_0<1$, and then $\eta_0\le\sqrt{2\epsilon_0}/(1-\epsilon_0)$.
5. *Narrow starts at rest: the leading terms.* **Class (N):** starts at rest with a smooth complex $g$ of any shape, along a sequence with $\delta p\to0$, $\delta p\ll p_c$, moments $\langle\lvert p\rvert^n\rangle_g=O(\delta p^n)$ for $n\le6$, $\lVert(\hat x-\bar x_g)^2g\rVert=O(\delta x_g^2)$ and $\lvert g(\pi)\rvert^2\to c_{\rm cut}\neq1$; **range (N2):** $\epsilon_0(\lambda)\ll1$ (in particular $\lvert\Delta\nu_{kk'}\rvert\lambda\,\delta p^2\ll1$, $\Delta\nu_{kk'}:=\nu_k-\nu_{k'}$). The start enters through $\bar x_g:=\langle g\vert\hat x\vert g\rangle$, $\delta x_g^2:=X_g^2-\bar x_g^2$ (the **spread of the start in the position**; the position variance of the level component is $\delta x_g^2+\tfrac14\langle\theta_k'^2\rangle_g$, with a level-dependent addition $\le t_k/4$), $C_g:=-\int\frac{\mathrm dp}{2\pi}\,p\,j$ (the covariance of position and label), $\langle p^3\rangle_g$ and $J_g:=\int\frac{\mathrm dp}{2\pi}(p^2-\delta p^2)\,j$. With $u=(\hat x-\bar x_g)g$, $w=pg$ and one integration by parts on the arc, $\operatorname{Im}\langle u\vert w\rangle=\tfrac12(1-\lvert g(\pi)\rvert^2)$ and $\operatorname{Re}\langle u\vert w\rangle=C_g$, so
$$
\delta x_g^2\,\delta p^2\ \ge\ C_g^2+\tfrac14\bigl(1-\lvert g(\pi)\rvert^2\bigr)^2,\qquad\text{in particular}\quad2\,\delta p\,X_g\ge2\,\delta p\,\delta x_g\ge\bigl\lvert1-\lvert g(\pi)\rvert^2\bigr\rvert\qquad(\text{bound, every start at rest}).
\tag{7.37}
$$
(For $g=1$ both sides vanish.) *Local form.* On the label weight, $\pi_{k,\sigma}(p)=\pi_k+O(\mathsf b\,p^2/p_c^2)$ with $\pi_k:=\mathcal F_k-\sigma G_k$ from here on, $\omega_k'=\nu_kp\,(1+O(p^2/p_c^2))$, $F=1-O(p^2/p_c^2)$. To leading order the branch part of part 4 is $\Omega_k=-\pi_k(\lambda\hat x+\tfrac12\sigma\nu_k\lambda^2\hat p)$ and $Z=e^{-\frac i2\sigma\Delta\nu\lambda\hat p^2}$ up to a phase, operators on $g$ with $[\hat x,\hat p]=i$. Then $Z^\dagger\hat xZ=\hat x+\sigma\Delta\nu\lambda\hat p$ and
$$
\Theta_1=-\bigl(K_{kk'}+\tfrac12\sigma\lambda^2\Delta\nu_{kk'}(\pi_k-\pi_{k'})\,\hat p\bigr),\qquad K_{kk'}:=\lambda\,(\pi_k-\pi_{k'})\,\hat x+\tfrac12\sigma\lambda^2\,\varpi_{kk'}\,\hat p,\qquad\varpi_{kk'}:=\nu_{k'}\pi_k-\nu_k\pi_{k'} ,
$$
and $\Theta_2$ is a number (commutators of operators linear in $\hat x,\hat p$). $K_{kk'}$ is the generator of the relative displacement of the two level components: $\lambda(\pi_k-\pi_{k'})$ in the label and $\tfrac12\sigma\lambda^2\varpi_{kk'}$ in the position at the common label. By (7.35), with $\operatorname{Var}_g(a\hat x+b\hat p)=a^2\delta x_g^2+2abC_g+b^2\delta p^2$ and (7.37):
$$
\boxed{\;Q_2\simeq-\tfrac12\operatorname{Var}_g(K_{kk'})=-\Bigl[\tfrac12\lambda^2(\pi_k-\pi_{k'})^2\,\delta x_g^2+\tfrac12\sigma\lambda^3(\pi_k-\pi_{k'})\,\varpi_{kk'}\,C_g+\tfrac18\lambda^4\varpi_{kk'}^2\,\delta p^2\Bigr]\;},\qquad\operatorname{Var}_g(K_{kk'})\ge\bigl(1-\lvert g(\pi)\rvert^2\bigr)^2\max\Bigl\{\frac{\lambda^2(\pi_k-\pi_{k'})^2}{4\,\delta p^2},\ \frac{\lambda^4\varpi_{kk'}^2}{16\,\delta x_g^2}\Bigr\}.
\tag{7.38}
$$
The lower bound is exact. So for the class (N) the second-order term is **different from zero iff $(\pi_k-\pi_{k'},\varpi_{kk'})\neq(0,0)$**, for every shape of $g$, starts of definite parity included. It depends on $\lambda$ as $\lambda^2$, $\lambda^3$, $\lambda^4$; on the level data through $\pi_k-\pi_{k'}$ and $\varpi_{kk'}$; on the start through $\delta x_g^2$, $C_g$ and $\delta p^2$: the $\lambda^2$ term grows with the spread in the position, the $\lambda^4$ term falls with it ($\delta p\ge\lvert1-c_{\rm cut}\rvert/(2\delta x_g)$). For the real Gaussian starts (E1), $\lvert g\rvert^2\propto e^{-p^2/2\delta p^2}$, $g$ real: $C_g=0$, $\delta p=1/(2\delta x)$, and (7.38) is
$$
1-\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\simeq\frac{\delta x^2}2\,(\pi_k-\pi_{k'})^2\lambda^2+\frac{\lambda^4}{32\,\delta x^2}\Bigl(\frac{\tau_{k'}^2}{M_{k'}}\pi_k-\frac{\tau_k^2}{M_k}\pi_{k'}\Bigr)^2,\qquad\pi_k=\mathcal F_k-\sigma G_k.
\tag{7.10}
$$
*First-order term.* In (7.35), $\chi_\perp\neq0$ only through $Z\neq1$. Expanding (7.24) with $\Delta\omega=\Delta\omega(0)+\tfrac12\Delta\nu\,p^2+\dots$: a real weight $h$ gives $\operatorname{Im}\bigl[\int Fe^{-i\sigma\Delta\omega\lambda}h\big/\int Fe^{-i\sigma\Delta\omega\lambda}\rho\bigr]=-\tfrac\sigma2\lambda\Delta\nu\int\frac{\mathrm dp}{2\pi}(p^2-\delta p^2)h$ at leading order; with $h=\rho\,\varpi_{kk'}p$ and $h=(\pi_k-\pi_{k'})j$:
$$
\operatorname{Re}\epsilon_1=\Bigl(\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\Bigr)_{1}\simeq\Delta\nu_{kk'}\Bigl[\tfrac14\lambda^3\,\varpi_{kk'}\,\langle p^3\rangle_g-\tfrac\sigma2\lambda^2\,(\pi_k-\pi_{k'})\,J_g\Bigr]
\tag{7.25}
$$
(the same follows from $\Theta_1$ above: $\operatorname{Re}\epsilon_1=\tfrac12\sigma\Delta\nu\lambda\,\operatorname{Cov}_g(K_{kk'},\hat p^2)$). $J_g=0$ for real $g$; $\langle p^3\rangle_g=J_g=0$ for definite parity. Its sign is not fixed. So, in (N), (N2): **(O1)** iff the bracket of (7.25) and $\Delta\nu_{kk'}$ are different from zero (a real $g$ with a skewed label weight suffices); otherwise **(O2)** iff $(\pi_k-\pi_{k'},\varpi_{kk'})\neq(0,0)$. To second order the deviation is $1-\lvert\Gamma_{kk'}\rvert/\lvert\Gamma^{(0)}_{kk'}\rvert\simeq-(7.25)+\tfrac12\operatorname{Var}_g(K_{kk'})$; the first-order term is the larger one where $\operatorname{Var}_g(K_{kk'})^{1/2}\ll\lvert\Delta\nu\rvert\lambda\,\delta p^2$ times the skewness of the start.
   *What (7.38) and (7.25) neglect (order estimates; $\tau/M$, $\nu$, $\mathsf b$ the larger of the two levels' values; $\mathcal V_{\max}:=\mathsf b^2(\lambda^2X_g^2+\nu^2\lambda^4\delta p^2)$ is the largest size of $\operatorname{Var}_g(K)$).* (i) *Local form.* The Taylor remainders of $\pi_{k,\sigma}$, $\omega_k'$, $F$ change $\Theta_1\varphi_k$ by a vector of norm $O(\delta p^2/p_c^2)\,\mathcal V_{\max}^{1/2}$ (moments $n\le6$). (ii) *The term $\tfrac12\sigma\lambda^2\Delta\nu(\pi_k-\pi_{k'})\hat p$ of $\Theta_1$* changes $\mathcal V^{1/2}$ by at most $\tfrac12\lambda^2\lvert\Delta\nu(\pi_k-\pi_{k'})\rvert\delta p\le\lvert\Delta\nu\rvert\lambda\,\delta p^2\operatorname{Var}_g(K)^{1/2}/\lvert1-c_{\rm cut}\rvert$, by (7.38). (iii) *Inter-branch components of $\Theta_1\varphi_k$*, orthogonal to the branch part, so they add to $\mathcal V$ in squares: $ia_{k,\perp}$ and its counterpart of $k'$, of norm $\le A_{k,\perp}+A_{k',\perp}$ (bound (7.29)); and the part of $\varphi_k$ outside the branch $\sigma$ of the level $k'$, of norm $\le\tfrac12(\tfrac{\tau_k}{M_k}+\tfrac{\tau_{k'}}{M_{k'}})\langle\sin^2p\rangle_g^{1/2}\le\tfrac\tau M\delta p$, on which $\Omega_{k'}$ acts as on a start of the other branch: $O\bigl(\tfrac\tau M\mathsf b\,\lambda\,(\delta p\,X_g+1+\nu\lambda\,\delta p^2)\bigr)$. (iv) *The terms with $\chi_\perp$:* $\tfrac12(\operatorname{Im}\epsilon_1)^2\le\tfrac12\eta_0^2\mathcal V$; $\tfrac12\eta_0\lVert\delta\Theta_1^2\varphi_k\rVert=O(\eta_0\mathcal V_{\max})$ (fourth moments); and $\eta_0\operatorname{Var}(\Theta_2)^{1/2}$, where only the operator part of $\Theta_2$ counts: from the $p$-dependence inside the branch, $\Theta_2\ni-\tfrac\sigma{12}\lambda^3\pi_k(\omega_k'\pi_k)'$ (double integral of $[\tilde C_1(s),\tilde C_1(s')]=i\sigma(s'-s)\pi_k(\omega_k'\pi_k)'$), with the fluctuation $O(\lambda^3\nu\mathsf b^2\delta p^2/p_c^2)$; and from the second-order inter-branch term $-\sigma\lambda\,(C_1)_{\sigma,-\sigma}(C_1)_{-\sigma,\sigma}/(2\omega_k)$, $(C_1)_{-\sigma,\sigma}=\tfrac i2(b_{0,k}+2b_{\perp,k}\partial_p)$, with the fluctuation $O\bigl(\lambda\tfrac{\tau^2\mathsf b^2}{M^3}(1+\delta pX_g)^2\bigr)$. Together:
$$
\bigl\lvert Q_2+\tfrac12\operatorname{Var}_g(K_{kk'})\bigr\rvert\lesssim\mathcal V_{\max}\Bigl[\frac{\delta p^2}{p_c^2}+\lvert\Delta\nu\rvert\lambda\,\delta p^2+\eta_0\Bigr]+\bigl(A_{k,\perp}+A_{k',\perp}\bigr)^2+\frac{\tau^2\mathsf b^2}{M^2}\lambda^2\bigl(1+\delta pX_g+\nu\lambda\,\delta p^2\bigr)^2+\eta_0\,\lambda\,\mathsf b^2\Bigl[\frac{\nu\lambda^2\delta p^2}{p_c^2}+\frac{\tau^2}{M^3}(1+\delta pX_g)^2\Bigr].
\tag{7.32}
$$
(7.38) is the leading second-order term where the right side of (7.32) is small against $\tfrac12\operatorname{Var}_g(K_{kk'})$. If the gradients and the data of the two levels differ by relative amounts of order one ($\operatorname{Var}_g(K)\sim\mathcal V_{\max}$) and $\delta pX_g=O(1)$, this is the range (N2) with $\lambda\gg\tau\,\delta p/M^2$ (from $A_\perp^2\sim\tau^2\mathsf b^2/M^4\ll\mathsf b^2\lambda^2/\delta p^2$; the other terms are then small by $\delta p\ll p_c$ and $\eta_0\ll1$). For (7.25): the Taylor remainders change its two terms by relative $O(\delta p^2/p_c^2)$; the next order of $e^{-i\sigma\Delta\omega\lambda}$ is real and enters at relative $O((\Delta\nu\lambda\delta p^2)^2)$; the term of (7.24) with $F'\simeq-\tfrac14(\theta_k'-\theta_{k'}')^2p$ is $O(\lambda\mathsf b\,t(1+t)\delta p^3)$; and $\lvert\Gamma^{(1),\perp}_{kk'}\rvert$ has the bound (7.30), with the growing part $O(\lambda\tfrac{\tau^4\mathsf b}{M^4}\delta p^3)$. If $\pi_k-\pi_{k'}$, $\varpi_{kk'}$ are small against $\mathsf b$, $\nu\mathsf b$, the ranges shrink accordingly; for $\pi_k=\pi_{k'}$, $\varpi_{kk'}=0$ the leading terms vanish and the deviation is of higher order in $\delta p$, given by (7.35).
6. *Starts at rest that are not narrow, and pairs with equal data.* For every start at rest outside (N), (N2) (fixed $g$ of any width, or $\epsilon_0\gtrsim1$) the order is given by (7.36) and the terms by (7.24) and (7.35), exactly; no reduction to finitely many moments is derived here: the results are the exact functionals (7.24), (7.35), into which the whole functions $\rho$, $j$ enter, with the case distinction (7.36). Their dependence on $\lambda$: $\Theta_1\varphi_k$ is a sum of vectors of the form $\lambda^n\times(\text{bounded in }\lambda)$, $n\le2$, by (7.29), so $\operatorname{Re}\epsilon_1$ grows at most as $\lambda^2$ and $\mathcal V$ at most as $\lambda^4$, as long as $\Gamma^{(0)}_{kk'}$ stays away from zero; at a zero, (OZ) applies. On the spread in the position: through $j$ and $g'$, at most linearly in $X_g$ per factor $\Theta_1$. For a pair with **equal data**, $(\tau_k,M_k)=(\tau_{k'},M_{k'})$, and every smooth $g$, every $p_0$ and $\lambda$: $Z=\mathbb 1$, $\varphi_k=\varphi_{k'}$, $\chi_\perp=0$, and (7.35) is, exactly,
$$
\lvert\Gamma_{kk'}\rvert=1-\tfrac12\Bigl[\lVert\Delta a\rVert^2+\lVert\Delta a_\perp\rVert^2-\lvert\langle g\vert\Delta a\rangle\rvert^2\Bigr]+O(\text{grad}^3),\qquad\Delta a:=a_k-a_{k'},\quad\Delta a_\perp:=a_{k,\perp}-a_{k',\perp}\ \ \text{of (7.29)},
\tag{7.39}
$$
which are (7.29) with the differences of the gradients. The bracket is the variance of $\Omega_k-\Omega_{k'}$ in the start: (O2) unless the start is an eigenvector of $\Omega_k-\Omega_{k'}$, and (O0) if the gradients are equal as well.
7. *Level-independent fall.* Let $a_{k,\sigma}(0)=\sigma\nu_k\pi_k=a$ for all $k$. Then $\pi_k-\pi_{k'}=-\sigma a\Delta\nu_{kk'}/(\nu_k\nu_{k'})$, $\varpi_{kk'}=-\sigma a\Delta\nu_{kk'}(\nu_k+\nu_{k'})/(\nu_k\nu_{k'})$, and (7.25), (7.38) become, in the class (N), range (N2), with the status of (7.25), (7.38):
$$
\Bigl(\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\Bigr)_{1}\simeq\frac{a\,\Delta\nu_{kk'}^2}{\nu_k\,\nu_{k'}}\Bigl[\tfrac12\lambda^2J_g-\tfrac\sigma4\lambda^3\,(\nu_k+\nu_{k'})\,\langle p^3\rangle_g\Bigr],
\tag{7.26}
$$
$$
Q_2\simeq-\frac{a^2\,\Delta\nu_{kk'}^2}{2\,\nu_k^2\,\nu_{k'}^2}\Bigl[\lambda^2\delta x_g^2+\sigma\lambda^3(\nu_k+\nu_{k'})\,C_g+\tfrac14\lambda^4(\nu_k+\nu_{k'})^2\delta p^2\Bigr]\ \le\ -\frac{a^2\,\Delta\nu_{kk'}^2\,(1-\lvert g(\pi)\rvert^2)^2\,\lambda^2}{8\,\nu_k^2\,\nu_{k'}^2\,\delta p^2}.
\tag{7.40}
$$
**Does the deviation vanish when the fall is the same for all levels? No**, with this case distinction:
   - *(N), (N2), $a\neq0$, $\nu_k\neq\nu_{k'}$:* it does not vanish; $Q_2<0$ by the bound of (7.40) for every shape of $g$, and the first-order term (7.26) is present in addition if $J_g\neq0$ or $\langle p^3\rangle_g\neq0$. In the class $\mathcal C$ ($\tau_k=\tau$, $M_k$ distinct) this is every pair of levels: with $a=-\tau^2g$, $a\Delta\nu/(\nu_k\nu_{k'})=g(d_k-d_{k'})$, $\nu_k+\nu_{k'}=\tau^2\tfrac{M_k+M_{k'}}{M_kM_{k'}}$; the $\lambda^2$ term is $\tfrac12\delta x_g^2g^2(d_k-d_{k'})^2\lambda^2$, and by (3.23) $g_\eta(d_k-d_{k'})=\tfrac{2\pi(k-k')}{N}\bigl(r(x+1)-r(x)\bigr)$: it is the difference of the clock rates across the spread of the start.
   - *(N), (N2), $a=0$ or $\nu_k=\nu_{k'}$:* the leading terms (7.26), (7.40) vanish ($\pi_k=\pi_{k'}$, $\varpi_{kk'}=0$). The deviation itself vanishes only in case (O0); otherwise it is of higher order in $\delta p$ and given by (7.35) (for equal data by (7.39), with $\Delta a\neq0$ as soon as $\pi_{k,\sigma}(p)\neq\pi_{k',\sigma}(p)$ on the label weight).
   - *Starts at rest outside (N), (N2):* (7.36) with (7.24), (7.35); a level-independent fall is one condition on the gradients at $p=0$ and makes neither $\operatorname{Re}\epsilon_1$ nor $\mathcal V$ vanish unless (O0) holds or the start is an eigenvector of $\Theta_1$.

### Step 6. Item 2(a): records at first order

By the Setup, $p^{[\xi]}$ of (4.1) belongs to the effective level profiles with $\mathcal F_k\to\mathcal F_k-\varepsilon_\xi$. The zeroth-order velocity (4.5) does not depend on $\xi$; the first-order velocity is linear in $(G_k,\mathcal F_k-\varepsilon_\xi,g_{\zeta,k})$. Averaging with $w_\xi$ and $\sum_\xi w_\xi\varepsilon_\xi=0$:
$$
v^{(1)}\bigl[G_k,\mathcal F_k,g_{\zeta,k};\chi\bigr]=\sum_\xi w_\xi\Bigl(v^{(1)}[G_k,\mathcal F_k,g_{\zeta,k}]-\varepsilon_\xi\,\partial_{\mathcal F_k}v^{(1)}\Bigr)=v^{(1)}[G_k,\mathcal F_k,g_{\zeta,k}].
\tag{7.11}
$$
**No change**: (4.7), (4.16), (4.19) hold at first order jointly in $G_k,\mathcal F_k,g_{\zeta,k},\kappa(\xi-\langle X\rangle)$ for every $\chi$, sharp $X$ or not. The reason is linearity and the vanishing mean of $\xi-\langle X\rangle$, which is what counting $\kappa\langle X\rangle$ in $V_{x,k}$ achieves.

### Step 7. Item 2(b): mean velocity to second and third order in $\kappa$

All gradients of the effective body contract vanish; fix $\xi$, a level and $\varepsilon:=\varepsilon_\xi$. The level block is $C_0(\hat p)+\varepsilon\hat x$ plus a constant.
1. *Exact reduction.* $e^{i\varepsilon\hat x\lambda}f(\hat p)e^{-i\varepsilon\hat x\lambda}=f(\hat p-\varepsilon\lambda)$, so $e^{-i(C_0+\varepsilon\hat x)\lambda}=e^{-i\varepsilon\hat x\lambda}W(\lambda)$ with $W$ acting at each $p$ as the solution of $i\partial_s\psi=C_0(q(s))\psi$, $q(s):=p-\varepsilon s$, $\psi(0)=\beta_\sigma(p)$. The velocity operator is $i[C,\hat x]=C_0'(\hat p)$ (the records commute with $\hat x$), so $v^{[\xi]}(\lambda)=\langle\,\psi^\dagger C_0'(q(\lambda))\psi\,\rangle_g$.
2. *Bloch form.* $C_0(q)=\omega(q)\,\mathbf n\cdot\boldsymbol\sigma$ with $\mathbf n=(\sin\theta,0,\cos\theta)$, $\mathbf e_\theta=\partial_\theta\mathbf n$, $\mathbf e_y=\mathbf n\times\mathbf e_\theta$. For $\mathbf S=\psi^\dagger\boldsymbol\sigma\psi=S_n\mathbf n+S_\theta\mathbf e_\theta+S_y\mathbf e_y$, $\dot{\mathbf S}=2\omega\,\mathbf n\times\mathbf S$ and $\dot{\mathbf n}=-\varepsilon\theta'\mathbf e_\theta$ give, with all functions at $q(s)$,
$$
\dot S_n=-\varepsilon\theta'S_\theta,\quad\dot S_\theta=\varepsilon\theta'S_n-2\omega S_y,\quad\dot S_y=2\omega S_\theta,\quad\mathbf S(0)=\sigma\mathbf n;\qquad v_p=\omega'(q)S_n+\omega(q)\theta'(q)S_\theta .
\tag{7.12}
$$
3. *Solution to second order.* For $z=S_\theta+iS_y$: $z(\lambda)=\varepsilon\int_0^\lambda\theta'S_n\,e^{i(\varphi(\lambda)-\varphi(s))}\mathrm ds$, $\varphi(s)=2\int_0^s\omega(q)$. One integration by parts and $\varphi(\lambda)=2\omega\lambda-\varepsilon\omega'\lambda^2+\tfrac13\varepsilon^2\omega''\lambda^3+O(\varepsilon^3)$ give, with all functions now at $p$: $S_\theta=\tfrac{\varepsilon\sigma\theta'}{2\omega}\sin2\omega\lambda-\varepsilon^2\sigma\bigl[\tfrac{\theta'\omega'}{2\omega}\lambda^2\cos2\omega\lambda+\tfrac1{4\omega}(\theta'/\omega)'(1-\cos2\omega\lambda)\bigr]$ and $S_n=\sigma-\varepsilon^2\sigma\tfrac{\theta'^2}{4\omega^2}(1-\cos2\omega\lambda)$. Expanding $\omega'(q),(\omega\theta')(q)$ about $p$: $v_p=\sigma\omega'+\varepsilon\sigma\bigl[-\lambda\omega''+\tfrac12\theta'^2\sin2\omega\lambda\bigr]+\varepsilon^2v_2+\varepsilon^3v_3+O(\varepsilon^4)$,
$$
v_2(p,\lambda)=\sigma\Bigl[\tfrac12\lambda^2\omega'''-\tfrac{\theta'\theta''}{4\omega}(1-\cos2\omega\lambda)-\tfrac{\lambda}{2\omega}\theta'(\omega\theta')'\sin2\omega\lambda-\tfrac12\lambda^2\theta'^2\omega'\cos2\omega\lambda\Bigr].
\tag{7.13}
$$
The first-order term agrees with (4.7) for $\mathcal F=-\varepsilon$ and averages to zero over $\xi$ (Step 6). The terms of $v_2$, $v_3$ with the highest powers of $\lambda$ and an oscillating factor come from the phase of the first-order oscillation, $\tfrac12\varepsilon\sigma\theta'^2\sin\varphi(\lambda)$: $-\tfrac12\lambda^2\theta'^2\omega'\cos2\omega\lambda$ in $v_2$, and in $v_3$
$$
v_3(p,\lambda)=\sigma\Bigl[-\tfrac16\lambda^3\omega''''-\tfrac14\lambda^4\theta'^2\omega'^2\sin2\omega\lambda+\tfrac16\lambda^3\theta'^2\omega''\cos2\omega\lambda+\lambda^3\omega'\,(\dots)+O(\lambda^2)\Bigr],
$$
where $(\dots)$ is smooth in $p$ times oscillating factors, and the only terms without an oscillating factor are $-\tfrac16\lambda^3\omega''''$ and one linear in $\lambda$ (the adiabatic series in $p-\varepsilon\lambda$). The Taylor expansion in $\varepsilon$ at a fixed label needs $\lvert\varepsilon\rvert\lambda\ll p_c$, $\lvert\varepsilon\,\omega'\rvert\lambda^2\ll1$ and $\varepsilon^2\nu\lambda^3\ll1$; beyond, the oscillating terms sum to $\tfrac12\varepsilon\sigma\theta'^2\sin\varphi(\lambda)$, bounded by $\tfrac12\lvert\varepsilon\rvert t$.
4. *Average over $\xi$ and $g$.* The $\lambda^2\cos$ term carries the factor $\omega'$: $\langle\theta'^2\omega'\cos2\omega\lambda\rangle_g=-\tfrac1{2\lambda}\int\tfrac{\mathrm dp}{2\pi}(\lvert g\rvert^2\theta'^2)'\sin2\omega\lambda$ (integration by parts on the circle). With $\int\tfrac{\mathrm dp}{2\pi}\lvert(\lvert g\rvert^2)'\rvert\le2X_g$ and $\lvert\theta'\rvert\le\tau/M$:
$$
\begin{aligned}
v(\lambda)&=\sigma\langle\omega_k'\rangle_g+\kappa^2\sigma_X^2\Bigl[\tfrac12\lambda^2\sigma\langle\omega_k'''\rangle_g+R_2(\lambda)\Bigr]+\kappa^3\mu_3\Bigl[-\tfrac16\lambda^3\sigma\langle\omega_k''''\rangle_g+R_3(\lambda)\Bigr]+O(\kappa^4),\\
\lvert R_2(\lambda)\rvert&\le\lambda\Bigl[\tfrac{\tau_k^2}{2M_k^2}X_g+\tfrac14\sup\lvert(\theta'^2)'\rvert+\tfrac12\sup\bigl\lvert\tfrac{\theta'(\omega\theta')'}{\omega}\bigr\rvert\Bigr]+\tfrac12\sup\bigl\lvert\tfrac{\theta'\theta''}{\omega}\bigr\rvert\ \ (\text{bound}),\qquad R_3(\lambda)=O(\lambda^{5/2})\ \ (\text{order estimate}).
\end{aligned}
\tag{7.14}
$$
The fastest-growing parts are the coefficients of the adiabatic term $\sigma\omega'(p-\varepsilon\lambda)$; their coefficients are exact. *Estimate of $R_3$ (order estimate at fixed smooth $g$).* Let $\mathcal D_k:=\lambda\bigl(\lvert\omega_k'(p_0)\rvert\delta p+\lvert\omega_k''(p_0)\rvert\delta p^2\bigr)$ be the **dephasing parameter**: the spread of the phase $2\omega_k\lambda$ over the label weight. In $\langle v_3\rangle_g$, each factor $\omega'$ of an oscillating term is removed by one integration by parts ($\omega'e^{2i\omega\lambda}=\partial_pe^{2i\omega\lambda}/(2i\lambda)$), which gains $1/\lambda$ and puts a derivative on $\lvert g\rvert^2\theta'^2$; an oscillating integral without a factor $\omega'$ over a weight of height $\sim1/\delta p$ is $O(\mathcal D_k^{-1/2})$ (stationary phase at the nondegenerate points $p\in\{0,\pm\tfrac\pi2,\pi\}$ of $\omega_k$; $O(\mathcal D_k^{-1})$ away from them). The $\lambda^4\omega'^2$ term gives $-\tfrac{\sigma\lambda^3}{8}\int\tfrac{\mathrm dp}{2\pi}\lvert g\rvert^2\theta'^2\omega''\cos2\omega\lambda$ plus terms smaller by $1/\mathcal D_k$; with the $\lambda^3\theta'^2\omega''$ term: $R_3=O\bigl(t_k\nu_k\lambda^3\mathcal D_k^{-1/2}\bigr)$, that is $O(\lambda^{5/2}t_k\nu_k^{1/2}/\delta p)$ at rest, for $\mathcal D_k\gg1$; for $\mathcal D_k\lesssim1$ the oscillating terms are not suppressed, and $R_3$ is of the order of the fastest-growing part (relative amplitude $t_k/(4+3t_k)<\tfrac13$ at rest, (7.43)). The bound on $R_2$ holds for all $\lambda\ge0$; it contains $X_g$ and is therefore not uniform in the spread ($X_g\ge\lvert1-\lvert g(p_0+\pi)\rvert^2\rvert/(2\delta p)$, (7.37) with the cut at $p_0+\pi$). Range: fixed smooth $g$, chain and $\lambda$, $\kappa\to0$, with the conditions of part 3 on the support of $w_\xi$. Controls: a sharp $X$ gives $\sigma_X=\mu_3=0$ and no term; a symmetric distribution of $X$ gives $\mu_3=0$.

### Step 8. Item 2(c),(d): change of the acceleration

1. *Exact acceleration for every start.* The change of the acceleration of $\bar x_{\rm lab}$ at the orders $\kappa^2$, $\kappa^3$ is $\kappa^2\sigma_X^2\,a_2[g](\lambda)+\kappa^3\mu_3\,a_3[g](\lambda)$ with $a_n[g]:=\tfrac{\mathrm d}{\mathrm d\lambda}\langle v_n\rangle_g=\langle\partial_\lambda v_n\rangle_g$. From the complete coefficient (7.13), exactly, for every smooth $g$ and $\lambda\ge0$:
$$
\partial_\lambda v_2(p,\lambda)=\sigma\Bigl[\lambda\,\omega'''+\lambda^2\,\omega\,\omega'\theta'^2\sin2\omega\lambda-\lambda\,\theta'\bigl(2\omega'\theta'+\omega\theta''\bigr)\cos2\omega\lambda-\frac{\theta'}{2\omega}\bigl(2\omega\theta''+\omega'\theta'\bigr)\sin2\omega\lambda\Bigr].
\tag{7.41}
$$
Pointwise in $p$ the second term grows faster than the first, and it oscillates at the level-dependent rate $2\omega_k(p)$. What the acceleration "for a sharp label" is depends on the order in which $\delta p\to0$ and the growth of $\lambda$ are taken; the two orders do not commute. They are separated by $\mathcal D_k$ of Step 7. At a fixed $g$, the average over a window and the instantaneous value are different quantities as well; each is stated below with the order of limits in which it is obtained. Averaged with $\lvert g\rvert^2$, the term $\lambda^2\omega\omega'\theta'^2\sin2\omega\lambda$ of (7.41) loses one power of $\lambda$: with $\omega'e^{2i\omega\lambda}=\partial_pe^{2i\omega\lambda}/(2i\lambda)$ and two integrations by parts on the circle, exactly, for every smooth $g$ and $\lambda\ge0$:
$$
\begin{aligned}
a_2[g](\lambda)&=\sigma\lambda\langle\omega_k'''\rangle_g+a_2^{\rm osc}[g](\lambda),\qquad a_2^{\rm osc}:=\sigma\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\Bigl\{\frac\lambda2\,(\lvert g\rvert^2)'\,\omega\theta'^2\cos2\omega\lambda+\Bigl[\tfrac34(\lvert g\rvert^2)'\theta'^2+\lvert g\rvert^2\Bigl(\tfrac12\theta'\theta''-\frac{\omega'\theta'^2}{2\omega}\Bigr)\Bigr]\sin2\omega\lambda\Bigr\},\\
\lvert a_2^{\rm osc}\rvert&\le\nu_kX_g\,\lambda+c_3[g]\ \ (\text{bound}),\quad c_3[g]:=\tfrac32t_kX_g+\tfrac12\sup\lvert\theta'\theta''\rvert+\tfrac12\sup\Bigl\lvert\frac{\omega'\theta'^2}{\omega}\Bigr\rvert;\qquad\lim_{\lambda\to\infty}\frac{a_2[g](\lambda)}{\lambda}=\sigma\langle\omega_k'''\rangle_g\ \ (\text{every fixed smooth }g).
\end{aligned}
\tag{7.47}
$$
(The bound uses $\omega\theta'^2\le\nu_k$, $\theta'^2\le t_k$ and $\int\tfrac{\mathrm dp}{2\pi}\lvert(\lvert g\rvert^2)'\rvert\le2X_g$. The limit is the Riemann–Lebesgue lemma: $\omega_k$ is analytic and not constant, so the integral of a fixed integrable weight against $e^{2i\omega_k\lambda}$ tends to zero.) The bound is of the same power of $\lambda$ as the first term and decides nothing by itself; the limit holds at fixed $g$, not uniformly in $g$.
2. *The two orders of limits.*
$$
\begin{array}{l|l|l|l}
\text{case}&\text{order of limits}&\text{order }\kappa^2&\text{order }\kappa^3\text{ at rest}\\\hline
\text{A (window average)}&\text{fixed smooth }g;\ \text{average over }[\lambda_1,\lambda_1+\Lambda];\ \text{then }\delta p\to0\text{ in the coefficient}&2s_2\bar\lambda+\text{rest, bound (7.33); coefficient (7.15) at }\bar\lambda&3s_3\bigl(\bar\lambda^2+\tfrac{\Lambda^2}{12}\bigr)+\text{rest, (7.33), (7.48); coefficient (7.16) at }\bar\lambda\text{ for }\Lambda\ll\bar\lambda\\
\text{A}'\text{ (instantaneous, dephased)}&\text{fixed smooth }g;\ \text{no window};\ \lambda\to\infty\ (\mathcal D_k\to\infty);\ \text{then }\delta p\to0\text{ in the coefficient}&a_2[g]/\lambda\to2s_2,\ \text{(7.47): coefficient (7.15); rates (7.49)}&\text{not (7.16): }a_3[g]=3s_3\lambda^2+a_3^{\rm osc},\ a_3^{\rm osc}\propto\lambda^{5/2},\ \text{(7.50)}\\
\text{B (coherent)}&\delta p\to0\text{ at fixed }\lambda\text{ or fixed window }(\mathcal D_k\to0)&\kappa^2\sigma_X^2\,\partial_\lambda v_2(p_0,\lambda),\ \text{(7.41), (7.42)}&\text{(7.43)}\\
\text{between}&\text{every smooth }g,\ \text{every }\lambda&\kappa^2\sigma_X^2\langle\partial_\lambda v_2\rangle_g,\ \text{(7.41), exact}&\kappa^3\mu_3\langle\partial_\lambda v_3\rangle_g
\end{array}
\tag{7.44}
$$
Case A is the order of limits of the drift acceleration (4.10), (4.11) (window at fixed $g$, then the sharp label).

   *Case A.* Write $\langle v_n\rangle_g=s_n\lambda^n+R_n$ with $s_2=\tfrac\sigma2\langle\omega_k'''\rangle_g$, $s_3=-\tfrac\sigma6\langle\omega_k''''\rangle_g$. For the average over a window $[\lambda_1,\lambda_1+\Lambda]$, $\overline{a_n}:=(\langle v_n\rangle_g(\lambda_1+\Lambda)-\langle v_n\rangle_g(\lambda_1))/\Lambda$, with the midpoint $\bar\lambda:=\lambda_1+\tfrac\Lambda2$, $c_1[g]$ the square bracket of the bound on $R_2$ in (7.14) and $c_0:=\tfrac12\sup\lvert\theta'\theta''/\omega\rvert$:
$$
\bigl\lvert\overline{a_2}-2s_2\,\bar\lambda\bigr\rvert\le\frac{2c_1[g]\,\bar\lambda+2c_0}{\Lambda}\ \ (\text{bound; every smooth }g,\ \lambda_1\ge0,\ \Lambda>0),\qquad\overline{a_3}=3s_3\Bigl(\bar\lambda^2+\frac{\Lambda^2}{12}\Bigr)+\frac{R_3(\lambda_1+\Lambda)-R_3(\lambda_1)}{\Lambda}.
\tag{7.33}
$$
A window $\Lambda\gg c_1[g]/\lvert\langle\omega_k'''\rangle_g\rvert$ is **sufficient** for the first term to dominate $\overline{a_2}$; a sequence of starts and windows with $\delta p\to0$ and $c_1[g]/\Lambda\to0$ is sufficient to combine the two limits (not shown to be necessary). *Order $\kappa^3$, starts at rest.* The second statement of (7.33) is an identity: $3s_3(\bar\lambda^2+\Lambda^2/12)$ is the average of the secular acceleration $3s_3\lambda^2$ over the window; the term $\Lambda^2/12$ is exact and of the relative size $\Lambda^2/(12\bar\lambda^2)\le\tfrac13$ (since $\Lambda\le2\bar\lambda$). For the rest, take the estimate of Step 7, part 4, with its coefficient: the two terms there add to $R_3\simeq\tfrac\sigma{24}\lambda^3\langle\theta'^2\omega''\cos2\omega\lambda\rangle_g$ ($-\tfrac18+\tfrac16$). For a narrow start at rest ($\delta p\ll p_c$) with $\mathcal D_k=\nu_k\lambda\,\delta p^2\gg1$ the stationary points $p=0,\pi$, where $\theta'^2\omega''=t_k\nu_k$, $\omega=M_k$, $\omega''=\nu_k$, give $\int\tfrac{\mathrm dp}{2\pi}\lvert g\rvert^2e^{2i\omega\lambda}\simeq\tfrac{w_0}{2\sqrt{\pi\nu_k\lambda}}\,e^{i(2M_k\lambda+\pi/4)}$ with $w_0:=\lvert g(0)\rvert^2+\lvert g(\pi)\rvert^2$ (for a Gaussian label weight $w_0=\sqrt{2\pi}/\delta p$, and the modulus is $(2\mathcal D_k)^{-1/2}$). Hence, with $\lambda_1+\Lambda\le2\bar\lambda$:
$$
R_3(\lambda)\simeq\frac{\sigma\,t_k\,\nu_k^{1/2}\,w_0}{48\sqrt\pi}\,\lambda^{5/2}\cos\bigl(2M_k\lambda+\tfrac\pi4\bigr),\qquad\varrho_{3,\rm win}:=\frac{\lvert R_3(\lambda_1+\Lambda)-R_3(\lambda_1)\rvert}{3\lvert s_3\rvert\,\bar\lambda^2\,\Lambda}\lesssim\frac{\sqrt2\,w_0}{3\,(4+3t_k)}\,\frac{1}{M_k\Lambda}\sqrt{\frac{\nu_k\bar\lambda}{\pi}}\qquad(\text{order estimates}).
\tag{7.48}
$$
(Twice the amplitude at the end of the window, at most $2^{5/2}$ times that at $\bar\lambda$, divided by $\Lambda\cdot\tfrac12\nu_k(4+3t_k)\bar\lambda^2$; $t_k/\nu_k^{1/2}=\nu_k^{1/2}/M_k$. Neglected: relative $O(1/\mathcal D_k)$, $O(\delta p^2/p_c^2)$, $O(1/(\nu_k\lambda))$.) So the window average of the third-order acceleration at rest is $3s_3(\bar\lambda^2+\Lambda^2/12)$ up to the relative $\varrho_{3,\rm win}$, and it is the coefficient $3s_3\bar\lambda^2$ up to the relative $\varrho_{3,\rm win}+\Lambda^2/(12\bar\lambda^2)$: this needs $\Lambda_3\ll\Lambda\ll\bar\lambda$, with $\Lambda_3:=\tfrac{\sqrt2\,w_0}{3(4+3t_k)M_k}\sqrt{\nu_k\bar\lambda/\pi}$, of the order $\tfrac{t_k}{4+3t_k}\,\bar\lambda\,\mathcal D_k(\bar\lambda)^{-1/2}$; such windows exist for $\mathcal D_k(\bar\lambda)\gg1$. The condition is sufficient at the level of the estimate. The coefficients have the sharp-label limits $\langle\omega_k'''\rangle_g\to\omega_k'''(p_0)$, $\langle\omega_k''''\rangle_g\to\omega_k''''(p_0)$ (mean and variance only; errors $\le\tfrac12\sup\lvert\omega_k^{(5)}\rvert\delta p^2$, $\le\tfrac12\sup\lvert\omega_k^{(6)}\rvert\delta p^2$). With $v=\sigma\omega_k'(p_0)$ of (4.17) and $\omega'''=-\omega'(4+3\omega''/\omega)$, for every $p_0$ and, in the last form, **near rest** ($\lvert p_0\rvert\ll p_c$, where $v=\sigma\nu_kp_0(1+O(p_0^2/p_c^2))$):
$$
\delta a^{(2)}_{k,\sigma}=\kappa^2\sigma_X^2\,\lambda\,\sigma\,\omega_k'''(p_0)=-\kappa^2\sigma_X^2\,\lambda\,v\Bigl(4+\frac{3\,\omega_k''(p_0)}{\omega_k(p_0)}\Bigr)=-\kappa^2\sigma_X^2\,\lambda\Bigl(4+\frac{3\tau_k^2}{M_k^2}\Bigr)v\,\Bigl(1+O\bigl(p_0^2/p_c^2\bigr)\Bigr).
\tag{7.15}
$$
It vanishes at rest, is of first order in $v$, opposes $v$, grows linearly in $\lambda$, and at given $v$ is the same in both branches. Small $v$ alone does not give the last form: near $p_0=\tfrac\pi2$ the velocity is small as well, and the bracket of the middle form tends to $4-3\tau_k^2/(M_k^2+\tau_k^2)$; near $p_0=\pi$ it is as near $0$. The third-order term does not vanish at rest:
$$
\delta a^{(3)}_{k,\sigma}=-\tfrac12\kappa^3\mu_3\,\lambda^2\,\sigma\,\omega_k''''(p_0),\qquad\delta a^{(3)}_{k,\sigma}\big\vert_{p_0=0}=\frac{\sigma}{2}\,\kappa^3\mu_3\,\lambda^2\,\frac{\tau_k^2}{M_k}\Bigl(4+\frac{3\tau_k^2}{M_k^2}\Bigr),
\tag{7.16}
$$
of zeroth order in $v$ (relative corrections $O(p_0^2/p_c^2)$ near rest), odd in the branch, present only for a skewed distribution of $X$. (7.15) and (7.16) are the sharp-label coefficients of the secular parts $2s_2\lambda$ and $3s_3\lambda^2$, that is the derivatives of the fastest-growing velocity terms of (7.14). As accelerations they are obtained in case A, at $\lambda=\bar\lambda$: (7.15) under the window condition after (7.33), (7.16) under (7.48) and $\Lambda\ll\bar\lambda$.

   *Case A′.* The instantaneous accelerations $a_n[g](\lambda)$ of a fixed smooth $g$ for $\lambda\to\infty$, without a window. *Order $\kappa^2$.* By (7.47), $a_2[g](\lambda)/\lambda\to2s_2$ for every fixed smooth $g$, whether or not the label weight contains a stationary point of $\omega_k$; the sharp-label coefficient of the limit is (7.15). How fast the oscillating part becomes small depends on the label weight (order estimates; $\lvert p_0\rvert,\delta p\ll p_c$):
$$
a_2^{\rm osc}[g](\lambda)=\begin{cases}O\bigl(\nu_kX_g\,\lambda\,\mathcal D_I^{-n}\bigr)\ \text{for every }n,\qquad\mathcal D_I:=\lambda\,w_I\min(\delta p,d_I)&\text{(F)}\\[4pt]\dfrac\sigma4\sqrt{\dfrac{\nu_k\lambda}{\pi}}\;\rho_s'\,\cos\bigl(2M_k\lambda+\tfrac\pi4\bigr)\Bigl(1+O\bigl(\mathcal D_k^{-1}\bigr)+O\bigl(\delta p^2/p_c^2\bigr)\Bigr),\qquad\rho_s':=(\lvert g\rvert^2)'(0)+(\lvert g\rvert^2)'(\pi)&\text{(S)}\end{cases}
\tag{7.49}
$$
(F): the label weight vanishes on a neighbourhood of every stationary point $0,\pm\tfrac\pi2,\pi$ of $\omega_k$; $d_I$ is the distance of its support $I$ from the nearest of them and $w_I:=\min_I\lvert\omega_k'\rvert$ ($\simeq\nu_kd_I$ near rest). Each of $n$ integrations by parts with $e^{2i\omega\lambda}=\partial_pe^{2i\omega\lambda}/(2i\lambda\omega')$ gives a factor $1/(\lambda w_I\delta p)$ from a derivative of the weight or $\lvert\omega''\rvert/(\lambda\omega'^2)\le1/(\lambda w_Id_I)$ from a derivative of $1/\omega'$; the constants depend on $n$ and on the shape of $g$. For a weight supported in $\lvert p-p_0\rvert\le b\,\delta p$ near rest this requires $\lvert p_0\rvert>b\,\delta p$, and the parameter is $\lambda\nu_k\lvert p_0\rvert\delta p$, the first term of $\mathcal D_k$, once $\lvert p_0\rvert-b\,\delta p\gtrsim\delta p$. (S): the weight contains the stationary point $0$ or $\pi$ (every start at rest with $w_0\neq0$, and starts near rest with $\lvert p_0\rvert\lesssim b\,\delta p$), and $\nu_k\lambda\,\delta p^2\gg1$: stationary phase in the first term of $a_2^{\rm osc}$, where $\omega\theta'^2=\nu_k$; the terms with $\sin2\omega\lambda$ are smaller by $t_k/(\nu_k\lambda)$; at $\pm\tfrac\pi2$, $\theta'=0$. In (S) the oscillating part grows as $\lambda^{1/2}$ against the secular $\lambda$: its relative size $\varrho_{2,\rm osc}:=\tfrac{\lvert\rho_s'\rvert}{4\lvert\langle\omega_k'''\rangle_g\rvert}\sqrt{\nu_k/(\pi\lambda)}$ tends to zero, but slowly. For a weight of the width $\delta p$ with $\lvert p_0\rvert\lesssim\delta p$ ($\lvert\rho_s'\rvert\sim\sqrt{2\pi}\lvert p_0\rvert/\delta p^3$, $\lvert\langle\omega_k'''\rangle_g\rvert\simeq\nu_k(4+3t_k)\lvert p_0\rvert$) it is $\sim\bigl(2\sqrt2\,(4+3t_k)\,\delta p^2\sqrt{\nu_k\lambda\,\delta p^2}\bigr)^{-1}$, small only for $\nu_k\lambda\,\delta p^2\gg1/(8(4+3t_k)^2\delta p^4)$, far beyond $\mathcal D_k\gg1$. For a label weight that is even about $p=0$, $a_2[g]\equiv0$ (every integrand of (7.47) is odd). *Order $\kappa^3$, starts at rest.* The differentiation in $\lambda$ multiplies an oscillating term by its rate $2\omega_k$, so an estimate of $R_3$ gives none of $\partial_\lambda R_3$; the acceleration is calculated from $\partial_\lambda v_3$. From $v_3$ of Step 7, part 3, the terms of $\partial_\lambda v_3$ with the highest powers of $\lambda$ and an oscillating factor are $\sigma\bigl[-\tfrac12\lambda^4\omega\theta'^2\omega'^2\cos2\omega\lambda-\tfrac13\lambda^3\omega\theta'^2\omega''\sin2\omega\lambda\bigr]$ (at $p=0$ the term $-\tfrac13\tau_k^2t_k\lambda^3\sin2M_k\lambda$ of (7.43)). Averaged with $\lvert g\rvert^2$, one integration by parts turns the first into $+\tfrac14\lambda^3\langle\omega\theta'^2\omega''\sin2\omega\lambda\rangle_g$ plus terms smaller by $1/\mathcal D_k$. With the second ($\tfrac14-\tfrac13$) and the stationary phase of (7.48), where $\omega\theta'^2\omega''=\tau_k^2t_k$, for a fixed smooth $g$ at rest with $\delta p\ll p_c$ and $\mathcal D_k=\nu_k\lambda\,\delta p^2\gg1$:
$$
a_3[g](\lambda)=3s_3\lambda^2+a_3^{\rm osc}[g](\lambda),\qquad a_3^{\rm osc}\simeq-\frac\sigma{12}\lambda^3\bigl\langle\omega_k\theta_k'^2\omega_k''\sin2\omega_k\lambda\bigr\rangle_g\simeq-\frac{\sigma\,\tau_k^3\,w_0}{24\sqrt\pi\,M_k^{3/2}}\,\lambda^{5/2}\sin\bigl(2M_k\lambda+\tfrac\pi4\bigr),\qquad\varrho_{3,\rm osc}(\lambda):=\frac{\text{amplitude of }a_3^{\rm osc}}{3\lvert s_3\rvert\lambda^2}\simeq\frac{w_0}{12\,(4+3t_k)}\sqrt{\frac{\nu_k\lambda}{\pi}}
\tag{7.50}
$$
(order estimates; $a_3^{\rm osc}\simeq\partial_\lambda R_3$ of (7.48), of the size $2M_k\lvert R_3\rvert$; neglected: relative $O(1/\mathcal D_k)$, $O(\delta p^2/p_c^2)$ and $O(1/(\nu_k\lambda))$, the last from the terms $\propto\lambda^2$ of (7.43)). The oscillating part grows as $\lambda^{5/2}$, faster than the secular $\lambda^2$. For a Gaussian label weight $\varrho_{3,\rm osc}=\sqrt{\mathcal D_k/2}\,/\bigl(6(4+3t_k)\delta p^2\bigr)$, which exceeds one for $\mathcal D_k>72(4+3t_k)^2\delta p^4$, that is on the whole dephased range of a narrow start. So for a fixed smooth $g$ at rest with $w_0\neq0$, $a_3[g](\lambda)/\lambda^2$ has **no limit** for $\lambda\to\infty$: dephasing alone does not lead to (7.16), and without a window the third-order acceleration at rest is (7.50), not (7.16). A window removes the oscillation: $\varrho_{3,\rm win}\lesssim2^{5/2}\varrho_{3,\rm osc}(\bar\lambda)/(M_k\Lambda)$, (7.48).

   *Case B.* Taylor's formula in $p$ (mean and variance of the label weight) gives $a_2[g](\lambda)\to\partial_\lambda v_2(p_0,\lambda)$ for $\delta p\to0$ at fixed $\lambda$. Exactly zero at rest (every term of (7.13) vanishes at $p=0$: $\omega'=\theta''=(\omega\theta')'=0$). Near rest, expanding the coefficients of (7.41) with the facts of the Setup ($\omega=\omega_k(p_0)$ in the trigonometric functions):
$$
\delta a^{(2)}_{k,\sigma}\Big\vert_{\rm B}=\kappa^2\sigma_X^2\,v\Bigl[-(4+3t_k)\lambda+\lambda\cos2\omega\lambda+\nu_k\lambda^2\sin2\omega\lambda+\frac{2+3t_k}{2M_k}\sin2\omega\lambda\Bigr]\Bigl(1+O\bigl(p_0^2/p_c^2\bigr)\Bigr);\qquad\Bigl\lvert\overline{a_2}(p_0)-\sigma\omega'''\bar\lambda\Bigr\rvert\le\frac1\Lambda\Bigl[\theta'^2\lvert\omega'\rvert\Bigl(\bar\lambda^2+\frac{\Lambda^2}4\Bigr)+\frac{\lvert\theta'(\omega\theta')'\rvert}{\omega}\bar\lambda+\frac{\lvert\theta'\theta''\rvert}{2\omega}\Bigr].
\tag{7.42}
$$
The second statement is an exact bound for the window average at a sharp label, every $p_0$ (the three oscillating terms of (7.13) at the two ends of the window). So in case B the change of the acceleration is still zero at rest and of first order in $v$, but it is not (7.15): the oscillating term $\nu_k\lambda^2\sin2\omega\lambda$ exceeds the secular one for $\nu_k\lambda>4+3t_k$, and in a window average its relative size is up to $t_k(\bar\lambda/\Lambda+\Lambda/4\bar\lambda)/(4+3t_k)$, which no choice of the window makes small unless $t_k\ll1$. At third order and at rest, the Bloch equations (7.12) at $p=0$ ($q=-\varepsilon s$, $\varphi(\lambda)-\varphi(s)=2M(\lambda-s)+\tfrac13\nu\varepsilon^2(\lambda^3-s^3)$, $z$ to the order $\varepsilon^3$ by elementary integrals) give, exactly:
$$
\begin{aligned}
v_3(0,\lambda)&=\sigma\Bigl\{\frac{\nu_k}{6}\lambda^3\bigl[(4+3t_k)+t_k\cos2M_k\lambda\bigr]-\frac{t_k(1+t_k)}4\lambda^2\sin2M_k\lambda-\frac{t_k}{4M_k}\lambda\bigl[(1+2t_k)+\tfrac12t_k\cos2M_k\lambda\bigr]+\frac{t_k(2+5t_k)}{16M_k^2}\sin2M_k\lambda\Bigr\},\\
\partial_\lambda v_3(0,\lambda)&=\sigma\Bigl\{\frac{\nu_k}{2}\lambda^2\bigl[(4+3t_k)-\cos2M_k\lambda\bigr]-\frac{\tau_k^2t_k}{3}\lambda^3\sin2M_k\lambda-\frac{t_k(2+t_k)}4\lambda\sin2M_k\lambda-\frac{t_k(1+2t_k)}{4M_k}\bigl(1-\cos2M_k\lambda\bigr)\Bigr\}.
\end{aligned}
\tag{7.43}
$$
$\delta a^{(3)}_{k,\sigma}\vert_{\rm B}=\kappa^3\mu_3\,\partial_\lambda v_3(0,\lambda)$ at rest: its non-oscillating part is (7.16), and it has in addition an oscillating part that grows as $\lambda^3$. For $p_0\neq0$ the highest power is $-\tfrac12\sigma\lambda^4\omega\,\theta'^2\omega'^2\cos2\omega\lambda$ (Step 7, part 3), small against (7.43) for $\lvert\omega_k'(p_0)\rvert\lambda\ll1$.
3. *Level and branch dependence.* Branch: at given $v$ the second-order terms (7.15), (7.42) are the same in both branches; the third-order terms (7.16), (7.43) are odd in $\sigma$. Level:
   - *Case A.* (7.15) is the same for all levels at leading order in $v$ near rest **iff** $\tau_k/M_k$ does not depend on $k$, and as a function of $p_0$ iff $(\tau_k,M_k)$ do not ($\omega_k-\omega_{k'}$ constant forces equal data). (7.16) at rest is level-independent iff $\nu_k(4+3t_k)$ is. These are statements about the secular parts of the window averages; the rests of (7.33) depend on the level and are small under the window conditions of part 2.
   - *Case A′.* Order $\kappa^2$: the limit of $a_2[g]/\lambda$ has the level dependence of (7.15); at finite $\lambda$ the oscillating part (7.49) has, in (S), the rate $2M_k$ and an amplitude $\propto\nu_k^{1/2}$, the same for all levels iff $(\tau_k,M_k)$ are, with a relative size that tends to zero. Order $\kappa^3$ at rest: (7.50) is the same for all levels, for all $\lambda$, **iff** $(\tau_k,M_k)$ do not depend on $k$ (the rate $2M_k$, then the amplitude $\propto\tau_k^3M_k^{-3/2}$); a common $\nu_k(4+3t_k)$, which suffices for the window average, does not suffice here. (7.50) is odd in $\sigma$.
   - *Case B.* (7.42) and (7.43) are the same for all levels, for all $\lambda$, **iff** $(\tau_k,M_k)$ do not depend on $k$: the oscillation rate $2M_k$ has to be common, and then $t_k$. A common $\tau_k/M_k$, which suffices in case A, does not suffice here.
   - In the class $\mathcal C$ ($\tau_k=\tau$, $M_k$ distinct): **no** in all three cases; for $M_k\ll\tau$ the case-A terms scale as $M_k^{-2}$ and $M_k^{-3}$.
4. *Background form and the weak-record regime (item 2(d)).* *The speed bound of one level.* In the level $k$ the velocity operator is $i[C^{(k)}+\varepsilon\hat x,\hat x]=-\tau_k\cos\hat p\,\sigma_1$ at zero gradients (with gradients, the same with the factors $\zeta_{x,k}$), of norm $\tau_k$. With $d_0=\kappa\sigma_X$, $\hat e$ and $\bar x=d_0\bar x_{\rm lab}\hat e$ of (17.15) of 03-ilang-space/17-motion:
$$
\Bigl\lVert\frac{\mathrm d\bar x}{\mathrm d\lambda}\Bigr\rVert\le c_k:=\tau_k\,d_0\ \ \text{for every state in the level }k;\qquad c_k\le d_0\max_j\tau_j\le v^{\rm full}_{\max}:=d_0\sum_{n\in\mathbb Z_N}\Bigl\lvert\frac1N\sum_j\tau_je^{2\pi ijn/N}\Bigr\rvert .
\tag{7.45}
$$
$c_k$ is a bound for one conserved level, derived here. $v^{\rm full}_{\max}$ is the row-sum expression of (5.11) of 04-ilang-time/05-moving-clock for the complete body contract in the place basis $(x,s,a)$, where the hopping contains $\sum_j\tau_j\lvert f_j\rangle\langle f_j\rvert$, not diagonal in the clock places (row sum of a circulant matrix; it is at least the largest eigenvalue). The two are different statements; they coincide, $c_k=v_{\max}=\tau d_0$, iff the $\tau_j$ are equal, as in the class $\mathcal C$. For $\Phi_\sigma[\alpha,g]$ the bound is $\sum_k\lvert\alpha_k\rvert^2c_k$, by (7.1). With $u:=d_0v$ and $\gamma_X:=\mu_3/\sigma_X^3$, the case-A terms (7.15), (7.16) near rest (the sharp-label coefficients of the secular parts of the window averages; $\lambda$ stands for $\bar\lambda$) are
$$
\delta\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}=\Bigl[-\lambda\Bigl(4d_0^2+\frac{3c_k^2}{M_k^2}\Bigr)u+\frac{\sigma\gamma_X}{2}\,\lambda^2\,\frac{c_k^2}{M_k}\Bigl(4d_0^2+\frac{3c_k^2}{M_k^2}\Bigr)\Bigr]\Bigl(1+O\bigl(p_0^2/p_c^2\bigr)\Bigr)\hat e,\qquad d_0\,a_{k,\sigma}(0)=\frac{c_k^2}{d_0M_k}\bigl(\sigma\mathcal F_k-G_k\bigr),
\tag{7.17}
$$
with $p_0^2/p_c^2=v^2/(\nu_kp_c)^2$ to this order. They are smaller than the drift acceleration (4.7) at rest where
$$
\varrho_2:=\frac{\lambda\lvert u\rvert M_k}{d_0\,\tau_k^2\lvert\sigma\mathcal F_k-G_k\rvert}\Bigl(4d_0^2+\frac{3c_k^2}{M_k^2}\Bigr)<1\quad\text{and}\quad\varrho_3:=\frac{\lvert\gamma_X\rvert d_0\lambda^2}{2\lvert\sigma\mathcal F_k-G_k\rvert}\Bigl(4d_0^2+\frac{3c_k^2}{M_k^2}\Bigr)<1,
\tag{7.18}
$$
that is $\lambda<\lambda_2:=\nu_k\lvert\sigma\mathcal F_k-G_k\rvert/(\kappa^2\sigma_X^2(4+3t_k)\lvert v\rvert)$ and $\lambda<\lambda_3:=\bigl(2\lvert\sigma\mathcal F_k-G_k\rvert/(\kappa^3\lvert\mu_3\rvert(4+3t_k))\bigr)^{1/2}$. The rests of the window averages (7.33), relative to the drift acceleration, are
$$
\varrho_{2,\rm rest}:=\kappa^2\sigma_X^2\,\frac{2c_1[g]\,\bar\lambda+2c_0}{\Lambda}\cdot\frac{M_k}{\tau_k^2\,\lvert\sigma\mathcal F_k-G_k\rvert}\ \ (\text{bound}),\qquad\varrho_{3,\rm rest}:=\frac{\kappa^3\lvert\mu_3\rvert\,\lvert R_3(\lambda_1+\Lambda)-R_3(\lambda_1)\rvert}{\Lambda\,\nu_k\lvert\sigma\mathcal F_k-G_k\rvert}=O\Bigl(\frac{\kappa^3\lvert\mu_3\rvert\,t_k\,\bar\lambda^3}{\Lambda\,\mathcal D_k^{1/2}\lvert\sigma\mathcal F_k-G_k\rvert}\Bigr).
\tag{7.34}
$$
*The accuracy range.* At the orders retained by the question (first order in the gradients at $\kappa^0$; $\kappa^2$ and $\kappa^3$ at zero gradients), the window-averaged acceleration of the level $k$ for a fixed smooth $g$ near rest is the drift term of (4.10) plus $\kappa^2\sigma_X^2\overline{a_2}+\kappa^3\mu_3\overline{a_3}$. Since $\lambda_1\ge0$ gives $\Lambda\le2\bar\lambda$, (7.33) yields: if, at $\lambda=\bar\lambda$ and for every level with $\alpha_k\neq0$,
$$
\varrho_2+\varrho_{2,\rm rest}+\tfrac43\varrho_3+\varrho_{3,\rm rest}\le\delta_{\rm acc},
\tag{7.46}
$$
then the record terms change the window-averaged acceleration of each level by at most $\delta_{\rm acc}$ times its drift acceleration, and two levels with the same drift acceleration fall alike to the relative accuracy $2\delta_{\rm acc}$. (7.46) is sufficient at the retained orders ($\varrho_2,\varrho_{2,\rm rest}$ controlled by bounds, $\varrho_{3,\rm rest}$ by the estimate of $R_3$); terms that mix $\kappa$ with the gradients, $\kappa^4$ and the second order in the gradients are outside the scope of the question and not part of the statement. It is **not necessary**: its left side adds, in absolute value, bounds and one estimate, while the rests can be far smaller than their bounds (for a label weight that is nearly even about $p_0=0$, $\langle v_2\rangle_g$ is nearly zero and $c_1[g]$ is not), and the terms of the orders $\kappa^2$ and $\kappa^3$ have signs. *What is necessary.* Let two levels $k,k'$ have the same first-order window average $\bar a:=\overline{a^{(1)}}\neq0$. At the retained orders their window-averaged accelerations differ by the difference of their record terms, so they agree to the relative accuracy $\delta$ **iff**
$$
\Bigl\lvert\kappa^2\sigma_X^2\bigl(\overline{a_2}^{(k)}-\overline{a_2}^{(k')}\bigr)+\kappa^3\mu_3\bigl(\overline{a_3}^{(k)}-\overline{a_3}^{(k')}\bigr)\Bigr\rvert\le\delta\,\lvert\bar a\rvert;\qquad\text{in the regime (R):}\quad\kappa^2\bar\lambda\,\Bigl\lvert\tfrac12\kappa\mu_3\bar\lambda-\sigma_X^2\,p_0\Bigr\rvert\,\lvert n_k-n_{k'}\rvert\le\delta\,\lvert\bar a\rvert,\qquad n_k:=\nu_k(4+3t_k).
\tag{7.51}
$$
The first statement is necessary and sufficient by the definition of the accuracy, for every smooth $g$ and every window. The second is its leading form in the **regime (R)**: the same $g$ in both levels (as in $\Phi_\sigma[\alpha,g]$), concentrated at $p_0$ near rest; the rests of (7.33) of both levels small against the left side; $\Lambda\ll\bar\lambda$; and the sharp-label coefficients $\langle\omega_k'''\rangle_g\simeq-n_kp_0$, $\langle\omega_k''''\rangle_g\simeq-n_k$ (errors $\tfrac12\sup\lvert\omega_k^{(5)}\rvert\delta p^2$, $\tfrac12\sup\lvert\omega_k^{(6)}\rvert\delta p^2$ and relative $O(p_0^2/p_c^2)$, small against the kept terms). There the record term of the level $k$ is $\sigma\,n_k\,\kappa^2\bar\lambda\bigl(\tfrac12\kappa\mu_3\bar\lambda-\sigma_X^2p_0\bigr)$, which is (7.15) plus (7.16) with $v=\sigma\nu_kp_0$. With $\bar a\simeq a_{k,\sigma}(0)$ and $\varrho_2,\varrho_3$ of the level $k$ at $\bar\lambda$, (7.18), the condition reads $\tfrac{\lvert n_k-n_{k'}\rvert}{n_k}\,\bigl\lvert\varrho_3\operatorname{sgn}\mu_3-\varrho_2\operatorname{sgn}p_0\bigr\rvert\le\delta$. Hence, in (R): (i) for $n_k=n_{k'}$ it holds for every $\kappa$, whatever $\varrho_2,\varrho_3$ are; (ii) at $p_0=\kappa\mu_3\bar\lambda/(2\sigma_X^2)$ the two orders cancel in every level; (iii) for $\mu_3p_0\le0$ (no cancellation) it reads $\varrho_2+\varrho_3\le\delta\,n_k/\lvert n_k-n_{k'}\rvert$, and with $\lvert n_k-n_{k'}\rvert\ge c_n\,n_k$ it implies $\varrho_2+\varrho_3\le\delta/c_n$. Only in case (iii), and only in (R), is a condition on $\varrho_2+\varrho_3$ necessary. In the class $\mathcal C$, $n_k=\tau^2(4M_k^2+3\tau^2)/M_k^3$ is strictly decreasing in $M_k$, so $n_k\neq n_{k'}$ for every pair. Asking each of the four terms of (7.46) to be $\le\delta_{\rm acc}/4$ gives the sufficient range
   - in $\lambda$: $\bar\lambda\le\tfrac14\delta_{\rm acc}\lambda_2$ and $\bar\lambda\le(\tfrac3{16}\delta_{\rm acc})^{1/2}\lambda_3$, with a window $\Lambda\ge4\kappa^2\sigma_X^2(2c_1[g]\bar\lambda+2c_0)/(\delta_{\rm acc}\nu_k\lvert\sigma\mathcal F_k-G_k\rvert)$, which exists inside $\Lambda\le2\bar\lambda$ only if $4\kappa^2\sigma_X^2c_1[g]\le\delta_{\rm acc}\nu_k\lvert\sigma\mathcal F_k-G_k\rvert$ (a condition on the spread of the start, through $X_g$), and, for the order $\kappa^3$ at rest, $\varrho_{3,\rm rest}\simeq\varrho_3\,\varrho_{3,\rm win}\le\tfrac14\delta_{\rm acc}$ with (7.48), which needs $\mathcal D_k\gg1$ or a window $\Lambda\gg\Lambda_3$;
   - in $\kappa$, at fixed $\bar\lambda$ and $v$: $\kappa^2\le\delta_{\rm acc}\nu_k\lvert\sigma\mathcal F_k-G_k\rvert/(4\sigma_X^2(4+3t_k)\lvert v\rvert\bar\lambda)$ and $\kappa^3\le3\delta_{\rm acc}\lvert\sigma\mathcal F_k-G_k\rvert/(8\lvert\mu_3\rvert(4+3t_k)\bar\lambda^2)$;
   - in the gradients: $\lvert\sigma\mathcal F_k-G_k\rvert\ge\tfrac4{\delta_{\rm acc}}\max\bigl\{\kappa^2\sigma_X^2(4+3t_k)\lvert v\rvert\bar\lambda/\nu_k,\ \tfrac23\kappa^3\lvert\mu_3\rvert(4+3t_k)\bar\lambda^2\bigr\}$.

   For a sharp label without dephasing (case B) the same comparison uses (7.42), (7.43): at rest the second-order term is zero, and the third-order term needs in addition $\kappa^3\lvert\mu_3\rvert\,\tfrac13\tau_k^2t_k\lambda^3\le\delta_{\rm acc}\nu_k\lvert\sigma\mathcal F_k-G_k\rvert$. For the instantaneous acceleration of a fixed $g$ without a window (case A′) the third-order term at rest needs in addition $\varrho_3\,\varrho_{3,\rm osc}(\lambda)\le\delta_{\rm acc}$, that is $\kappa^3\lvert\mu_3\rvert\,\tau_k^3w_0\,\lambda^{5/2}/(24\sqrt\pi\,M_k^{3/2})\le\delta_{\rm acc}\nu_k\lvert\sigma\mathcal F_k-G_k\rvert$, by (7.50) (order estimate); (7.46) does not control it. So, for the fall to be the same for all levels to the relative accuracy $\delta$ (window averages, retained orders), **"weak records" has to mean** (7.51) for every pair of levels involved, on the whole range of $\lambda$, windows and $v$ observed. (7.46) with $\delta_{\rm acc}=\delta/2$ for every level involved (and the drift acceleration for $\bar a$) is a **sufficient** condition for it in single-level quantities: a joint condition on $\kappa^2\lambda\lvert v\rvert$, $\kappa^3\lvert\mu_3\rvert\lambda^2$ and the gradients. Neither is uniform in $\lambda$. (7.46) depends on the start (through $X_g$ and $\mathcal D_k$) and fails if the gradients are sent to zero at fixed $\kappa$, and so does (7.51) unless its left side vanishes; "$\kappa\to0$ at fixed $\lambda$, chain and start" satisfies both for every $\delta>0$.

### Step 9. Item 3: the operator under the factor

1. *Fall from rest.* $(1+gx)[M_k\sigma_3+u_k]-\mathcal F_Ux=M_k\sigma_3+gM_kx\,\sigma_3+u_k-(\mathcal F_U-gu_k)x$, a linear level profile with $G_k=gM_k$, $\mathcal F_k=\mathcal F_U-gu_k$, $V_k=u_k$. The hopping factors do not enter (4.16):
$$
a_{k,\pm}(0)=-g\,\tau_k^2\pm\frac{\tau_k^2}{M_k}\bigl(\mathcal F_U-g\,u_k\bigr).
\tag{7.19}
$$
$$
\begin{aligned}
\text{upper: }&\tau_k^2\Bigl[\tfrac{\mathcal F_U-gu_k}{M_k}-g\Bigr]=a_+\ \forall k;\qquad\text{lower: }\tau_k^2\Bigl[-\tfrac{\mathcal F_U-gu_k}{M_k}-g\Bigr]=a_-\ \forall k;\\
\text{both, }a_+=a_-=a:\ &g\,u_k=\mathcal F_U\ \text{ and }\ g\,\tau_k^2=-a\quad\forall k.
\end{aligned}
\tag{7.20}
$$
(Sum and difference of the two lines give the third.) For $\tau_k=\tau$ the upper and the lower condition coincide, for every $g$: $(\mathcal F_U-gu_k)/M_k=:y_U$ independent of $k$, and then $a_\pm=\tau^2(-g\pm y_U)$; one value iff $y_U=0$. For $g\neq0$, with $y_U=:-g\,w$, this is $u_k=u_*+wM_k$ with $gu_*=\mathcal F_U$, and $a_\pm=-\tau^2g(1\pm w)$; one value iff $w=0$. For $g=0$ the numbers $u_k$ drop out and the condition is on $\mathcal F_U/M_k$ alone: it holds iff $\mathcal F_U=0$ (then $a_\pm=0$) or all $M_k$ are equal (then $a_\pm=\pm\tau^2\mathcal F_U/M_k$; one value iff $\mathcal F_U=0$).
2. *Well-defined statements (M6).* By (8.1) and (8.20) of 04-ilang-time/08-maximal-speed the effective body contract is defined modulo $c\,\mathbb 1$ with one real $c$ for all $x$, branches and levels (a level-dependent constant is not diagonal in the places and is not a replacement of (8.1)). In the parametrization of item 3,
$$
C_{\rm eff}\mapsto C_{\rm eff}+c\,\mathbb 1\iff(u_k,\mathcal F_U)\mapsto(u_k+c,\ \mathcal F_U+g\,c);\qquad\text{invariants: }g,\ M_k,\ u_k-u_{k'},\ \mathcal F_k=\mathcal F_U-g\,u_k .
\tag{7.21}
$$
Hence: $u_k$ — **not** well defined; $u_k-u_{k'}$ — **well defined**; "$u_k=0$ for all $k$" — **not** well defined (well defined are "$u_k$ is the same for all $k$" and "$\mathcal F_U=gu_k$ for all $k$"); "$u_k\propto M_k$" — **not** well defined (well defined is "$u_k$ is affine in $M_k$", with the slope $w$ and $\mathcal F_U-gu_*$). The conditions (7.20) contain only invariants.
3. *Evaluations.* $u_k=uM_k$: $a_{k,\sigma}(0)=-\tau_k^2g(1+\sigma u)+\sigma\tau_k^2\mathcal F_U/M_k$; for $\tau_k=\tau$ and distinct $M_k$ it is level-independent iff $\mathcal F_U=0$, with the branch-dependent value $-\tau^2g(1+\sigma u)$. Class $\mathcal C$, variant (f) with a plain gradient ($u_k=c_{\rm f}$): $a_{k,\sigma}(0)=-\tau^2g+\sigma\tau^2(\mathcal F_U-gc_{\rm f})/M_k$, level-independent iff $\mathcal F_U=gc_{\rm f}$, then $-\tau^2g$ in both branches; $\mathcal F_U=0$ reproduces row (f) of (4.25). Control: $\mathcal F_U-gu_k=0$ (e.g. $u_k=0$, $\mathcal F_U=0$) against $u_k=c_{\rm f}\neq0$, $\mathcal F_U=0$.

### Step 10. Item 4: table of orders

Orders are those of the leading deviation; "grad" is the joint order in the gradients; the entries on $M_k/\tau_k$ are relative to the fall; orders in $v$ are orders near rest ($\lvert p_0\rvert\ll p_c$). (i): the fall is the same for all levels; (ii): the relation (4.24), a statement at rest in the class $\mathcal C$.

| Source | (i) level-independent fall | (ii) relation (4.24) | Tags |
|---|---|---|---|
| Lattice | grad$^1$, $v^2$, relative $(M_k/\tau_k)^2$, $\kappa^0$, $\lambda^0$: $c_{2,k,+}-c_{2,k',+}=-2a_0\bigl(\tfrac{M_k^2}{\tau_k^4}-\tfrac{M_{k'}^2}{\tau_{k'}^4}\bigr)$; none at $v=0$ | none at $v=0$; at $v\neq0$ level-dependent at grad$^1$, $v^2$, $(M_k/\tau)^2$ | (4.20), (4.21), (4.23), (4.25) |
| Drift velocity | grad$^1$, $v^2$, $(M_k/\tau_k)^0$ unless $c^{(0)}_{2,k}$ is level-independent; in $\mathcal C$ only for $\mathcal F=0$, $g_m=g_\eta$; $O(v^4)$ an order estimate | the acceleration is $a(0)+c_2v^2$: grad$^1$, relative $c_2v^2/a(0)$ (variant (a): $-\ell_kv^2/\tau^2$), $\kappa^0$, $\lambda^0$ | (4.17), (4.20), (4.21), (4.23) |
| Superposition of levels | no new condition (limits of (7.2)); if the fall is level-dependent: mean acceleration $\alpha$-dependent at grad$^1$; between-level variance grad$^2$, $\lambda^4$ (coefficient in the sharp-label limit); at $p_0\neq0$ separation at grad$^0$, $v^1$, $\lambda^1$ for $\delta p<\delta p_*$ of (7.27) | position side unchanged (exact mixture); label rates level-dependent at grad$^1$ (sequences with $c_{\rm cut}\neq1$, (7.28)); $\lvert\Gamma_{kk'}\rvert$, order by (7.36): grad$^1$ iff $\operatorname{Re}\epsilon_1\neq0$, for narrow starts at rest $\Delta\nu_{kk'}[\lambda^3\varpi_{kk'}\langle p^3\rangle_g,\ \lambda^2(\pi_k-\pi_{k'})J_g]$; grad$^2$ always present for narrow starts with $(\pi_k-\pi_{k'},\varpi_{kk'})\neq0$: $\lambda^2\delta x_g^2$, $\lambda^3C_g$, $\lambda^4\delta p^2$, and the only order for definite parity or equal data; both also for a level-independent fall $a\neq0$ unless $\nu_k=\nu_{k'}$ | (7.1)–(7.5), (7.27), (7.7), (7.28), (7.8), (7.9), (7.23)–(7.26), (7.29)–(7.32), (7.35)–(7.40) |
| Records | $\kappa^1$: none. $\kappa^2\sigma_X^2$: grad$^0$, zero at $v=0$, $v^1$; case A (window average) and case A′ (instantaneous value for $\lambda\to\infty$ at fixed $g$): $\lambda^1$, factor $4+3t_k$, level-independent iff $\tau_k/M_k$ is; in A′ in addition an oscillating part at the rate $2M_k$, at most $\propto\lambda^{1/2}$ when the label weight contains a stationary point, (7.49); case B: in addition oscillating terms up to $\nu_k\lambda^2$ at the rate $2\omega_k$, level-independent iff $(\tau_k,M_k)$ are. $\kappa^3\mu_3$: grad$^0$, $v^0$; case A (window average under (7.48)): $\lambda^2$, factor $\nu_k(4+3t_k)$; case A′: in addition an oscillating part $\propto\tau_k^3M_k^{-3/2}\lambda^{5/2}$ that outgrows the $\lambda^2$ term, (7.50), level-independent iff $(\tau_k,M_k)$ are; case B: in addition oscillating terms up to $\tfrac13\tau_k^2t_k\lambda^3$ | the same terms add to the left side; for two levels the relative accuracy $\delta$ holds iff (7.51) (window averages, retained orders); (7.46) is sufficient for the accuracy $2\delta_{\rm acc}$, not necessary | (7.11), (7.13)–(7.18), (7.33), (7.34), (7.41)–(7.51) |
| Operator under the factor | grad$^1$, $v^0$, $\kappa^0$, $\lambda^0$, relative $(\mathcal F_U-gu_k)/(gM_k)$: the same order as the fall, no small parameter | in $\mathcal C$ the hypothesis holds iff $\mathcal F_U=gc_{\rm f}$; then unchanged | (7.19)–(7.21), (4.25) row (f) |
| Plain gradient | grad$^1$, $v^0$: $\sigma\tau_k^2\mathcal F_k/M_k$, level-dependent and odd in the branch; in $\mathcal C$ admissible at rest only for $\mathcal F=\pm m(g_m-g_\eta)$, through $v^2$ only $\mathcal F=0$ (leading order in $M_k/\tau$) | the hypothesis fixes $\mathcal F=m(g_m-g_\eta)$; any other $\mathcal F$ breaks it at grad$^1$, $v^0$ | (4.16), (4.19), (4.22), (4.23), (4.25) row (d) |
$$
\text{(the table above is the result of item 4)}
\tag{7.22}
$$

## Result

Common class, unless an item restricts it: a body of $\mathcal C_{\rm lev}$ with linear level profiles, both branches $\sigma$, every level, $\lambda\ge0$, the infinite chain as the limit of long chains. "First order" is the first order in the gradients at fixed $\lambda$, start and chain, followed by the long chain. "Leading order in $\kappa$" is $\kappa\to0$ at fixed $\lambda$, chain and start (the effective body contract). A *sharp-label limit* is a limit along a sequence of starts concentrated at $p_0$ with $\delta p\to0$; where nothing else is said it uses only the mean and the variance of the label weight.

**Item 1, starts $\Phi_\sigma[\alpha,g]$.**
- **(7.1)** Exact in the gradients and in $\kappa$; every $\alpha$, smooth $g$, $\chi$, $\lambda$, chain. $p_x=\sum_k\lvert\alpha_k\rvert^2p^{(k)}_x$, $\bar x_{\rm lab}=\sum_k\lvert\alpha_k\rvert^2\bar x_k$; rests on the conservation of the level and the orthonormality of the $\lvert f_k\rangle$.
- **(7.2)** Every $\alpha$, every smooth $g$ concentrated at $p_0$. Limits: first order, long chain, $\Lambda\to\infty$ at fixed $g$, then sharp label; leading order in $\kappa$. Drift acceleration $\sum_k\lvert\alpha_k\rvert^2a_{k,\sigma}(p_0)$ (the bound is exact for every $\Lambda>0$). The same for all $\alpha$ iff $a_{k,\sigma}(p_0)$ is level-independent; at rest line 1 or 2 of (4.19), no new condition.
- **(7.3), (7.4)** Starts at rest, fixed smooth $g$; first order, long chain. $D=\tfrac14\lambda^4\operatorname{Var}_\alpha(\langle a_{\cdot,\sigma}\rangle_g)+E$ with a bound on $E$; the sharp-label limit is taken on the coefficient of $\lambda^4$, which tends to $\tfrac14\operatorname{Var}_\alpha(a_{\cdot,\sigma}(0))$. (7.4), class $\mathcal C$, $\lvert\alpha_k\rvert^2=1/N$: zero for (a), (c); $\tfrac14\lambda^4\tau^4g^2m^2\mathcal V_N$ for (b1), (b2); $\tfrac14\lambda^4\tau^4g^2c_{\rm f}^2\mathcal V_N$ for (f).
- **(7.5), (7.27)** Zero gradients. Exact, every smooth $g$: $v_k=\sigma\langle\omega_k'\rangle_g$; bound (7.27); sharp-label velocities (7.5). Item 1(d): a start of finite width can have equal velocities in all levels (example, every member of $\mathcal C_{\rm lev}$); in $\mathcal C$, for $\sin2p_0\neq0$, no start with $\delta p<\delta p_*$ has equal velocities in two levels; in $\mathcal C_{\rm lev}$ equality at one such $p_0$ is one condition and needs $\tau_k\neq\tau_{k'}$, at all $p_0$ iff the data are equal.
- **(7.6), (7.7), (7.28)** (7.6) exact in the gradients. (7.7) first order, every smooth $g$, no window. (7.28) bound, every start at rest. For sequences of starts at rest with $\delta p\to0$, $\lvert g(\pi)\rvert^2\to c_{\rm cut}$: limiting rate $(1-c_{\rm cut})(\mathcal F_k-\sigma G_k)$. For $c_{\rm cut}\neq1$: level-independent iff $\mathcal F_k-\sigma G_k$ is; together with a level-independent drift acceleration $a\neq0$ iff also $\tau_k^2/M_k$ is; impossible in $\mathcal C$. For $c_{\rm cut}=1$ the limiting rate is zero in every level.
- **(7.8), (7.9), (7.23)** (7.8) exact with the medium in the level components; at leading order in $\kappa$ the $\Gamma_{kk'}$ of the effective body contract. (7.9) exact, every smooth $g$. (7.23) exact, every $g$, $p_0$, $\lambda$: $\Gamma_{kk'}[g;\text{grad}]=\Gamma_{kk'}[g^-;-\text{grad}]$.
- **(7.35), (7.36)** Exact Taylor expansion in the gradients at fixed $\lambda$ and start; every smooth $g$, $p_0$, $\lambda$ with $\Gamma^{(0)}_{kk'}\neq0$; leading order in $\kappa$. $\lvert\Gamma_{kk'}\rvert/\lvert\Gamma^{(0)}_{kk'}\rvert=1+\operatorname{Re}\epsilon_1+Q_2+O(\text{grad}^3)$, $Q_2=-\tfrac12\mathcal V+\tfrac12(\operatorname{Im}\epsilon_1)^2+\operatorname{Re}\epsilon_2$; bound $\lvert\epsilon_1\rvert\le\eta_0\mathcal V^{1/2}$. **Order of the first deviation**, complete by (7.36): none (equal blocks); degree one at a zero of $\Gamma^{(0)}_{kk'}$; first order iff $\operatorname{Re}\epsilon_1\neq0$; second order iff $\operatorname{Re}\epsilon_1=0\neq Q_2$; otherwise third or higher. Starts of definite parity and pairs with equal data have $\operatorname{Re}\epsilon_1=0$ (exact).
- **(7.24), (7.29), (7.30), (7.31)** Exact, every smooth complex $g$, $p_0$, $\lambda\ge0$: the complete first-order term $\Gamma^{(1)}_{kk'}$, branch part and inter-branch part, with the bounds $A_{k,\perp}$ and (7.30). (7.31): bound, every start at rest; $\Gamma^{(0)}_{kk'}\neq0$ where $\epsilon_0<1$.
- **(7.37)** Bound, every start at rest: $\delta x_g^2\delta p^2\ge C_g^2+\tfrac14(1-\lvert g(\pi)\rvert^2)^2$, hence $2\delta pX_g\ge\lvert1-\lvert g(\pi)\rvert^2\rvert$.
- **(7.38), (7.10), (7.25), (7.32)** Class (N): narrow starts at rest of any shape (smooth complex $g$, $\delta p\ll p_c$, moment conditions, $c_{\rm cut}\neq1$); range (N2): $\epsilon_0\ll1$. Status: leading terms, order estimates; the coefficients are exact in the local form, the lower bound of (7.38) is exact, and (7.32) gives the size of every neglected contribution with its calculation. Second order: $Q_2\simeq-\tfrac12\operatorname{Var}_g(K_{kk'})$, different from zero iff $(\pi_k-\pi_{k'},\varpi_{kk'})\neq(0,0)$; dependence $\lambda^2\delta x_g^2$, $\lambda^3C_g$, $\lambda^4\delta p^2$; (7.10) is the real Gaussian case. First order: (7.25), present iff $\Delta\nu_{kk'}\neq0$ and its bracket is not zero. Where (7.38) leads, for data that differ by order one: (N2) with $\lambda\gg\tau\delta p/M^2$.
- **(7.39)** Exact through second order; pairs with equal $(\tau,M)$, every smooth $g$, $p_0$, $\lambda$: $1-\lvert\Gamma_{kk'}\rvert=\tfrac12\operatorname{Var}(\Omega_k-\Omega_{k'})$. Starts at rest that are not narrow, with unequal data: (7.36), (7.24), (7.35), exact; growth at most $\lambda^2$ (first order) and $\lambda^4$ (second order).
- **(7.26), (7.40)** Class (N), range (N2), status of (7.25), (7.38). With a level-independent fall $a$: the deviation **does not vanish** for $a\neq0$ and $\nu_k\neq\nu_{k'}$ ($Q_2<0$ by an exact lower bound, for every shape of $g$); in $\mathcal C$ this is every pair. For $a=0$ or $\nu_k=\nu_{k'}$ the leading terms vanish; the deviation vanishes identically only for equal blocks.

**Item 2, branch starts $B_{k,\sigma}[g]$.**
- **(7.11)** Exact at this order; every smooth $g$, every $\chi$; first order jointly in $G_k$, $\mathcal F_k$, $g_{\zeta,k}$, $\kappa(\xi-\langle X\rangle)$: no change of the first-order velocity, nor of (4.7), (4.16), (4.19).
- **(7.13), (7.14)** Zero gradients; every fixed smooth $g$, every $\chi$ with finite moments of $X$; Taylor expansion in $\kappa$ at fixed $\lambda$, start, chain. (7.13) exact pointwise in $p$. Fastest-growing parts $\tfrac12\lambda^2\sigma\langle\omega_k'''\rangle_g$ and $-\tfrac16\lambda^3\sigma\langle\omega_k''''\rangle_g$, coefficients exact. $R_2$: bound, linear in $\lambda$. $R_3=O(t_k\nu_k\lambda^3\mathcal D_k^{-1/2})$: order estimate for $\mathcal D_k\gg1$, with its calculation.
- **(7.41), (7.44), (7.47)** Exact, every smooth $g$, every $\lambda$: the change of the acceleration at order $\kappa^2$ is $\kappa^2\sigma_X^2\langle\partial_\lambda v_2\rangle_g$. (7.47) is its averaged form: $\sigma\lambda\langle\omega_k'''\rangle_g$ plus an oscillating part with the bound $\nu_kX_g\lambda+c_3[g]$, and $a_2[g]/\lambda\to\sigma\langle\omega_k'''\rangle_g$ for $\lambda\to\infty$ at every fixed smooth $g$. The sharp-label limit and the growth of $\lambda$ do not commute, and at fixed $g$ a window average differs from the instantaneous value. (7.44) lists the cases A (window average), A′ (instantaneous value after dephasing) and B (sharp label at fixed $\lambda$) with their orders of limits, separately for the orders $\kappa^2$ and $\kappa^3$.
- **(7.15), (7.16), (7.33), (7.48)** Case A (fixed smooth $g$, average over a window, then sharp label in the coefficient; the order of limits of the drift acceleration). (7.33): bound at order $\kappa^2$, identity at order $\kappa^3$. Order $\kappa^2$: $\delta a^{(2)}=-\kappa^2\sigma_X^2\lambda v(4+3\omega_k''/\omega_k)$ at $\lambda=\bar\lambda$, exact coefficient for every $p_0$, under the sufficient window condition $\Lambda\gg c_1[g]/\lvert\langle\omega_k'''\rangle_g\rvert$: zero at rest, first order in $v$, same in both branches; near rest ($\lvert p_0\rvert\ll p_c$) the factor is $4+3t_k$. Order $\kappa^3$ at rest: the window average is $\tfrac\sigma2\kappa^3\mu_3\nu_k(4+3t_k)(\bar\lambda^2+\Lambda^2/12)$ up to the relative $\varrho_{3,\rm win}$ of (7.48) (order estimate; narrow start at rest, $\mathcal D_k\gg1$); it is (7.16) at $\bar\lambda$ for $\Lambda_3\ll\Lambda\ll\bar\lambda$: zeroth order in $v$, odd in the branch, skewed $X$ only. The secular parts are level-independent iff $\tau_k/M_k$, respectively $\nu_k(4+3t_k)$, is; not in $\mathcal C$.
- **(7.49), (7.50)** Case A′ (fixed smooth $g$, no window, $\lambda\to\infty$). Order $\kappa^2$: $a_2[g]/\lambda\to2s_2$, with the sharp-label coefficient (7.15), for every label weight. Order estimates for the oscillating part: smaller than every power of $1/\mathcal D_I$ if the label weight is free of stationary points (near rest $\lvert p_0\rvert>b\,\delta p$ for a support of the half-width $b\,\delta p$); of the amplitude $\tfrac14\sqrt{\nu_k\lambda/\pi}\,\lvert\rho_s'\rvert$, growing as $\lambda^{1/2}$, if it contains the stationary point $0$ or $\pi$. Order $\kappa^3$ at rest (order estimate; narrow start, $\mathcal D_k\gg1$, $w_0\neq0$): $a_3[g]=3s_3\lambda^2+a_3^{\rm osc}$, with an oscillating part of the amplitude $\tau_k^3w_0\lambda^{5/2}/(24\sqrt\pi M_k^{3/2})$ at the rate $2M_k$ that outgrows the secular part; $a_3[g]/\lambda^2$ has no limit, and the instantaneous acceleration after dephasing is **not** (7.16). Level-independent iff $(\tau_k,M_k)$ are; not in $\mathcal C$.
- **(7.42), (7.43)** Case B (sharp label at fixed $\lambda$ or fixed window). Exact: $\kappa^2\sigma_X^2\partial_\lambda v_2(p_0,\lambda)$, zero at rest, first order in $v$, with oscillating terms up to $\nu_k\lambda^2$; exact bound for the window average; exact third-order velocity and acceleration at rest (7.43), with an oscillating part $\propto\lambda^3$. Level-independent iff $(\tau_k,M_k)$ are; not in $\mathcal C$.
- **(7.45)** Bound: $\lVert\mathrm d\bar x/\mathrm d\lambda\rVert\le c_k=\tau_kd_0$ in the level $k$; it equals the bound (5.11) of 04-ilang-time/05-moving-clock iff the $\tau_k$ are equal.
- **(7.17), (7.18), (7.34), (7.46), (7.51)** Case A, near rest, background of (17.15) of 03-ilang-space/17-motion. The record terms are smaller than the drift acceleration for $\varrho_2,\varrho_3<1$. (7.46) for every level involved is **sufficient** for a fall that is level-independent to the relative accuracy $2\delta_{\rm acc}$ (window averages, retained orders; bounds at order $\kappa^2$, estimate of $R_3$ at order $\kappa^3$), with the ranges in $\lambda$, $\kappa$ and the gradients of Step 8, part 4; it is not necessary. **Weak records** has to mean (7.51): by definition necessary and sufficient for the relative accuracy $\delta$ between two levels with the same first-order window average. In the regime (R) it is $\kappa^2\bar\lambda\,\lvert\tfrac12\kappa\mu_3\bar\lambda-\sigma_X^2p_0\rvert\,\lvert n_k-n_{k'}\rvert\le\delta\lvert\bar a\rvert$ with $n_k=\nu_k(4+3t_k)$ (leading form), which holds for every $\kappa$ if $n_k=n_{k'}$ and is a condition on $\varrho_2+\varrho_3$ only for $\mu_3p_0\le0$. Neither condition is uniform in $\lambda$; both fail for gradients $\to0$ at fixed $\kappa$ unless the left side of (7.51) vanishes. Instantaneous accelerations (cases A′, B) need the additional conditions of Step 8, part 4.

**Item 3, on-site block $(1+gx)[M_k\sigma_3+u_k\mathbb 1_2]-\mathcal F_Ux\,\mathbb 1_2$, any $\tau_k$, $g_{\zeta,k}$; drift acceleration from rest, leading order in $\kappa$.**
- **(7.19), (7.20)** Exact in these limits. $a_{k,\pm}(0)=-g\tau_k^2\pm\tfrac{\tau_k^2}{M_k}(\mathcal F_U-gu_k)$ and the three equivalences; one value in both branches iff $gu_k=\mathcal F_U$ and $g\tau_k^2=-a$ for all $k$.
- **(7.21)** Exact. Well defined: $g$, $M_k$, $u_k-u_{k'}$, $\mathcal F_U-gu_k$. Not well defined: $u_k$, "$u_k=0$ for all $k$", "$u_k\propto M_k$".

**Item 4.** **(7.22)** the table of orders.

## Consistency checks

1. **Dimensions.** With costs and $\tau,M,G,\mathcal F,\kappa\sigma_X$ in units of $1/\lambda$ and labels dimensionless: (7.15) is $\lambda^{-2}\cdot\lambda\cdot\lambda^{-1}=\lambda^{-2}$; (7.16) and each term of (7.43) times $\kappa^3\mu_3$ are $\lambda^{-2}$, as (4.16); every term of (7.38), (7.40) and the ratios of (7.18), (7.34) are dimensionless. Passed.
2. **One level, a sharp $X$, a plain gradient only, a Gaussian start.** $\alpha_k=\delta_{kk_0}$: (7.1), (7.2) reduce to the single-level start, $D=0$, and (7.8) gives $P(a)=1/N$. Sharp $X$: $\varepsilon_\xi=0$, (7.11) is trivial and (7.14)–(7.17), (7.41)–(7.43) give no term. Plain gradient only: the transport of Step 4 is a rigid shift at the rate $\mathcal F_k$, and (7.29) gives $b_{\perp,k}=0$ and $S_\theta=-\tfrac{\sigma\theta'\mathcal F}{2\omega}\sin2\omega\lambda$, the first-order term of Step 7 with $\mathcal F=-\varepsilon$. Real Gaussian start: (7.38) with $C_g=0$, $\delta p=1/(2\delta x)$ is (7.10); and $\operatorname{Re}\epsilon_1=\tfrac12\sigma\Delta\nu\lambda\operatorname{Cov}_g(K,\hat p^2)$ from $\Theta_1$ reproduces (7.25), which was obtained from (7.24). Passed.
3. **Short-$\lambda$ expansions.** (7.13): the $\varepsilon^2$ term of $v_p$ at small $\lambda$ is $\tfrac12\lambda^2\sigma(\omega'''-3\omega'\theta'^2-3\omega\theta'\theta'')$ (double commutator with $\varepsilon\hat x$), and (7.13) expanded to $\lambda^2$ gives the same; near rest this is $-\tfrac12\lambda^2v$, and (7.42) at small $\lambda$ gives $-\lambda v$ ($-4-3t+1+2+3t=-1$), its derivative. (7.43): at $p=0$ the $\varepsilon^3$ term of the velocity has no term in $\lambda$ and none in $\lambda^3$ ($\langle C_0''''\rangle=0$ at $p=0$); in (7.43) the coefficients of $\lambda$, $-\tfrac{t}{4M}(1+\tfrac52t)+\tfrac{t(2+5t)}{8M}$, and of $\lambda^3$, $tM[\tfrac23(1+t)-\tfrac12(1+t)+\tfrac t4-\tfrac{2+5t}{12}]$, both vanish, and so does the coefficient of $\lambda^2$ in $\partial_\lambda v_3$. The new displays: (7.47) in the sharp-label limit near rest at small $\lambda$ gives $-\lambda v$ as well ($-(4+3t)+\tfrac12(2+3t)+\tfrac32(2+3t)-(1+3t)=-1$, with $(\omega\theta'^2)'\simeq-\nu p(2+3t)$ and $\omega\theta'\theta''-\omega'\theta'^2\simeq-\nu p(1+3t)$); the pointwise coefficient $-\tfrac13\lambda^3\omega\theta'^2\omega''$ behind (7.50) is at $p=0$ the term $-\tfrac13\tau_k^2t_k\lambda^3\sin2M_k\lambda$ of (7.43), and $\partial_\lambda$ of the leading term of (7.48) is the leading term of (7.50); (7.49), (7.50) are of the dimensions $\lambda^0$ and $\lambda^1$, as $2s_2\lambda$ and $3s_3\lambda^2$. Passed.

## Open issues

- The third order in the gradients of $\lvert\Gamma_{kk'}\rvert$ (case (O3) of (7.36)) is outside the scope and not derived.
- (7.32) and the remainder of (7.25) are order estimates with their calculations; no constants and no bounds are derived for them, apart from (7.29), (7.30), (7.31), (7.37).
- $R_3$ in (7.14), its leading term and $\varrho_{3,\rm win}$ in (7.48), and the oscillating parts (7.49), (7.50) are order estimates (leading stationary-phase terms, with their calculation); no bounds are derived for them, and none for $\partial_\lambda R_3$. For $p_0\neq0$ in case B only the term of $v_3$ with the highest power of $\lambda$ is given.
- The instantaneous acceleration of order $\kappa^3$ of a fixed $g$ away from rest (case A′, $p_0\neq0$) is not determined; (7.50) is for starts at rest.
- The second form of (7.51) is a leading-order statement in the regime (R). Outside (R) only the first form is available, and no necessary condition in single-level quantities is derived.
- Terms that mix $\kappa$ with the gradients, the order $\kappa^4$ and the second order in the gradients of the acceleration are outside the scope; (7.46) is a statement at the retained orders.
- (7.3): the constant $r$ of the bound on $E$ depends on $g$; nothing is stated on $D$ at fixed $\lambda$ in the sharp-label limit.
- The effect of an operator under the factor on the relation (4.24) outside the class $\mathcal C$ needs the clock rate of that body, which is not among the quoted inputs.

## Methods used

- Block decomposition by a conserved label; witness rule for the clock label
- First-order perturbation theory in the gradients; linearity of mixtures
- Continuity equation for the plane-wave label weight, with the boundary term at the cut of the arc
- Taylor's formula with the mean and the variance of the label weight for sharp-label limits
- Reflection symmetry (parity) argument for the order of a deviation
- Interaction picture, Magnus and Baker–Campbell–Hausdorff expansion of a product of unitaries; variance formula for the modulus of an overlap; Cauchy–Schwarz bounds
- Local (quadratic) form of the branch dynamics; Weyl displacement operators; uncertainty inequality with covariance and boundary term
- Exact reduction to a driven two-level problem; Bloch equations; adiabatic expansion with integration by parts
- Window averages; stationary phase and non-stationary phase estimates; dephasing parameter
- Riemann–Lebesgue lemma for the instantaneous value after dephasing
- Operator norm of the velocity operator; row-sum bound of a circulant matrix
- Invariance under replacements of contract terms (well-definedness)

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