Version 3 · earlier version; the current one is v5 · External review, round 2: major errors

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 07-limits
- **Version:** v3
- **Mode:** external regeneration
- **Date:** 2026-10-11

## Changes from previous version

External regeneration after `verification.md` (round 1 on v2, issues I1–I6, all accepted). All tags (7.1)–(7.26) are kept; new results carry (7.27)–(7.34). Tagged statements that changed and keep their tag: **(7.5)** (the equivalence is now stated for the sharp-label velocities $\sigma\omega_k'(p_0)$, not for $v_k$ of a start of finite width); **(7.7)** (the sharp-label limit is now stated for sequences with $\lvert g(p_0+\pi)\rvert^2\to0$); **(7.24)** (the condition $\Gamma^{(0)}_{kk'}\neq0$ is added and the inter-branch part is now explicit, (7.29), (7.30)); **(7.15), (7.16)** (formulas unchanged; they are now defined as the $\lambda$-derivatives of the secular velocity terms of (7.14), not as window-averaged accelerations of a sharp label); **(7.18)** (formula unchanged; restricted to the secular terms); **(7.22)** (rows "Superposition of levels" and "Records"). Changed passages, in the order of the file:

1. **Header.** Version and mode.
2. **Step 1, part 3, first sentence (class check).** v2: "position readings of $\Phi_\sigma[\alpha,g]$ show the drift of one level". Now restricted to the drift acceleration in the limits of (7.2).
3. **Step 2, the paragraph after (7.3) and the last paragraph (class check).** The limit $\delta p\to0$ is a limit of the coefficient of $\lambda^4$; the properties of the sequence that it uses are named; "$O(\delta p^4)$ at fixed $g$" is replaced by the bound. (7.3), (7.4) unchanged.
4. **Step 3 (I1).** Rewritten after its first sentence: (7.5) restated, the bound (7.27) added, the answer to item 1(d) split into starts of finite width (yes, with an example) and the sharp-label limit.
5. **Step 4, parts 2 and 3 (I2).** (7.7) restated; the bound (7.28) with the weight at the cut added; the equivalences restated for the sequences for which they are proven. Part 1 and (7.6) unchanged.
6. **Step 5.** Part 2, last two sentences (I4: "has no first-order term"). Part 3, condition (E2) (I3), with the new estimate (7.32); (7.10) unchanged. Part 4: class qualifier. Part 5: wording "no first-order term". Part 6 (I4, I6): the inter-branch amplitude (7.29), the inter-branch part (7.30), the bound (7.31), the nonzero-reference condition and the case $\Gamma^{(0)}_{kk'}=0$. Part 7: (F2), (F3) and the bullets on the classes revised; (7.25), (7.26) unchanged as formulas. A new part 8 states what of item 1(f) is resolved and what is open.
7. **Step 7, part 4, from "The third-order secular term" on (I5).** The claim on a window average for a sharp label (relative size $O(1/(M_k\Lambda))$) is withdrawn; the derivation of the estimate of $R_3$ is given. (7.12)–(7.14) unchanged.
8. **Step 8 (I5).** Part 1: the definition of the secular terms and the window statement (7.33) are new; the text around (7.15), (7.16) is adapted. Part 3: the text after (7.18) is rewritten, with (7.34). Part 2 and (7.17) unchanged.
9. **Step 10.** Rows "Superposition of levels" and "Records".
10. **Result, Consistency checks (check 2 extended), Open issues, Methods used.** Rewritten or extended: every result states its class of starts or of sequences of starts, the order of the limits and its status (exact, bound, order estimate).

Unchanged, word for word: "Setup and assumptions"; Step 1, parts 1 and 2; Step 4, part 1; Step 6; Step 7, parts 1–3; Step 8, part 2; Step 9; the other rows of the table; checks 1 and 3.

## Response to verification

- **I1: Accepted and fixed in Step 3.** The strict ordering is a statement on the sharp-label velocities. For starts of finite width the general result $v_k=\sigma\langle\omega_k'\rangle_g$ is kept, the verifier's example is included, and (7.27) gives the spread below which the ordering of the sharp label carries over.
- **I2: Accepted and fixed in Step 4, parts 2 and 3.** The weight at the cut is kept in the limit, (7.28). The rate $\mathcal F_k-\sigma G_k$ and the equivalences are stated for sequences with $\delta p\to0$ and $\lvert g(\pi)\rvert^2\to0$; the general limit and the exceptional case are stated.
- **I3: Accepted and fixed in Step 5, parts 3 and 6.** The first-order inter-branch amplitude is derived, (7.29); it grows linearly in $\lambda$, with a coefficient suppressed by $\langle\sin^4p\rangle_g^{1/2}$. (E2) is replaced by the estimate (7.32) and the range in which it is smaller than the terms of (7.10).
- **I4: Accepted and fixed in Step 5, parts 2, 5–8, and in the Result.** The first-order functional is now complete, (7.24) with (7.29), (7.30). "Is of second order" is replaced by "has no first-order term". The Result states that item 1(f) is resolved in part, for which classes, and what is open.
- **I5: Accepted and fixed in Step 7, part 4, and Step 8, parts 1 and 3.** The secular terms are defined; the window-averaged acceleration is given with an exact bound and its order of limits, (7.33); the statement for a sharp label at a fixed window is withdrawn; (7.18) is restricted to the secular terms and the rest enters the accuracy condition, (7.34).
- **I6: Accepted and fixed in Step 5, parts 6 and 7.** The normalized expansion carries the condition $\Gamma^{(0)}_{kk'}\neq0$, with the sufficient condition (7.31); at a zero the unnormalized modulus is used.

## 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 $\chi$.
- **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 $\sum_\xi w_\xi\varepsilon_\xi=0$, $\sum_\xi w_\xi\varepsilon_\xi^2=\kappa^2\sigma_X^2$, $\sum_\xi w_\xi\varepsilon_\xi^3=\kappa^3\mu_3$ with $\mu_3:=\langle(X-\langle X\rangle)^3\rangle$. 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. $\operatorname{Var}_\alpha(y):=\sum_k\lvert\alpha_k\rvert^2\bigl(y_k-\sum_j\lvert\alpha_j\rvert^2y_j\bigr)^2$.
- **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 $\omega''(0)=\tau^2/M$ and, by two more derivatives, $\omega'''=-\omega'(4+3\omega''/\omega)$, $\omega''''(0)=-\tfrac{\tau^2}{M}\bigl(4+\tfrac{3\tau^2}{M^2}\bigr)$; $\theta'=-\tau M\cos p/\omega^2$, $\lvert\theta'\rvert\le\tau/M$.
- **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 leading term dominates is stated with it. Statements marked *order estimate* are not bounds.

## Derivation

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

1. *Exact mixture.* $C=\sum_k\bigl(C^{(k)}\otimes\mathbb 1_c+\kappa\hat x\otimes X\bigr)\otimes\lvert f_k\rangle\langle f_k\rvert$ 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 $\lvert f_k\rangle$. 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 $g$, and no bound on $r$ uniform in $\delta p$ is derived. $D$ is of second order in the gradients (a square of first-order displacements). 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, $\tau_k=\tau_{k'}$ and $M_k=M_{k'}$, the velocities are equal for every $g$, and this is the only such case at the sharp label: equality at all $p_0$ holds iff $\tau_k=\tau_{k'}$ and $M_k=M_{k'}$ (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 $\ge0$ (a bump at the cut of fixed height and vanishing width). 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 $c_{\rm cut}=0$** (for example for starts that vanish in a neighbourhood of the cut). 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), $a_{k,\sigma}(0)=\sigma\,\omega_k''(0)\,\pi_{k,\sigma}(0)$, a relation between the two coefficients for every start; the drift acceleration (4.11) has no cut term.
3. *Equivalences.* They are proven 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 $a$. For a start of finite width the rate is (7.7), and its level-independence is a condition on the averages $\langle\pi_{k,\sigma}\rangle_g$ that is not reduced further here.

### 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). 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, and the order of the first deviation.* From (5.6) of 04-ilang-time/05-moving-clock, $\langle\beta_{k',\sigma}\vert\beta_{k,\sigma}\rangle=\cos\tfrac{\theta_k-\theta_{k'}}2$, 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 $\mathsf PB_{k,\sigma}[g]=\sigma B_{k,\sigma}[g]$ for even $g$, with the same sign in both levels; $\mathsf P$ is unitary. Hence for a start with even $g$ (a start at rest), $\Gamma_{kk'}$ and $\lvert\Gamma_{kk'}\rvert$ **have no first-order term** in the gradients: the first deviation is of second order or higher. The symmetry alone does not show that the second-order term is different from zero. For a start at rest that is not even, the first order is determined in parts 5–7.
3. *Leading term (order estimate).* Conditions: (E1) $\lvert g\rvert^2\propto e^{-p^2/2\delta p^2}$, $g$ real, $\delta p\ll\min(1,M_k/\tau_k)$, so the spread of the start in the position label is $\delta x=1/(2\delta p)$; (E2) inside the branch $\sigma$, at first order, the level component is transported by (7.7): $c_k(p,\lambda)\simeq g(p-\pi_k\lambda)\,e^{-i\int_0^\lambda E_k(p-\pi_ks)\mathrm ds}\beta_{k,\sigma}$, $E_k=V_k+\sigma\omega_k$, $\pi_k:=\pi_{k,\sigma}(0)$, and the inter-branch part is neglected, see below; (E3) $\lvert\pi_k\rvert\lambda\ll M_k/\tau_k$ and $\lvert\omega_k''-\omega_{k'}''\rvert\lambda\,\delta p^2\ll1$, with $\omega_k'':=\omega_k''(0)=\tau_k^2/M_k$ (then $\Gamma^{(0)}_{kk'}\neq0$, by (7.31)). Then the overlap is a Gaussian integral: the product $g(p-\pi_{k'}\lambda)g(p-\pi_k\lambda)$ gives $e^{-(\pi_k-\pi_{k'})^2\lambda^2/8\delta p^2}$, and the phase linear in $p$ about the common centre gives a relative position shift $\Delta X=\tfrac12\lambda^2\sigma(\omega_{k'}''\pi_k-\omega_k''\pi_{k'})$ and the factor $e^{-\delta p^2\Delta X^2/2}$:
$$
1-\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\simeq\frac{\delta x^2}2\,(\pi_k-\pi_{k'})^2\lambda^2+\frac{\lambda^4}{32\,\delta x^2}\Bigl(\frac{\tau_{k'}^2}{M_{k'}}\pi_k-\frac{\tau_k^2}{M_k}\pi_{k'}\Bigr)^2,\qquad\pi_k=\mathcal F_k-\sigma G_k.
\tag{7.10}
$$
   *The neglected inter-branch part (E2).* It is **not** bounded in $\lambda$. By (7.29) of part 6 the first-order inter-branch amplitude has a norm $\le A_{k,\perp}(\lambda)$ that grows linearly in $\lambda$, with a coefficient suppressed by the width of the start, $\langle\sin^4p\rangle_g^{1/2}\simeq\sqrt3\,\delta p^2$ for (E1). At second order it enters $1-\lvert\Gamma_{kk'}\rvert/\lvert\Gamma^{(0)}_{kk'}\rvert$ through (i) the loss of norm of the two in-branch components and the overlap of the two inter-branch amplitudes, of size at most $\tfrac12(A_{k,\perp}+A_{k',\perp})^2$; (ii) products of an in-branch and an inter-branch first-order amplitude, with the overlap $\mp\sigma\sin\tfrac{\theta_k-\theta_{k'}}2=O(\tfrac\tau M\lvert p\rvert)$; (iii) the second-order inter-branch amplitude with the same overlap, and the second-order phase inside the branch. (ii) and (iii) are not derived. With $\tau/M$, $\nu:=\tau^2/M$, $\mathsf b$ the larger of the two levels' values:
$$
N_\perp\sim\frac{\tau^2\,\mathsf b^2}{M^4}\,\bigl(1+M\lambda\bigr)\bigl(1+\nu\,\lambda\,\delta p^2\bigr)^2\qquad(\text{order estimate of the neglected part of (7.10)}).
\tag{7.32}
$$
   (7.10) is claimed only where $N_\perp$ is small against its right side. If the gradients and the level data of the two levels differ by relative amounts of order one ($\lvert\pi_k-\pi_{k'}\rvert\sim\mathsf b$ and $\lvert\varpi_{kk'}\rvert\sim\nu\,\mathsf b$, with $\varpi_{kk'}:=\tfrac{\tau_{k'}^2}{M_{k'}}\pi_k-\tfrac{\tau_k^2}{M_k}\pi_{k'}$ the bracket of the second term of (7.10)) and $\nu\lambda\,\delta p^2\lesssim1$, this is $\lambda\gg\tau\,\delta p/M^2$; for $\nu\lambda\,\delta p^2\gg1$ it is $\lambda\gg\tau^2\delta p^2/M^3$. If $\pi_k-\pi_{k'}$ and $\varpi_{kk'}$ are small against $\mathsf b$ and $\nu\mathsf b$, the range shrinks accordingly, and for $\pi_k=\pi_{k'}$, $\varpi_{kk'}=0$ (7.10) gives no leading term.
4. *Level-independent fall.* For the starts (E1) in the range (E3), (7.32). With $a_{k,\sigma}(0)=\sigma\omega_k''\pi_k=a$ for all $k$: $\pi_k-\pi_{k'}=\sigma a\bigl(\tfrac{M_k}{\tau_k^2}-\tfrac{M_{k'}}{\tau_{k'}^2}\bigr)$. The deviation (7.10) vanishes iff $a=0$ or $\tau_k^2/M_k=\tau_{k'}^2/M_{k'}$. So **no**: in the class $\mathcal C$ with $a\neq0$ it does not vanish. Variants (a), (c): $1-\lvert\Gamma\rvert/\lvert\Gamma^{(0)}\rvert\simeq\tfrac12\delta x^2g^2(d_k-d_{k'})^2\lambda^2+\tfrac{\lambda^4\tau^4g^2}{32\delta x^2}\bigl(\tfrac{M_k}{M_{k'}}-\tfrac{M_{k'}}{M_k}\bigr)^2$. In $\mathcal C$, $\pi_k-\pi_{k'}=-\sigma g_\eta(d_k-d_{k'})$ is, by (3.23), $-\sigma\tfrac{2\pi(k-k')}{N}\bigl(r(x+1)-r(x)\bigr)$: the $\lambda^2$ term is the difference of the clock rates across the spread $\delta x$.
5. *Reflection relation for every start.* Let $g^-(p):=g(-p)$. The proof of (7.9) uses that $g$ is even only in its last step; for every smooth $g$ it gives $\mathsf PB_{k,\sigma}[g]=\sigma B_{k,\sigma}[g^-]$, with the same sign in both levels. Hence, exactly, for the coherence of the effective body contract, every member of $\mathcal C_{\rm lev}$ with linear level profiles, both branches, every $p_0$ and every $\lambda$:
$$
\Gamma_{kk'}\bigl[g;\,\text{gradients}\bigr]=\Gamma_{kk'}\bigl[g^-;\,-\text{gradients}\bigr].
\tag{7.23}
$$
The first-order term of $\Gamma_{kk'}$ is therefore odd under $g\to g^-$. It vanishes for every start of **definite parity**, $g^-=\pm g$ (an odd $g$ is a start at rest as well), and for these starts $\lvert\Gamma_{kk'}\rvert$ is exactly even in the gradients; for other starts (7.23) does not exclude a first-order term. Independently of the start: if $(\tau_k,M_k)=(\tau_{k'},M_{k'})$, then $\lvert\Gamma^{(0)}_{kk'}\rvert=1$ by (7.9), the largest value that (7.8) allows for any gradients, so $\lvert\Gamma_{kk'}\rvert$ has no first-order term for any $g$. In both cases the statement is "no first-order term"; a second-order term different from zero is not shown by these arguments.
6. *First-order term for a general $g$.* Class: every smooth complex $g$. Limits: first order in the gradients of the levels $k$, $k'$ at fixed $\lambda$ and start, infinite chain, leading order in $\kappa$. Notation: $(\cdot)_1$ is the first-order term in the gradients; $\rho:=\lvert g\rvert^2$; $j:=\operatorname{Im}(g^{*}g')$, so that $-j/\rho$ is the position label carried by the label component $p$ of the start ($\langle\hat x\rangle=-\int\frac{\mathrm dp}{2\pi}j$); $F:=\cos\tfrac{\theta_k-\theta_{k'}}2$; $\Delta\omega:=\omega_k-\omega_{k'}$; $\pi_k:=\pi_{k,\sigma}(p)$ of (7.7); $\lVert f\rVert^2:=\int\frac{\mathrm dp}{2\pi}\lvert f\rvert^2$. In the interaction picture, $c_k=e^{-i(V_k+C^{(k)}_0)\lambda}\tilde c_k$, first-order perturbation theory in $C_1$ of (4.2), with $C_1\psi=\tfrac i2(B_k'\psi+2B_k\psi')$, $\beta_{k,\sigma}'=\tfrac\sigma2\theta_k'\beta_{k,-\sigma}$, $\langle\beta_{k,\sigma}\vert B_k\vert\beta_{k,\sigma}\rangle=-\pi_k$ and real $\beta_{k,\sigma}$, gives $\tilde c_k=(g+a_k)\beta_{k,\sigma}+a_{k,\perp}\beta_{k,-\sigma}$ with, exactly, $a_k=-\lambda\bigl(\pi_kg'+\tfrac12\pi_k'g\bigr)+\tfrac{i\sigma}2\lambda^2\omega_k'\pi_k\,g$ (the first order of the transport (E2); it conserves the norm and reproduces (7.7)) and the inter-branch amplitude
$$
\begin{aligned}
a_{k,\perp}&=\tfrac12\bigl(b_{0,k}\,g+2\,b_{\perp,k}\,g'\bigr)I_{0,k}-i\sigma\,\omega_k'\,b_{\perp,k}\,g\,I_{1,k},\qquad I_{n,k}:=\int_0^\lambda s^n e^{-2i\sigma\omega_ks}\,\mathrm ds,\quad I_{1,k}=\tfrac{i\sigma}{2\omega_k}\bigl(\lambda e^{-2i\sigma\omega_k\lambda}-I_{0,k}\bigr),\\
b_{\perp,k}&:=\langle\beta_{k,-\sigma}\vert B_k\vert\beta_{k,\sigma}\rangle=\frac{\tau_k\sin p}{\omega_k}\bigl(G_k-M_kg_{\zeta,k}\bigr),\qquad b_{0,k}:=-g_{\zeta,k}\tau_k\cos p\,\frac{M_k}{\omega_k}-\sigma\,\theta_k'\,\pi_{k,-\sigma},\\
\lVert a_{k,\perp}\rVert&\le A_{k,\perp}(\lambda):=\frac{\tau_k\mathsf b_k}{2M_k^2}\Bigl(2+\frac{\tau_k}{M_k}+2\lVert\sin p\;g'\rVert\Bigr)+\frac{\tau_k^3\mathsf b_k}{2M_k^3}\Bigl(\lambda+\frac1{M_k}\Bigr)\langle\sin^4p\rangle_g^{1/2}\qquad(\text{exact}).
\end{aligned}
\tag{7.29}
$$
The bound uses $\lvert I_{0,k}\rvert\le1/\omega_k$, $\lvert I_{1,k}\rvert\le\lambda/(2\omega_k)+1/(2\omega_k^2)$, $\omega_k\ge M_k$, $\lvert b_{\perp,k}\rvert\le\tfrac{\tau_k}{M_k}\mathsf b_k\lvert\sin p\rvert$, $\lvert\omega_k'b_{\perp,k}\rvert\le\tfrac{\tau_k^3}{M_k^2}\mathsf b_k\sin^2p$ and $\lvert b_{0,k}\rvert\le\tfrac{\tau_k}{M_k}\mathsf b_k\bigl(2+\tfrac{\tau_k}{M_k}\bigr)$. The plain gradient $\mathcal F_k$ does not enter $b_{\perp,k}$; the part of $a_{k,\perp}$ that grows with $\lambda$ is $\tfrac{\sigma}{2\omega_k}\lambda\,\omega_k'b_{\perp,k}\,g\,e^{-2i\sigma\omega_k\lambda}$, which vanishes at $p=0$ as $p^2$. Inserting $c_k=e^{-iV_k\lambda}\bigl[e^{-i\sigma\omega_k\lambda}(g+a_k)\beta_{k,\sigma}+e^{i\sigma\omega_k\lambda}a_{k,\perp}\beta_{k,-\sigma}\bigr]$ 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:
$$
\Gamma^{(1)}_{kk'}=e^{-i(V_k-V_{k'})\lambda}\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,e^{-i\sigma\Delta\omega\lambda}\Bigl\{\rho\Bigl[\tfrac\lambda2(\pi_k+\pi_{k'})F'+\tfrac{i\sigma}2\lambda^2\bigl(\omega_{k'}'\pi_k-\omega_k'\pi_{k'}\bigr)F\Bigr]-i\lambda(\pi_k-\pi_{k'})F\,j\Bigr\}+\Gamma^{(1),\perp}_{kk'},\qquad\Bigl(\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\Bigr)_{1}=\operatorname{Re}\frac{\Gamma^{(1)}_{kk'}}{\Gamma^{(0)}_{kk'}}\ \ \text{where }\Gamma^{(0)}_{kk'}(\lambda)\neq0.
\tag{7.24}
$$
The displayed integral is the part inside the branch $\sigma$. 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 of $\Gamma_{kk'}$, exact for every smooth $g$, every $p_0$ and every $\lambda\ge0$; the bound of (7.30) is exact ($\lvert\theta_k\rvert\le\tfrac{\tau_k}{M_k}\lvert\sin p\rvert$ and the Cauchy–Schwarz inequality). The first-order term vanishes for $k=k'$ and, by (7.23), for $g^-=\pm g$. *The reference coherence.* The second equation of (7.24) needs $\Gamma^{(0)}_{kk'}(\lambda)\neq0$; phase averaging in (7.9) can make it vanish. At a zero, $\lvert\Gamma_{kk'}\rvert=\lvert\Gamma^{(1)}_{kk'}\rvert$ up to second order: it is of degree one in the gradients but not linear (for a common scale $\epsilon$ of the gradients it is $\lvert\epsilon\rvert\,\lvert\Gamma^{(1)}_{kk'}\rvert$), and the normalized expressions (7.10), (7.25), (7.26) are not defined. A sufficient condition 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):
$$
1-\bigl\lvert\Gamma^{(0)}_{kk'}(\lambda)\bigr\rvert\le\Bigl[\tfrac12\lambda\sup_p\lvert\omega_k''-\omega_{k'}''\rvert+\tfrac18\Bigl(\frac{\tau_k}{M_k}+\frac{\tau_{k'}}{M_{k'}}\Bigr)^2\Bigr]\delta p^2\qquad(\text{exact, every start at rest}),
\tag{7.31}
$$
so $\Gamma^{(0)}_{kk'}(\lambda)\neq0$ wherever the right side is smaller than $1$.
7. *Leading part for a start at rest (order estimate), the two classes, and a level-independent fall.* Conditions: (F1) a start at rest (label weight with the mean $0$ and the variance $\delta p^2$ on the arc $(-\pi,\pi]$), $g$ smooth and complex, in general neither even nor real, with $\delta p\ll\min(1,M_k/\tau_k,M_{k'}/\tau_{k'})$ and $\langle\lvert p\rvert^n\rangle_g=O(\delta p^n)$ for $n\le6$ (a property of the sequence of starts beyond its mean and variance); (F2) the right side of (7.31) is small against $1$, in particular $\lvert\Delta\nu_{kk'}\rvert\,\lambda\,\delta p^2\ll1$, with $\nu_k:=\omega_k''(0)=\tau_k^2/M_k$ and $\Delta\nu_{kk'}:=\nu_k-\nu_{k'}$, so that $\Gamma^{(0)}_{kk'}\neq0$ and $\lvert\Gamma^{(0)}_{kk'}\rvert$ is close to $1$. Order of limits: the first order is a Taylor coefficient at fixed $\lambda$ and start (no condition on $\pi_k\lambda$), long chain, leading order in $\kappa$; then the expansion in $\delta p$ at fixed $\lambda$ within (F2). With $\Delta\omega=\Delta\omega(0)+\tfrac12\Delta\nu\,p^2+\dots$ in (7.24), 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. Applied to the two terms of (7.24) with the factor $i$ ($h=\rho\,(\omega_{k'}'\pi_k-\omega_k'\pi_{k'})\simeq\rho\,\varpi_{kk'}p$ and $h=(\pi_k-\pi_{k'})j$), with $\pi_k:=\pi_{k,\sigma}(0)=\mathcal F_k-\sigma G_k$ from here on:
$$
\Bigl(\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\Bigr)_{1}\simeq\Delta\nu_{kk'}\Bigl[\tfrac14\lambda^3\,\varpi_{kk'}\,\langle p^3\rangle_g-\tfrac\sigma2\lambda^2\,(\pi_k-\pi_{k'})\,J_g\Bigr],\qquad\varpi_{kk'}:=\nu_{k'}\pi_k-\nu_k\pi_{k'},\quad J_g:=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,(p^2-\delta p^2)\,\operatorname{Im}\bigl(g^{*}g'\bigr).
\tag{7.25}
$$
   - *The start enters through two numbers.* $\langle p^3\rangle_g$ is the third moment of the label weight (its skewness times $\delta p^3$). $J_g$ is minus the covariance, in the start, of $p^2$ with the position label: $J_g=0$ for real $g$, and $J_g=3c\operatorname{Var}_g(p^2)$ for $g=\lvert g\rvert e^{icp^3}$; a shift of the start in the position does not change it. $\varpi_{kk'}$ is the bracket of (7.10). The sign of (7.25) is not fixed: $\lvert\Gamma_{kk'}\rvert$ can rise or fall at first order, which is possible only because $\lvert\Gamma^{(0)}_{kk'}\rvert<1$ (factor $\Delta\nu_{kk'}$).
   - *(F3) Remainder ($\tau/M$, $\nu$, $\mathsf b$ are the larger of the two levels' values).* (a) Order estimate: corrections to the two terms of (7.25) that are smaller than their largest possible sizes, $\lambda^3\lvert\Delta\nu\rvert\,\nu\,\mathsf b\,\delta p^3$ and $\lambda^2\lvert\Delta\nu\rvert\,\mathsf b\,\delta p^2X_g$ (from $\lvert J_g\rvert\le\sqrt{\operatorname{Var}_g(p^2)}\,X_g$), by the factors $O\bigl(\delta p^2(1+\tau^2/M^2)\bigr)+O\bigl((\Delta\nu\,\lambda\,\delta p^2)^2\bigr)$; and the term of (7.24) with $F'$, $O\bigl(\lambda\,\mathsf b\,\tfrac{\tau^2}{M^2}(1+\tfrac{\tau^2}{M^2})\,\delta p^3\bigr)$. (b) Exact bound, from (7.30) with $\langle\sin^2p\rangle_g\le\delta p^2$: $\lvert\Gamma^{(1),\perp}_{kk'}\rvert\le\tfrac{\tau^2\mathsf b}{M^3}\,\delta p\,\Sigma_g+\tfrac{\tau^4\mathsf b}{M^4}\bigl(\lambda+\tfrac1M\bigr)\delta p\,\langle\sin^4p\rangle_g^{1/2}$, with $\Sigma_g:=2+\tfrac\tau M+2\lVert\sin p\;g'\rVert$; its growing part is $O(\lambda\,\tfrac{\tau^4\mathsf b}{M^4}\delta p^3)$. So (7.25) is the part of the first-order term that grows fastest with $\lambda$. For level data that differ by relative amounts of order one, and $\langle p^3\rangle_g\sim\delta p^3$ or $J_g\sim\delta p^2X_g$, it exceeds the remainder for $\lambda_{\rm F}\ll\lambda\ll1/(\lvert\Delta\nu\rvert\,\delta p^2)$, with $\lambda_{\rm F}:=\max\bigl\{\tfrac1M+\tfrac1\tau,\ \tfrac1M\Sigma_g^{1/2},\ (\Sigma_g/(M\tau^2\delta p^2))^{1/3}\bigr\}$; the last entry is needed for the $\lambda^3$ term only. No bound on the remainder (a) is derived.
   - *Class with no first-order term.* Starts of definite parity, $g(-p)=\pm g(p)$, for all level data and all $\lambda$; and every start for a pair of levels with equal $(\tau,M)$. Both statements are exact (part 5). The second-order term is determined only for the starts (E1): its leading part is (7.10), an order estimate.
   - *Class with a first-order term.* For every smooth $g$ and every $\lambda$ with $\Gamma^{(0)}_{kk'}\neq0$ the first deviation of $\lvert\Gamma_{kk'}\rvert$ is of first order iff $\operatorname{Re}\bigl(\Gamma^{(1)}_{kk'}/\Gamma^{(0)}_{kk'}\bigr)\neq0$, with $\Gamma^{(1)}_{kk'}$ of (7.24), (7.29), (7.30) (exact). Within (F1)–(F3) its leading part is (7.25) (order estimate): starts at rest that are not of definite parity, for a pair of levels with $\Delta\nu_{kk'}\neq0$ and with $\varpi_{kk'}\langle p^3\rangle_g\neq0$ or $(\pi_k-\pi_{k'})J_g\neq0$. A real $g$ with a skewed label weight suffices. Order estimate: (7.25) exceeds second-order terms of the size of (7.10) where $\lvert\varpi_{kk'}\rvert\lambda\,\delta p^2\ll\lvert\Delta\nu\,\langle p^3\rangle_g\rvert$, respectively $\lvert\pi_k-\pi_{k'}\rvert\,\delta x^2\ll\lvert\Delta\nu\,J_g\rvert$. For a start that is not of definite parity with $\langle p^3\rangle_g=J_g=0$, and for a pair with $\Delta\nu_{kk'}=0$ and unequal data, the first-order term is at most of the size of the remainder (F3); whether it vanishes is not determined.
   - *Level-independent fall.* With $a_{k,\sigma}(0)=\sigma\nu_k\pi_k=a$ for all $k$: $\pi_k=\sigma a/\nu_k$, and (7.25) becomes
$$
\Bigl(\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\Bigr)_{1}\simeq\frac{a\,\Delta\nu_{kk'}^2}{\nu_k\,\nu_{k'}}\Bigl[\tfrac12\lambda^2J_g-\tfrac\sigma4\lambda^3\,(\nu_k+\nu_{k'})\,\langle p^3\rangle_g\Bigr].
\tag{7.26}
$$
   For a start with $J_g\neq0$ or $\langle p^3\rangle_g\neq0$ it vanishes for all $\lambda$ iff $a=0$ or $\nu_k=\nu_{k'}$, the condition of part 4. So **no**, within (F1)–(F3): a level-independent fall with $a\neq0$ does not remove the leading first-order term, and in the class $\mathcal C$ ($\tau_k=\tau$, $M_k$ distinct) it never does. Variants (a), (c): $a=-\tau^2g$, $\tfrac{\Delta\nu^2}{\nu_k\nu_{k'}}=\tfrac{(d_k-d_{k'})^2}{M_kM_{k'}}$, $\nu_k+\nu_{k'}=\tau^2\tfrac{M_k+M_{k'}}{M_kM_{k'}}$. (7.26) has the class, the range and the status (order estimate) of (7.25).
8. *What of item 1(f) is resolved.* Resolved for every smooth $g$: the formula (7.8); the coherence at zero gradients (7.9); the complete first-order term (7.24), (7.29), (7.30), and with it the criterion for a first deviation of first order where $\Gamma^{(0)}_{kk'}\neq0$; the absence of a first-order term for starts of definite parity and for pairs with equal data. Resolved as order estimates, for narrow starts at rest only: the leading first-order term (7.25), (7.26) in (F1)–(F3), and the leading second-order term (7.10) for the real Gaussian starts (E1) in (E3), (7.32). **Open:** the second-order term for every start other than (E1), in particular whether it is different from zero for a given start of definite parity, and for starts that are not narrow; a closed leading form of the first-order term for starts at rest that are not narrow or outside (F2); the cases named in part 7 in which (7.25) vanishes; the dependence on the spread in the position outside (E1). The answer "the deviation does not vanish when the fall is the same for all levels" is therefore established only for the classes (E1) and (F1), as an order estimate.

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

In (4.1), $p^{[\xi]}$ belongs to the level profiles of the effective body contract 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 $\varphi(\lambda)=2\omega\lambda-\varepsilon\omega'\lambda^2+O(\varepsilon^2)$ 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 $p$: $v_p=\sigma\omega'+\varepsilon\sigma\bigl[-\lambda\omega''+\tfrac12\theta'^2\sin2\omega\lambda\bigr]+\varepsilon^2v_2+O(\varepsilon^3)$,
$$
v_2(p,\lambda)=\sigma\Bigl[\tfrac12\lambda^2\omega'''-\tfrac{\theta'\theta''}{4\omega}(1-\cos2\omega\lambda)-\tfrac{\lambda}{2\omega}\theta'(\omega\theta')'\sin2\omega\lambda-\tfrac12\lambda^2\theta'^2\omega'\cos2\omega\lambda\Bigr].
\tag{7.13}
$$
The first-order term agrees with (4.7) for $\mathcal F=-\varepsilon$ ($a=\sigma\mathcal F\omega''$, bounded rest) and averages to zero over $\xi$ (Step 6).
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 $\lvert\theta'\rvert\le\tau/M$:
$$
\begin{aligned}
v(\lambda)&=\sigma\langle\omega_k'\rangle_g+\kappa^2\sigma_X^2\Bigl[\tfrac12\lambda^2\sigma\langle\omega_k'''\rangle_g+R_2(\lambda)\Bigr]+\kappa^3\mu_3\Bigl[-\tfrac16\lambda^3\sigma\langle\omega_k''''\rangle_g+R_3(\lambda)\Bigr]+O(\kappa^4),\\
\lvert R_2(\lambda)\rvert&\le\lambda\Bigl[\tfrac{\tau_k^2}{2M_k^2}X_g+\tfrac14\sup\lvert(\theta'^2)'\rvert+\tfrac12\sup\bigl\lvert\tfrac{\theta'(\omega\theta')'}{\omega}\bigr\rvert\Bigr]+\tfrac12\sup\bigl\lvert\tfrac{\theta'\theta''}{\omega}\bigr\rvert\ \ (\text{exact}),\qquad R_3(\lambda)=O(\lambda^{5/2})\ \ (\text{order estimate}).
\end{aligned}
\tag{7.14}
$$
The third-order secular term is the $\varepsilon^3$ coefficient of the adiabatic part $\sigma\omega'(p-\varepsilon\lambda)$ of (7.12). *Estimate of $R_3$ (order estimate at fixed smooth $g$, obtained by counting powers; no constant is derived).* Iterating part 3 once more, with $\varphi(\lambda)-\varphi(s)=2\omega(\lambda-s)-\varepsilon\omega'(\lambda^2-s^2)+\tfrac13\varepsilon^2\omega''(\lambda^3-s^3)+O(\varepsilon^3)$ and repeated integration by parts in $s$, the $\varepsilon^3$ coefficient of $v_p$ is the adiabatic term plus (i) terms without an oscillating factor, all $O(\lambda)$; (ii) terms oscillating at $2\omega_k(p)$, with the amplitudes $\tfrac14\lambda^4\theta'^2\omega'^2$ and $\lambda^3\omega'\times(\text{smooth})$, with the term $\tfrac\sigma6\lambda^3\theta'^2\omega''\cos2\omega\lambda$ that carries no factor $\omega'$, and with lower powers of $\lambda$. In the average $\langle\cdot\rangle_g$ each factor $\omega'$ of an oscillating term is removed by one integration by parts on the circle, which gains $1/\lambda$ and puts a derivative on $\lvert g\rvert^2$; the oscillating integrals that are left without a factor $\omega'$ are $O(\lambda^{-1/2})$ by stationary phase (the stationary points of $\omega_k$, $p\in\{0,\pm\tfrac\pi2,\pi\}$, are nondegenerate). Hence $R_3=O(\lambda^{5/2})$, with a constant that depends on $\lvert g\rvert^2$ and its first two derivatives and is not uniform for $\delta p\to0$. Range: fixed smooth $g$, chain and $\lambda$, $\kappa\to0$; the leading parts dominate for $\lambda\gg X_g\tau_k^2/(M_k^2\lvert\langle\omega_k'''\rangle_g\rvert)$ within $\kappa\lvert\xi-\langle X\rangle\rvert\lambda\ll\min(1,M_k/\tau_k)$ on the support of $w_\xi$. The bound on $R_2$ is not uniform in $\delta p\to0$ ($X_g\ge1/(2\delta p)$). **For a sharp label at fixed $\lambda$ the secular terms do not dominate:** pointwise in $p$, the term $-\tfrac\sigma2\lambda^2\theta'^2\omega'\cos2\omega\lambda$ of (7.13) is of the same order in $\lambda$ as $\tfrac\sigma2\lambda^2\omega'''$, and a window average of the acceleration does not remove it (Step 8, part 1). The statement of v2 that such an average suppresses it by $O(1/(M_k\Lambda))$ is withdrawn. 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. *Secular terms, window average, closed form.* Write the coefficients of $\kappa^2\sigma_X^2$ and $\kappa^3\mu_3$ in (7.14) as $v_2(\lambda)=s_2\lambda^2+R_2(\lambda)$ and $v_3(\lambda)=s_3\lambda^3+R_3(\lambda)$. The **secular coefficients** are $s_n[g]:=\lim_{\lambda\to\infty}v_n(\lambda)/\lambda^n$ at fixed smooth $g$: $s_2=\tfrac\sigma2\langle\omega_k'''\rangle_g$ (exact, by the bound on $R_2$) and $s_3=-\tfrac\sigma6\langle\omega_k''''\rangle_g$ (by the order estimate of $R_3$). The **secular acceleration** of order $n$ is the $\lambda$-derivative of the secular velocity term, $n\,s_n\lambda^{n-1}$ times $\kappa^2\sigma_X^2$ or $\kappa^3\mu_3$. It is related to the average of the acceleration over a window $[\lambda_1,\lambda_1+\Lambda]$, $\overline{a_n}:=\bigl(v_n(\lambda_1+\Lambda)-v_n(\lambda_1)\bigr)/\Lambda$, as follows, 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 $c_0:=\tfrac12\sup\lvert\theta'\theta''/\omega\rvert$:
$$
\bigl\lvert\overline{a_2}-2s_2\,\bar\lambda\bigr\rvert\le\frac{2c_1[g]\,\bar\lambda+2c_0}{\Lambda}\ \ (\text{exact; every smooth }g,\ \lambda_1\ge0,\ \Lambda>0),\qquad\overline{a_3}=3s_3\Bigl(\bar\lambda^2+\frac{\Lambda^2}{12}\Bigr)+\frac{R_3(\lambda_1+\Lambda)-R_3(\lambda_1)}{\Lambda}.
\tag{7.33}
$$
Hence $\lim_{\Lambda\to\infty}\overline{a_2}/\bar\lambda=\sigma\langle\omega_k'''\rangle_g$ at fixed $g$, for every $\lambda_1\ge0$ (it may grow with $\Lambda$); the window has to be long, $\Lambda\gg c_1[g]/\lvert\langle\omega_k'''\rangle_g\rvert$, where $c_1[g]\ge\tfrac{\tau_k^2}{2M_k^2}X_g\ge\tfrac{\tau_k^2}{4M_k^2\delta p}$. The coefficients have the sharp-label limits $\langle\omega_k'''\rangle_g\to\omega_k'''(p_0)$, $\langle\omega_k''''\rangle_g\to\omega_k''''(p_0)$ for every sequence of starts concentrated at $p_0$ with $\delta p\to0$ (mean and variance only; errors $\le\tfrac12\sup\lvert\omega_k^{(5)}\rvert\delta p^2$ and $\le\tfrac12\sup\lvert\omega_k^{(6)}\rvert\delta p^2$). The sharp-label limit can be combined with the long window only along sequences of starts and windows with $\delta p\to0$ and $X_g/\Lambda\to0$ (so $\Lambda\,\delta p\to\infty$). At a fixed window and $\delta p\to0$ the statement fails: pointwise at a label $p_0$ with $\omega_k'(p_0)\neq0$, (7.13) gives $\overline{a_2}=\sigma\bar\lambda\,\omega'''-\tfrac{\sigma}{2\Lambda}\bigl[\lambda^2\theta'^2\omega'\cos2\omega\lambda\bigr]_{\lambda_1}^{\lambda_1+\Lambda}+O(\bar\lambda/\Lambda)$, with an oscillating term of the order of the secular one; and pointwise at $p_0=0$ the third-order coefficient is $v_3=\sigma\lambda^3\bigl[-\tfrac16\omega''''+\tfrac16\theta'^2\omega''\cos2\omega\lambda\bigr]+O(\lambda^2)$, with the relative amplitude $\tfrac{\tau^2/M^2}{4+3\tau^2/M^2}<\tfrac13$ of the oscillating term. So (7.15), (7.16) below are the secular accelerations in this order of limits: Taylor coefficient in $\kappa$ at fixed $\lambda$ and $g$; secular coefficient at fixed $g$; then the sharp-label limit of the coefficient. They are not the acceleration of a sharp label at a fixed $\lambda$ or a fixed window. With $v=\sigma\omega_k'(p_0)$ of (4.17) and $\omega'''=-\omega'(4+3\omega''/\omega)$:
$$
\delta a^{(2)}_{k,\sigma}=\kappa^2\sigma_X^2\,\lambda\,\sigma\,\omega_k'''(p_0)=-\kappa^2\sigma_X^2\,\lambda\,v\Bigl(4+\frac{3\,\omega_k''(p_0)}{\omega_k(p_0)}\Bigr)=-\kappa^2\sigma_X^2\,\lambda\Bigl(4+\frac{3\tau_k^2}{M_k^2}\Bigr)v+O(v^3).
\tag{7.15}
$$
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. The third-order term does not vanish at rest:
$$
\delta a^{(3)}_{k,\sigma}=-\tfrac12\kappa^3\mu_3\,\lambda^2\,\sigma\,\omega_k''''(p_0),\qquad\delta a^{(3)}_{k,\sigma}\big\vert_{p_0=0}=\frac{\sigma}{2}\,\kappa^3\mu_3\,\lambda^2\,\frac{\tau_k^2}{M_k}\Bigl(4+\frac{3\tau_k^2}{M_k^2}\Bigr).
\tag{7.16}
$$
It is odd in the branch and present only for a skewed distribution of $X$.
2. *Level dependence.* (7.15) is the same for all levels at leading order in $v$ **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, as in Step 3). (7.16) at rest is level-independent iff $\tfrac{\tau_k^2}{M_k}(4+\tfrac{3\tau_k^2}{M_k^2})$ is. In the class $\mathcal C$ ($\tau_k=\tau$, $M_k$ distinct): **no** in both cases; for $M_k\ll\tau$ the terms scale as $M_k^{-2}$ and $M_k^{-3}$.
3. *Background form (item 2(d)).* With $d_0=\kappa\sigma_X$, $v_{\max,k}:=\tau_kd_0$ ((5.11) of 04-ilang-time/05-moving-clock with $\tau\to\tau_k$; $=v_{\max}$ in $\mathcal C$), $\hat e$ and $\bar x=d_0\bar x_{\rm lab}\hat e$ of (17.15) of 03-ilang-space/17-motion, $u:=d_0v$ and $\gamma_X:=\mu_3/\sigma_X^3$, the secular terms (7.15), (7.16) are
$$
\delta\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}=\Bigl[-\lambda\Bigl(4d_0^2+\frac{3v_{\max,k}^2}{M_k^2}\Bigr)u+\frac{\sigma\gamma_X}{2}\,\lambda^2\,\frac{v_{\max,k}^2}{M_k}\Bigl(4d_0^2+\frac{3v_{\max,k}^2}{M_k^2}\Bigr)\Bigr]\hat e,\qquad d_0\,a_{k,\sigma}(0)=\frac{v_{\max,k}^2}{d_0M_k}\bigl(\sigma\mathcal F_k-G_k\bigr).
\tag{7.17}
$$
The secular record terms are smaller than the drift acceleration (4.7) at rest where
$$
\varrho_2:=\frac{\lambda\lvert u\rvert M_k}{d_0\,\tau_k^2\lvert\sigma\mathcal F_k-G_k\rvert}\Bigl(4d_0^2+\frac{3v_{\max,k}^2}{M_k^2}\Bigr)<1\quad\text{and}\quad\varrho_3:=\frac{\lvert\gamma_X\rvert d_0\lambda^2}{2\lvert\sigma\mathcal F_k-G_k\rvert}\Bigl(4d_0^2+\frac{3v_{\max,k}^2}{M_k^2}\Bigr)<1,
\tag{7.18}
$$
that is $\lambda<\lambda_2\propto\lvert\sigma\mathcal F_k-G_k\rvert/(\kappa^2\lvert v\rvert)$ and $\lambda<\lambda_3\propto(\lvert\sigma\mathcal F_k-G_k\rvert/\kappa^3)^{1/2}$. (7.18) compares the selected secular terms, derived at zero gradients, with a term of first order in the gradients at leading order in $\kappa$; it does not control the whole acceleration. The rest of the window-averaged acceleration is controlled at second order in $\kappa$ by the exact bound (7.33),
$$
\varrho_{2,\rm rest}:=\kappa^2\sigma_X^2\,\frac{2c_1[g]\,\bar\lambda+2c_0}{\Lambda}\cdot\frac{M_k}{\tau_k^2\,\lvert\sigma\mathcal F_k-G_k\rvert},
\tag{7.34}
$$
and at third order by $\kappa^3\lvert\mu_3\rvert\,\lvert R_3(\lambda_1+\Lambda)-R_3(\lambda_1)\rvert/\Lambda$, for which only the order estimate $O(\bar\lambda^{5/2}/\Lambda)$ at fixed $g$ is available; terms that mix $\kappa$ with the gradients are out of scope and not controlled. Order estimate for a fall from rest ($\lvert v\rvert\le\lvert a\rvert\lambda$; this mixes $\kappa^2$ with a gradient and is outside the derived orders): $\varrho_2\lesssim\lambda^2(4d_0^2+3v_{\max,k}^2/M_k^2)$, independent of the gradients. All ratios are level-dependent at their own order. So "weak records" must mean at least $\varrho_2,\varrho_3,\varrho_{2,\rm rest}\le\delta_{\rm acc}$ for every level involved, on the whole range of windows and of $v$ observed, for the fall to be level-independent to the relative accuracy $\delta_{\rm acc}$: a joint condition $\kappa^2\lambda\lvert v\rvert\ll$ gradient, $\kappa^3\lvert\mu_3\rvert\lambda^2\ll$ gradient and $\kappa^2\sigma_X^2\,c_1[g]\,\bar\lambda/\Lambda\ll\tau_k^2\times$gradient$/M_k$, not uniform in $\lambda$, not uniform in the spread of the start ($c_1[g]\to\infty$ for $\delta p\to0$), and violated if the gradients are sent to zero at fixed $\kappa$. These conditions are necessary for the terms considered; that they are sufficient is not shown (third-order rest, mixed terms).

### 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. (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) (starts of finite width can have equal velocities) | position side unchanged (exact mixture); label rates level-dependent at grad$^1$ (sequences with vanishing weight at the cut, (7.28)); $\lvert\Gamma_{kk'}\rvert$: for starts of definite parity, $g(-p)=\pm g(p)$, no grad$^1$ term (exact), and for the starts (E1) grad$^2$ with $\lambda^2\delta x^2$ and $\lambda^4/\delta x^2$ (order estimate, range (E3), (7.32)); for starts that are not of definite parity grad$^1$ iff $\operatorname{Re}(\Gamma^{(1)}_{kk'}/\Gamma^{(0)}_{kk'})\neq0$ (exact, $\Gamma^{(0)}_{kk'}\neq0$), with the leading part $\lambda^3\Delta\nu_{kk'}\langle p^3\rangle_g$ and $\lambda^2\Delta\nu_{kk'}J_g$ for narrow starts at rest (order estimate, range (F1)–(F3)); in the classes (E1), (F1) also for a level-independent fall with $a\neq0$, unless $\tau_k^2/M_k=\tau_{k'}^2/M_{k'}$; other starts: open | (7.1)–(7.5), (7.27), (7.7), (7.28), (7.9), (7.10), (7.23)–(7.26), (7.29)–(7.32) |
| Records | $\kappa^1$: none; secular terms (fixed smooth $g$, long window (7.33); not a sharp label at a fixed window): $\kappa^2\sigma_X^2$: grad$^0$, $v^1$, $\lambda^1$, factor $4+3\tau_k^2/M_k^2$; $\kappa^3\mu_3$: grad$^0$, $v^0$, $\lambda^2$, factor $\tfrac{\tau_k^2}{M_k}(4+\tfrac{3\tau_k^2}{M_k^2})$ (rests on the order estimate of $R_3$) | the same secular terms add to the left side; small on the range (7.18), with the rest (7.34) | (7.11), (7.14)–(7.18), (7.33), (7.34) |
| 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). Status words: *exact* (an identity), *bound* (an exact inequality), *order estimate* (not a bound). 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 $g$, $\chi$, $\lambda$ and chain. $p_x=\sum_k\lvert\alpha_k\rvert^2p^{(k)}_x$ and $\bar x_{\rm lab}=\sum_k\lvert\alpha_k\rvert^2\bar x_k$. It rests on the conservation of the level and the orthonormality of the $\lvert f_k\rangle$.
- **(7.2)** Class: every $\alpha$ and every smooth $g$ concentrated at $p_0$. Limits, in this order: first order, long chain, $\Lambda\to\infty$ at fixed $g$, then the sharp-label limit; leading order in $\kappa$. Drift acceleration $\sum_k\lvert\alpha_k\rvert^2a_{k,\sigma}(p_0)$; the bound of (7.2) is exact for every window $\Lambda>0$ and not uniform in $\delta p$ (it contains $X_g$). The drift acceleration is the same for all $\alpha$ iff $a_{k,\sigma}(p_0)$ is level-independent; at rest this is line 1 or 2 of (4.19), with no new condition.
- **(7.3), (7.4)** Class: 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 $E$ for all $\lambda\ge0$ whose constant $r$ depends on $g$; the $\lambda^4$ part dominates for $\lambda\gg r/\sqrt{\operatorname{Var}_\alpha}$ at fixed $g$. The sharp-label limit is taken on the coefficient of $\lambda^4$ only, which tends to $\tfrac14\operatorname{Var}_\alpha(a_{\cdot,\sigma}(0))$; no statement is made on $D$ at fixed $\lambda$ in that limit. (7.4), class $\mathcal C$ with $\lvert\alpha_k\rvert^2=1/N$: 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 $g$ of any width: $v_k=\sigma\langle\omega_k'\rangle_g$. Bound (7.27), every start concentrated at $p_0$: $\lvert v_k-\sigma\omega_k'(p_0)\rvert\le\tfrac12\sup\lvert\omega_k'''\rvert\delta p^2$. Sharp-label velocities $\sigma\omega_k'(p_0)$, (7.5). **Item 1(d):** a start of finite width *can* have level components with equal velocities, in every member of $\mathcal C_{\rm lev}$ (example with all velocities zero); in $\mathcal C$, for $\sin2p_0\neq0$, no start with $\delta p<\delta p_*$ of (7.27) has equal velocities in two levels. In $\mathcal C_{\rm lev}$ the sharp-label velocities of two levels with different data are equal at one such $p_0$ under one condition, which 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, any normalized level component. (7.7): first order at fixed $\lambda$ and start, long chain, every smooth $g$; the rate of the arc mean contains the weight at the cut; no window. (7.28): bound, every start at rest. Limit: for sequences of starts at rest with $\delta p\to0$ and $\lvert g(\pi)\rvert^2\to c_{\rm cut}$ the rate tends to $(1-c_{\rm cut})(\mathcal F_k-\sigma G_k)$; it is $\mathcal F_k-\sigma G_k$ only for $c_{\rm cut}=0$, and the spread alone does not give $c_{\rm cut}=0$. For sequences with $c_{\rm cut}\neq1$: the limiting rate is 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 level-independent; impossible in $\mathcal C$. For $c_{\rm cut}=1$ the limiting rate is zero in every level. For starts of finite width the level-independence of the rate is not reduced to a condition on the level data.
- **(7.8), (7.9)** (7.8): exact with the medium in the level components; clock-label statistics through $\alpha$ and $\Gamma_{kk'}$; at leading order in $\kappa$ the $\Gamma_{kk'}$ of the effective body contract. (7.9): the zero-gradient coherence, exact for every smooth $g$; for even $g$, $\Gamma_{kk'}$ is exactly even in the gradients.
- **(7.23)** Exact; every smooth $g$, every $p_0$ and $\lambda$; effective body contract. $\Gamma_{kk'}[g;\text{gradients}]=\Gamma_{kk'}[g^-;-\text{gradients}]$. **$\lvert\Gamma_{kk'}\rvert$ has no first-order term** for every start of definite parity, $g(-p)=\pm g(p)$, and for every start if $(\tau_k,M_k)=(\tau_{k'},M_{k'})$. That the second-order term is different from zero is not shown by this.
- **(7.24), (7.29), (7.30)** Exact; every smooth complex $g$, every $p_0$, every $\lambda\ge0$; first order at fixed $\lambda$ and start; leading order in $\kappa$. The complete first-order term of $\Gamma_{kk'}$: the part inside the branch, and the inter-branch part through the amplitude (7.29), whose norm has the bound $A_{k,\perp}(\lambda)$, linear in $\lambda$ with a coefficient $\propto\langle\sin^4p\rangle_g^{1/2}$. Where $\Gamma^{(0)}_{kk'}(\lambda)\neq0$ the first deviation of $\lvert\Gamma_{kk'}\rvert$ is of first order iff $\operatorname{Re}(\Gamma^{(1)}_{kk'}/\Gamma^{(0)}_{kk'})\neq0$; at a zero of $\Gamma^{(0)}_{kk'}$, $\lvert\Gamma_{kk'}\rvert=\lvert\Gamma^{(1)}_{kk'}\rvert$ to first order.
- **(7.31)** Bound; every start at rest. $1-\lvert\Gamma^{(0)}_{kk'}\rvert\le\bigl[\tfrac12\lambda\sup\lvert\omega_k''-\omega_{k'}''\rvert+\tfrac18(\tfrac{\tau_k}{M_k}+\tfrac{\tau_{k'}}{M_{k'}})^2\bigr]\delta p^2$; a sufficient condition for $\Gamma^{(0)}_{kk'}\neq0$.
- **(7.10), (7.32)** Order estimates. Class: real Gaussian starts at rest (E1), $\delta p\ll\min(1,M_k/\tau_k)$. Range (E3): $\lvert\pi_k\rvert\lambda\ll M_k/\tau_k$, $\lvert\nu_k-\nu_{k'}\rvert\lambda\,\delta p^2\ll1$, and the neglected inter-branch part (7.32) small against the right side of (7.10) (for data and gradients that differ by order one: $\lambda\gg\tau\delta p/M^2$). Second order in the gradients at fixed $\lambda$ and start; leading order in $\kappa$. $1-\lvert\Gamma_{kk'}\rvert/\lvert\Gamma^{(0)}_{kk'}\rvert\simeq\tfrac12\delta x^2(\pi_k-\pi_{k'})^2\lambda^2+\tfrac{\lambda^4}{32\delta x^2}\varpi_{kk'}^2$. With a level-independent fall $a$ it vanishes iff $a=0$ or $\tau_k^2/M_k=\tau_{k'}^2/M_{k'}$. (7.32) rests on the exact bound (7.29) and on second-order inter-branch terms that are estimated, not derived.
- **(7.25)** Order estimate. Class (F1): starts at rest with a smooth complex $g$ that need not be even or real, $\delta p\ll\min(1,M_k/\tau_k,M_{k'}/\tau_{k'})$, moments $\langle\lvert p\rvert^n\rangle_g=O(\delta p^n)$, $n\le6$. Range (F2), (F3): right side of (7.31) small, $\lambda_{\rm F}\ll\lambda\ll1/(\lvert\Delta\nu_{kk'}\rvert\delta p^2)$ when the level data differ by relative amounts of order one. Limits: first order at fixed $\lambda$ and start, long chain, leading order in $\kappa$, then leading order in $\delta p$. Fastest-growing part of the first-order term: $\Delta\nu_{kk'}\bigl[\tfrac14\lambda^3\varpi_{kk'}\langle p^3\rangle_g-\tfrac\sigma2\lambda^2(\pi_k-\pi_{k'})J_g\bigr]$; it needs $\Delta\nu_{kk'}\neq0$ and a skewed label weight or $J_g\neq0$. Its inter-branch remainder has the exact bound (7.30); its other remainders are estimated.
- **(7.26)** Same class, range and status as (7.25). A level-independent fall $a\neq0$ does not remove the leading first-order term unless $\tau_k^2/M_k=\tau_{k'}^2/M_{k'}$; in $\mathcal C$ it never does.
- **Item 1(f) is resolved in part** (Step 5, part 8). Resolved for every smooth $g$: (7.8), (7.9), the complete first-order term and the criterion for a first-order deviation, the absence of a first-order term for starts of definite parity. Resolved as order estimates for narrow starts at rest: the leading first-order term (F1) and, for real Gaussian starts (E1) only, the leading second-order term with its dependence on $\lambda$, the level data and $\delta x$. Open: the second-order term for all other starts; the leading form for starts that are not narrow or outside (F2); the cases in which (7.25) vanishes. "The deviation does not vanish for a level-independent fall with $a\neq0$" is established for (E1) and (F1) only, as an order estimate.

**Item 2, branch starts $B_{k,\sigma}[g]$.**
- **(7.11)** Exact at this order; every smooth $g$ and every $\chi$; first order jointly in $G_k$, $\mathcal F_k$, $g_{\zeta,k}$ and $\kappa(\xi-\langle X\rangle)$. The first-order velocity does not change when $X$ is not sharp; (4.7), (4.16), (4.19) are unchanged.
- **(7.13), (7.14)** Zero gradients of the effective body contract; every fixed smooth $g$ and every $\chi$ with finite moments of $X$; Taylor expansion in $\kappa$ at fixed $\lambda$, start and chain. $v=\sigma\langle\omega_k'\rangle_g+\kappa^2\sigma_X^2\bigl[\tfrac12\lambda^2\sigma\langle\omega_k'''\rangle_g+R_2\bigr]+\kappa^3\mu_3\bigl[-\tfrac16\lambda^3\sigma\langle\omega_k''''\rangle_g+R_3\bigr]+O(\kappa^4)$. (7.13) is exact pointwise in $p$. The bound on $R_2$ is exact, linear in $\lambda$, for all $\lambda\ge0$ at fixed $g$, and not uniform for $\delta p\to0$. $R_3=O(\lambda^{5/2})$ is an order estimate at fixed smooth $g$ (power counting and stationary phase), without a constant. The leading parts dominate for $\lambda\gg X_g\tau_k^2/(M_k^2\lvert\langle\omega_k'''\rangle_g\rvert)$ within $\kappa\lvert\xi-\langle X\rangle\rvert\lambda\ll\min(1,M_k/\tau_k)$. For a sharp label at fixed $\lambda$ the secular terms do not dominate.
- **(7.33)** Bound; every smooth $g$, $\lambda_1\ge0$, $\Lambda>0$. The second-order coefficient of the window-averaged acceleration is $\sigma\langle\omega_k'''\rangle_g\bar\lambda$ up to $(2c_1[g]\bar\lambda+2c_0)/\Lambda$; it needs $\Lambda\gg c_1[g]/\lvert\langle\omega_k'''\rangle_g\rvert$, and the sharp-label limit can be combined with it only if $X_g/\Lambda\to0$. Third order: an identity with the rest $R_3$.
- **(7.15), (7.16)** Secular accelerations: the $\lambda$-derivatives of the secular velocity terms of (7.14). Order of limits: expansion in $\kappa$ at fixed $\lambda$ and $g$; secular coefficient at fixed smooth $g$; then the sharp-label limit of the coefficient. Not the acceleration of a sharp label at fixed $\lambda$ or at a fixed window. $\delta a^{(2)}_{k,\sigma}=-\kappa^2\sigma_X^2\lambda\,v\,(4+3\omega_k''/\omega_k)$ (its coefficient is exact): zero at rest, of first order in $v$, the same in both branches at given $v$. $\delta a^{(3)}_{k,\sigma}$ at rest is $\tfrac\sigma2\kappa^3\mu_3\lambda^2\tfrac{\tau_k^2}{M_k}(4+\tfrac{3\tau_k^2}{M_k^2})$ (the coefficient of the adiabatic term is exact; that it is the fastest-growing part rests on the order estimate of $R_3$): odd in the branch, only for a skewed $X$. Level-independent iff $\tau_k/M_k$, respectively $\tfrac{\tau_k^2}{M_k}(4+\tfrac{3\tau_k^2}{M_k^2})$, is; not in $\mathcal C$.
- **(7.17), (7.18), (7.34)** The secular terms in the background of (17.15) of 03-ilang-space/17-motion, with $d_0=\kappa\sigma_X$ and $v_{\max,k}=\tau_kd_0$ from (5.11) of 04-ilang-time/05-moving-clock. They are smaller than the drift acceleration (4.7) at rest where $\varrho_2<1$ and $\varrho_3<1$; the rest of the window-averaged acceleration is bounded at second order through $\varrho_{2,\rm rest}$ (7.34). "Weak records" has to mean at least $\varrho_2,\varrho_3,\varrho_{2,\rm rest}\le\delta_{\rm acc}$ for every level on the whole observed range of windows and $v$. **Item 2(c), (d) is resolved for the secular terms**; the third-order rest and the terms that mix $\kappa$ with the gradients are not bounded, so the conditions are necessary for the terms considered and not shown to be sufficient.

**Item 3, on-site block $(1+gx)[M_k\sigma_3+u_k\mathbb 1_2]-\mathcal F_Ux\,\mathbb 1_2$, any $\tau_k$ and $g_{\zeta,k}$; branch starts at rest; drift acceleration (first order, long chain, long window at fixed $g$, then the sharp-label limit), 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; the rows on the superposition of levels and on the records state the class of starts and the status of each entry.

## 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) is $\lambda^{-3}\lambda^2\lambda^{-1}=\lambda^{-2}$, as (4.16); both terms of (7.10) and $\varrho_2,\varrho_3$ are dimensionless; (7.3) is dimensionless (labels squared). Passed.
2. **One level, a sharp $X$, and a plain gradient only.** $\alpha_k=\delta_{kk_0}$: (7.1), (7.2) reduce to the single-level start, $D=0$ in (7.3), and (7.8) gives $P(a)=1/N$, the uniform clock-label statistics of $\lvert f_{k_0}\rangle$. Sharp $X$: $w_\xi=\delta_{\xi\xi_0}$, $\varepsilon_\xi=0$, so (7.11) is trivial and (7.14)–(7.17) reduce to (4.5). Plain gradient only ($G_k=g_{\zeta,k}=0$): $B_k=-\mathcal F_k$, the transport of Step 4 is a rigid shift of the label weight at the rate $\mathcal F_k$, and the arc mean changes at the rate $\mathcal F_k(1-\lvert g(\pi)\rvert^2)$, which is (7.28) with $\pi_{k,\sigma}''=0$ (zero for the uniform weight). In the same case (7.29) gives $b_{\perp,k}=0$, $a_{k,\perp}=-\tfrac\sigma2\theta_k'\mathcal F_k\,g\,I_{0,k}$ and $S_\theta=2\operatorname{Re}\bigl((a_{k,\perp}/g)\,e^{2i\sigma\omega_k\lambda}\bigr)=-\tfrac{\sigma\theta'\mathcal F}{2\omega}\sin2\omega\lambda$, the first-order term of Step 7, part 3 with $\mathcal F=-\varepsilon$. Passed.
3. **Short-$\lambda$ expansion of (7.13).** Directly, the $\varepsilon^2$ term of $v_p$ at small $\lambda$ is $\tfrac12\lambda^2\varepsilon^2\langle\beta_\sigma\vert C_0'''\vert\beta_\sigma\rangle=\tfrac12\lambda^2\varepsilon^2\sigma(\omega'''-3\omega'\theta'^2-3\omega\theta'\theta'')$ (double commutator with $\varepsilon\hat x$; $\mathbf n$-component of $(\omega\mathbf n)'''$). Expanding (7.13) to $\lambda^2$ gives $\sigma\lambda^2[\tfrac12\omega'''-\tfrac12\omega\theta'\theta''-\theta'(\omega\theta')'-\tfrac12\theta'^2\omega']$, the same. Also, with zero gradients except $\mathcal F$, (7.7) and (4.16) agree with $a=\sigma\omega''(0)\,\mathrm d\bar p/\mathrm d\lambda$. Passed.

## Open issues

- **Item 1(f), open part.** The second-order term of $\lvert\Gamma_{kk'}\rvert$ for every start other than the real Gaussian starts (E1), including whether it is different from zero for a given start of definite parity; the leading form of the first-order term for starts at rest that are not narrow, or outside (F2); the cases in which (7.25) vanishes ($\langle p^3\rangle_g=J_g=0$ without definite parity; $\Delta\nu_{kk'}=0$ with unequal data).
- (7.10) is an order estimate under (E1)–(E3). Its inter-branch estimate (7.32) rests on the exact first-order bound (7.29), but the second-order inter-branch amplitude and the mixed terms are estimated, not derived; no bound.
- (7.25) and (7.26) are order estimates under (F1)–(F3): the inter-branch remainder is bounded (7.30), the remainder of the expansion in $\delta p$ is estimated, not bounded.
- Item 1(e): for starts of finite width the level-independence of the rate (7.7) is not reduced to a condition on the level data; for sequences with $c_{\rm cut}=1$ the limiting rate of the arc mean vanishes in every level.
- (7.3): the constant $r$ of the bound on $E$ depends on $g$; no bound uniform in $\delta p$ is derived, so nothing is stated on $D$ at fixed $\lambda$ in the sharp-label limit.
- $R_3=O(\lambda^{5/2})$ in (7.14) is an order estimate without a constant, and (7.16) as the fastest-growing third-order part rests on it. The acceleration of a sharp label at a fixed window is not characterized: oscillating terms of the order of the secular ones remain. That the conditions of Step 8, part 3 are sufficient for a given accuracy is not shown.
- The estimate $\varrho_2\lesssim\lambda^2(4d_0^2+3v_{\max,k}^2/M_k^2)$ for a fall from rest mixes $\kappa^2$ with a gradient (third order, out of scope) and is not derived.
- (7.17) uses $v_{\max,k}=\tau_kd_0$ for level-dependent $\tau_k$; the quoted bound (5.11) of 04-ilang-time/05-moving-clock is stated for one $\tau$.
- The effect of an operator under the factor on the relation (4.24) outside the class $\mathcal C$ (for example $u_k=u_*+wM_k$ with $w\neq0$) 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 first-order perturbation theory for the overlap of two level components, in-branch and inter-branch parts; Cauchy–Schwarz bounds; moment expansion of an oscillating average
- Gaussian overlap of displaced wave packets (order estimate)
- Exact reduction to a driven two-level problem; Bloch equations; adiabatic expansion with integration by parts
- Secular coefficients and window averages; stationary phase and Riemann–Lebesgue estimates for oscillating averages
- Invariance under replacements of contract terms (well-definedness)

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