Version 1 · earlier version; the current one is v5 · Review of v1

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 07-limits
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## 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, position readings of $\Phi_\sigma[\alpha,g]$ show the drift of one 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 $g$; then $\delta p\to0$, error $\le\tfrac12\sup\lvert a''_{k,\sigma}\rvert\delta p^2$ per level (4.11). $D$ 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 $O(\delta p^4)$ at fixed $g$, by (4.11), 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, the level component $k$ has the constant velocity
$$
v_k=\sigma\langle\omega_k'\rangle_g\ \xrightarrow{\ \delta p\to0\ }\ \sigma\,\omega_k'(p_0)=\sigma\,\frac{\tau_k^2\sin p_0\cos p_0}{\sqrt{M_k^2+\tau_k^2\sin^2p_0}};\qquad v_k=v_{k'}\ (\sin2p_0\neq0)\iff\frac{\tau_k^4}{M_k^2+\tau_k^2\sin^2p_0}=\frac{\tau_{k'}^4}{M_{k'}^2+\tau_{k'}^2\sin^2p_0}.
\tag{7.5}
$$
In the class $\mathcal C$ ($\tau_k=\tau$, $M_k$ strictly increasing) the velocities are strictly ordered for $\sin2p_0\neq0$: **no** start of this class with $p_0\notin\{0,\pm\tfrac\pi2,\pi\}$ has two level components with equal velocities; the components separate as $\lvert v_k-v_{k'}\rvert\lambda$ already at zeroth order in the gradients. In $\mathcal C_{\rm lev}$ equality at one $p_0$ is one condition on $(\tau_k,M_k,\tau_{k'},M_{k'})$ and needs $\tau_k\neq\tau_{k'}$; 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 $B_k=-g_{\zeta,k}\tau_k\sin p\,\sigma_1+G_k\sigma_3-\mathcal F_k$:
$$
\frac{\mathrm d\bar p_k}{\mathrm d\lambda}\Big\vert_{1}=\langle\pi_{k,\sigma}\rangle_g-\lvert g(p_0+\pi)\rvert^2\pi_{k,\sigma}(p_0+\pi),\quad\pi_{k,\sigma}(p):=\mathcal F_k-\sigma\frac{G_kM_k+g_{\zeta,k}\tau_k^2\sin^2p}{\omega_k(p)};\qquad\lim_{\delta p\to0}=\pi_{k,\sigma}(0)=\mathcal F_k-\sigma G_k.
\tag{7.7}
$$
Order of limits: first order in the gradients at fixed $\lambda$, start; long chain; $\delta p\to0$ (which also removes the cut term). With (4.16), $a_{k,\sigma}(0)=\sigma\,\omega_k''(0)\,\pi_{k,\sigma}(0)$.
3. *Equivalences.* At rest the 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 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 $\tau_k^2\propto M_k$.

### 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 at rest with even $g$ the deviation of $\Gamma_{kk'}$, and of $\lvert\Gamma_{kk'}\rvert$, starts at **second order** in the gradients. For a start that is not even a first-order term is not excluded.
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): $c_k(p,\lambda)\simeq g(p-\pi_k\lambda)\,e^{-i\int_0^\lambda E_k(p-\pi_ks)\mathrm ds}\beta_{k,\sigma}$, $E_k=V_k+\sigma\omega_k$, $\pi_k:=\pi_{k,\sigma}(0)$; the neglected inter-branch part gives a second-order term bounded in $\lambda$, negligible for $\lambda\gg\tau_k/M_k^2$; (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 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 $e^{-\delta p^2\Delta X^2/2}$:
$$
1-\frac{\lvert\Gamma_{kk'}\rvert}{\lvert\Gamma^{(0)}_{kk'}\rvert}\simeq\frac{\delta x^2}2\,(\pi_k-\pi_{k'})^2\lambda^2+\frac{\lambda^4}{32\,\delta x^2}\Bigl(\frac{\tau_{k'}^2}{M_{k'}}\pi_k-\frac{\tau_k^2}{M_k}\pi_{k'}\Bigr)^2,\qquad\pi_k=\mathcal F_k-\sigma G_k.
\tag{7.10}
$$
4. *Level-independent fall.* 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 $\delta x$.

### 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). $R_3$ consists of terms oscillating at $2\omega_k(p)$ with amplitudes up to $\lambda^4\theta'^2\omega'^2$ and $\lambda^3\theta'^2\omega''$ (from the expansion of $\varphi$), reduced to $O(\lambda^{5/2})$ by integration by parts and stationary phase at $\omega_k'=0$, and of non-oscillating terms $O(\lambda)$. 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 the oscillating terms of (7.13) are removed by a window average instead, with relative size $O(1/(M_k\Lambda))$ (order estimate). 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. *Closed form.* For a sharp label $p_0$ (order: $\kappa$-expansion at fixed $\lambda$; window average; $\delta p\to0$), the $\lambda$-derivative of the secular parts of (7.14), 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 $\gamma_X:=\mu_3/\sigma_X^3$:
$$
\delta\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}=\Bigl[-\lambda\Bigl(4d_0^2+\frac{3v_{\max,k}^2}{M_k^2}\Bigr)u+\frac{\sigma\gamma_X}{2}\,\lambda^2\,\frac{v_{\max,k}^2}{M_k}\Bigl(4d_0^2+\frac{3v_{\max,k}^2}{M_k^2}\Bigr)\Bigr]\hat e,\qquad d_0\,a_{k,\sigma}(0)=\frac{v_{\max,k}^2}{d_0M_k}\bigl(\sigma\mathcal F_k-G_k\bigr).
\tag{7.17}
$$
The 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 $\lambda<\lambda_3\propto(\lvert\sigma\mathcal F_k-G_k\rvert/\kappa^3)^{1/2}$. 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. Both ratios are level-dependent at their own order. So "weak records" must mean $\varrho_2,\varrho_3\le\delta_{\rm acc}$ for every level involved, on the whole range of $\lambda$ 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 and $\kappa^3\lvert\mu_3\rvert\lambda^2\ll$ gradient, not uniform in $\lambda$ and violated if the gradients are sent to zero at fixed $\kappa$.

### 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: $(\mathcal F_U-gu_k)/M_k=:-gw$ independent of $k$, i.e. $u_k=u_*+wM_k$ with $gu_*=\mathcal F_U$, and then $a_\pm=-\tau^2g(1\pm w)$; one value iff $gw=0$.
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; if the fall is level-dependent: mean acceleration $\alpha$-dependent at grad$^1$; between-level variance grad$^2$, $\lambda^4$; at $p_0\neq0$ separation at grad$^0$, $v^1$, $\lambda^1$ | position side unchanged (exact mixture); label rates level-dependent at grad$^1$; $\lvert\Gamma_{kk'}\rvert$ deviates at grad$^2$, $\lambda^2\delta x^2$ and $\lambda^4/\delta x^2$, also for a level-independent fall | (7.1)–(7.5), (7.7), (7.9), (7.10) |
| Records | $\kappa^1$: none; $\kappa^2\sigma_X^2$: 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 $\tfrac{\tau_k^2}{M_k}(4+\tfrac{3\tau_k^2}{M_k^2})$ | the same terms add to the left side; negligible on the range (7.18) | (7.11), (7.14)–(7.18) |
| 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

- **(7.1)** exact mixture: $p_x=\sum_k\lvert\alpha_k\rvert^2p^{(k)}_x$, exact in the gradients and in $\kappa$. **(7.2)** drift acceleration $\sum_k\lvert\alpha_k\rvert^2a_{k,\sigma}(p_0)$; $\alpha$-independent iff $a_{k,\sigma}(p_0)$ is level-independent, at rest the condition (4.19).
- **(7.3), (7.4)** between-level variance $\tfrac14\lambda^4\operatorname{Var}_\alpha(a_{\cdot,\sigma}(0))$, with the variants. **(7.5)** velocities $\sigma\omega_k'(p_0)$; never equal in $\mathcal C$ for $\sin2p_0\neq0$.
- **(7.6), (7.7)** label rate $\mathcal F_k-\sigma G_k$ at rest; level-independent together with a nonzero level-independent fall iff also $\tau_k^2/M_k$ is level-independent; impossible in $\mathcal C$.
- **(7.8)–(7.10)** clock statistics through the coherences; $\lvert\Gamma_{kk'}\rvert$ deviates at second order in the gradients (exactly even for even $g$), leading term (7.10) (order estimate), not removed by a level-independent fall.
- **(7.11)** records do not change the first-order velocity. **(7.13), (7.14)** second- and third-order terms in $\kappa$. **(7.15), (7.16)** change of the acceleration; level-independent only for level-independent $\tau_k/M_k$, not in $\mathcal C$. **(7.17), (7.18)** background form and range.
- **(7.19)–(7.21)** fall from rest for a general operator under the factor, the three equivalences, and the invariants.
- **(7.22)** 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) 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 and a sharp $X$.** $\alpha_k=\delta_{kk_0}$: (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). 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

- The coefficients of (7.10) are an order estimate under (E1)–(E3) (Gaussian even start, within-branch transport, equal spreading); a bound for general $g$, and the first-order term for starts that are not even, are not derived.
- $R_3=O(\lambda^{5/2})$ in (7.14) and the window suppression $O(1/(M_k\Lambda))$ of the oscillating terms for a sharp label are order estimates; only $R_2$ has an exact bound, and it is not uniform in $\delta p\to0$.
- 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
- Reflection symmetry (parity) argument for the order of a deviation
- Gaussian overlap of displaced wave packets (order estimate)
- Exact reduction to a driven two-level problem; Bloch equations; adiabatic expansion with integration by parts
- 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