Version 2 · earlier version; the current one is v3 · External review, round 1: minor issues

# Fall of a branch start and level-independent fall

- **Subproject:** 05-ilang-free-fall
- **Package:** 04-branch-fall
- **Version:** v2
- **Mode:** internal regeneration
- **Date:** 2026-10-10

## Changes from previous version

Trigger: `review.v1.md`, issue I1 (the sharp-label limit at fixed $\lambda$ and the commutation criterion were stated beyond the class of starts for which they were proven).

- **Step 4 reworked for I1.** The data of a family of starts that $\delta p\to0$ does not fix is now explicit: the functional $\Xi_g$ and the class $\mathcal K_1(x_c,\beta)$ of sharp-label families, new (4.26). The centred families of v1 are $\mathcal K_1(x_c,0)$.
- **(4.13), statement changed, tag kept.** It names its class. It is proven for $\mathcal K_1(x_c,\beta)$ (the term with $\beta$ is new; at $\beta=0$ it is the formula of v1) and, if $G_k=g_{\zeta,k}M_k$, for every sharp-label family.
- **(4.14), statement changed, tag kept.** It names its class and is proven for $\mathcal K_1(x_c,\beta)$, which is wider than the centred class of v1. The remainder is now exact, and the amplitude of the oscillation is $\lvert n_k\rvert\sqrt{1+4\beta^2}$.
- **(4.15), statement changed, tag kept.** It names its class and contains the term with $\beta$; at $\beta=0$ it is the formula of v1.
- **Removed.** The sentence "For a family that is not centred, the term with $\xi_g$ need not converge unless $\theta'B_\perp(p_0)=0$" was not proven and is false. It is replaced by the exact statement (4.27) for every sharp-label family and by a proven example, the family $E_\infty$, for which the limit at fixed $\lambda$ does not exist, at rest included.
- **New tags.** (4.26) definitions; (4.27) exact first-order acceleration at fixed $g$ and the criterion for the existence of its sharp-label limit, for every family; (4.28) exact window average at fixed $g$; (4.29) table: which families the two limits commute for, at rest and in motion, and where this is not determined.
- **Follow-up edits.** Step 5 (bullets and (4.15)), Result item 1, Controls, consistency check 2, Open issues and the JSON twin name the classes. The Setup lists the new notation and two standard facts, and two notes separate the new symbols $w$, $h$ of Step 4 from the branch weights $w_\pm$ of Step 5 and the function $h(M_k)$ of Step 7. "The limits commute at rest" is now stated for $\mathcal K_1$ (and for every family if $G_k=g_{\zeta,k}M_k$), not for every family.
- Everything else is unchanged, with the tags (4.1)–(4.12) and (4.16)–(4.25). The drift acceleration (4.11) was and is proven for every $g$.

## Setup and assumptions

- **Description (assumption, M8).** A body $b$ of the class $\mathcal C_{\rm lev}$ with linear level profiles and a medium $c$; total contract $C=C_b\otimes\mathbb 1_c+\kappa\,\hat x\otimes X$, $\hat x=\sum_xx\,P_x$. The single-object term $C_b$ is counted in the one pair term $(b,c)$ (A5); the position dependence is written into the body's contract (A10). Every control used below is a member of this class with the same medium, hence admissible (A1–A13). $b$ and $c$ are the only instances of their types, so no swap condition applies. $\lambda\ge0$.
- **Start (assumption, M9).** $\phi_L\otimes\chi$, with $\phi_L$ the branch start $B_{k,\sigma}[g]$ placed on the chain as in Step 1(iv). The start is a branch vector of the zeroth-order block and is not adapted to the gradients. A sharp-label limit is taken along a family of such starts (Step 4); where a result needs a class of families, the class is an assumption on the starts and is named in the statement.
- **Inputs, as quoted in question.md.** (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), (3.5), (3.9), (3.15), (3.23), (3.26)–(3.28) of 03-clock-and-fall; (3.6) and (3.36) are quoted there as well and are not needed. Steps of those packages needed for $\mathcal C_{\rm lev}$ are rederived here.
- **Notation.** Inside one level the index $k$ is dropped: $\tau,M,G,\mathcal F,g_\zeta,\omega,\theta$. $\mathsf b:=\lvert G\rvert+M\lvert g_\zeta\rvert+\lvert\mathcal F\rvert$, $\mathsf b':=\lvert G\rvert+\tau\lvert g_\zeta\rvert+\lvert\mathcal F\rvert$. $\xi_g(p):=-\operatorname{Im}(g^*g')$, the position density of the start, $\int\frac{\mathrm dp}{2\pi}\xi_g=\bar x_{\rm lab}(0)$. $X_g:=\bigl(\int\frac{\mathrm dp}{2\pi}\lvert g'\rvert^2\bigr)^{1/2}$. A prime is $\partial_p$. $\Xi_g$, sharp-label families and the classes $\mathcal K_1(x_c,\beta)$, $E$, $E_\infty$: Step 4, (4.26).
- **Standard facts used.** First-order perturbation theory (Duhamel formula) for $e^{-i(C_0+C_1)\lambda}$; on plane-wave amplitudes $\hat x=i\partial_p$; Riemann–Lebesgue and stationary phase for $\int e^{2i\omega\lambda}R\,\mathrm dp$ with smooth $R$ and non-degenerate stationary points; weak convergence of measures; the zeros of an entire function that is not identically zero are isolated.

## Derivation

### Step 1. Reduction to one level block

**(i) Records.** With the spectral decomposition $X=\sum_\xi\xi\,Q_\xi$, $C=\sum_\xi(C_b+\kappa\xi\hat x)\otimes Q_\xi$, hence exactly

$$
p_x(\lambda)=\sum_\xi\lVert Q_\xi\chi\rVert^2\,p^{[\xi]}_x(\lambda),
\tag{4.1}
$$

where $p^{[\xi]}$ are the place weights of the body alone under $C_b+\kappa\xi\hat x$, that is with the plain gradients $\mathcal F^{\rm body}_k-\kappa\xi$. Expanding in $\xi-\langle X\rangle$, the linear term averages to zero. Therefore $\bar x_{\rm lab}=\bar x_{\rm lab}[C_{\rm eff}]+O(\kappa^2\sigma_X^2)$ at fixed $\lambda$, chain and start, with $C_{\rm eff}=C_b+\kappa\langle X\rangle\hat x$ of (8.19); the term of first order in $\kappa$ beyond $C_{\rm eff}$ vanishes exactly, and what is neglected is $\tfrac12\kappa^2\sigma_X^2\,\partial_{\mathcal F}^2\bar x_{\rm lab}$ plus higher central moments of $X$ (order estimate). From here on all profiles are those of $C_{\rm eff}$: $\mathcal F_k=\mathcal F_k^{\rm body}-\kappa\langle X\rangle$. $C_{\rm eff}$ is the same operator for every listing of the contract terms related by (8.1), place-dependent $\alpha_h$ included, and changes by a multiple of $\mathbb 1$ under $Z$ (8.20).

**(ii) Level block.** $C_{\rm eff}$ and the records conserve the level, so a start with the clock label in $\lvert f_k\rangle$ evolves by $C^{(k)}$. On the infinite chain $\sum_xR_x=e^{-i\hat p}$ and $\sum_x(x+\tfrac12)R_x=\tfrac12\{\hat x,e^{-i\hat p}\}$, so for the linear level profiles, exactly (this is (3.9) with $\tau\to\tau_k$, $g_\zeta\to g_{\zeta,k}$, $\mathcal F\to\mathcal F_k$),

$$
C^{(k)}=C_0+C_1,\quad C_0=-\tau\sin\hat p\,\sigma_1+M\sigma_3,\quad C_1=\tfrac12\{\hat x,B(\hat p)\},\quad B(p)=-g_\zeta\tau\sin p\,\sigma_1+G\sigma_3-\mathcal F .
\tag{4.2}
$$

**(iii) Velocity.** The records commute with $\hat x$, so $\mathrm d\bar x_{\rm lab}/\mathrm d\lambda=\langle\hat v\otimes\mathbb 1_c\rangle$ with $\hat v=i[C^{(k)},\hat x]$ on the level $k$. With $\hat x=i\partial_p$, $i[F(\hat p),\hat x]=F'(\hat p)$, and $\partial_pB=g_\zeta V_0$:

$$
\hat v=V_0(\hat p)+\hat v_1,\qquad V_0(p):=\partial_pC_0(p)=-\tau\cos p\,\sigma_1,\qquad \hat v_1=\tfrac{g_\zeta}{2}\{\hat x,V_0(\hat p)\}.
\tag{4.3}
$$

**(iv) Finite chain and long-chain limit.** The position amplitudes $\phi(x)$ of $B_{k,\sigma}[g]$ decay faster than any power of $\lvert x\rvert$ ($g\beta_{k,\sigma}$ is smooth and periodic). On the chain $\lvert x\rvert\le L$ (open ends, hopping terms $x=-L,\dots,L-1$) the start is $\phi_L:=\mathbb 1_{\lvert x\rvert\le L}\phi/\lVert\mathbb 1_{\lvert x\rvert\le L}\phi\rVert$. The velocity $v_L(\lambda)$ is an entire function of the gradients; $v_L^{(0)}$ is its value and $v_L^{(1)}$ its differential at zero gradients, a linear form in $(G,\mathcal F,g_\zeta)$. By the finite-propagation estimate $\lvert\langle x\rvert e^{-iC_0s}\lvert y\rangle\rvert\le e^{(\tau+M)s}\,((\tau+M)s)^{\lvert x-y\rvert}/\lvert x-y\rvert!$ and the decay of $\phi$, the Duhamel expressions of Step 3 on the finite chain converge to those on the infinite chain: $v^{(j)}_L(\lambda)\to v^{(j)}(\lambda)$, $j=0,1$, for $L\to\infty$ at fixed $\lambda$ and $g$, faster than any power of $1/L$, not uniformly in $\lambda$. All formulas below are these limits.

### Step 2. Branch basis

Write a two-component amplitude as $\sum_{\sigma}c_\sigma(p)\beta_{k,\sigma}(p)$ and let $\rho_1,\rho_2,\rho_3$ be the Pauli matrices on $(c_+,c_-)$. The $\beta_{k,\pm}$ of (5.6) are real, $\beta_+'=\tfrac{\theta'}2\beta_-$, $\beta_-'=-\tfrac{\theta'}2\beta_+$, and $\langle\beta_\pm\rvert\vec\sigma\lvert\beta_\pm\rangle=\pm(\sin\theta,0,\cos\theta)$, $\langle\beta_+\rvert\sigma_1\lvert\beta_-\rangle=\cos\theta$, $\langle\beta_+\rvert\sigma_3\lvert\beta_-\rangle=-\sin\theta$, with $\cos\theta=M/\omega$, $\sin\theta=-\tau\sin p/\omega$. Hence, exactly,

$$
\hat x=i\partial_p+\tfrac{\theta'}2\rho_2,\qquad C_0=\omega\rho_3,\qquad V_0=\omega'\rho_3+u\,\rho_1,\qquad B=-\mathcal F+B_\parallel\rho_3+B_\perp\rho_1,
\tag{4.4}
$$

$$
\omega'=\frac{\tau^2\sin p\cos p}{\omega},\quad \theta'=-\frac{\tau M\cos p}{\omega^2},\quad u=\omega\theta',\quad B_\parallel=\frac{g_\zeta\tau^2\sin^2p+GM}{\omega},\quad B_\perp=\frac{\tau\sin p\,(G-g_\zeta M)}{\omega},
$$

and, since $\rho_2$ anticommutes with $\rho_1,\rho_3$, $C_1=\tfrac12\{i\partial_p,-\mathcal F+B_\parallel\rho_3\}+C_1^{\rm od}$ with the branch-changing part $C_1^{\rm od}=\tfrac12\{i\partial_p,B_\perp\}\rho_1-\tfrac{\theta'}2\mathcal F\rho_2$. Differentiating $B_\parallel=\langle\beta_+\rvert B\lvert\beta_+\rangle+\mathcal F$ gives the identity $B_\parallel'=g_\zeta\omega'+\theta'B_\perp$.

### Step 3. Velocity of a branch start to first order (item 1a)

The start is $c(0)=g\,e_\sigma$. By the Duhamel formula, $\psi=\psi_0+\psi_1+O(2)$ with $\psi_0(\lambda)=g\,e^{-i\sigma\omega\lambda}e_\sigma$, $\psi_1(\lambda)=-i\int_0^\lambda\mathrm ds\,e^{-iC_0(\lambda-s)}C_1\psi_0(s)$, and $v^{(0)}=\langle\psi_0\rvert V_0\lvert\psi_0\rangle$, $v^{(1)}=\langle\psi_0\rvert\hat v_1\lvert\psi_0\rangle+2\operatorname{Re}\langle\psi_0\rvert V_0\lvert\psi_1\rangle$.

1. Zeroth order, exactly and for all $\lambda$ (only $\omega'\rho_3$ has a diagonal element):
$$
v^{(0)}(\lambda)=\sigma\,\langle\omega_k'\rangle_g\qquad(\text{constant in }\lambda).
\tag{4.5}
$$
2. With $i\partial_p(e^{-i\sigma\omega s}g)=e^{-i\sigma\omega s}(ig'+\sigma\omega'sg)$: $\langle\psi_0\rvert\hat v_1\lvert\psi_0\rangle=g_\zeta\bigl[\lambda\langle\omega'^2\rangle_g+\sigma\int\frac{\mathrm dp}{2\pi}\omega'\xi_g\bigr]$.
3. Same-branch part of $\psi_1$, with $b_\sigma:=-\mathcal F+\sigma B_\parallel$: $\psi_1^{\sigma}=-ie^{-i\sigma\omega\lambda}\bigl[\lambda(ib_\sigma g'+\tfrac i2b_\sigma'g)+\tfrac{\lambda^2}2\sigma\omega'b_\sigma g\bigr]$. Its contribution is $2\sigma\operatorname{Re}\langle\psi_0\rvert\omega'\lvert\psi_1^\sigma\rangle=\sigma\lambda\int\frac{\mathrm dp}{2\pi}\omega'(b_\sigma\lvert g\rvert^2)'=-\sigma\lambda\langle\omega''b_\sigma\rangle_g$; the $\lambda^2$ term is imaginary and drops out.
4. Other-branch part: $\psi_1^{-\sigma}=-ie^{i\sigma\omega\lambda}\int_0^\lambda\mathrm ds\,e^{-2i\sigma\omega s}\bigl[q+\sigma\omega'B_\perp g\,s\bigr]$, $q:=iB_\perp g'+\tfrac i2(B_\perp'-\sigma\theta'\mathcal F)g$. With $\int_0^\lambda e^{-2i\sigma\omega s}\mathrm ds=\frac{1-e^{-2i\sigma\omega\lambda}}{2i\sigma\omega}$ and $\int_0^\lambda s\,e^{-2i\sigma\omega s}\mathrm ds=\frac{i\sigma\lambda e^{-2i\sigma\omega\lambda}}{2\omega}-\frac{1-e^{-2i\sigma\omega\lambda}}{4\omega^2}$, the contribution is $2\operatorname{Re}\langle\psi_0\rvert u\lvert\psi_1^{-\sigma}\rangle=\lambda\langle\theta'\omega'B_\perp\rangle_g+\sigma\operatorname{Re}\int\frac{\mathrm dp}{2\pi}(e^{2i\sigma\omega\lambda}-1)R_\sigma$. The term $s\,e^{-2i\sigma\omega s}$ meets the phase $e^{2i\sigma\omega\lambda}$ of $\langle\psi_0\rvert\cdots e^{i\sigma\omega\lambda}$ and gives a part that is linear in $\lambda$ and does not oscillate.

Adding 2–4 and using $B_\parallel'=g_\zeta\omega'+\theta'B_\perp$, exactly for all $\lambda\ge0$ and every smooth $g$:

$$
v^{(1)}(\lambda)=\lambda\,\langle a_{k,\sigma}\rangle_g+\rho(\lambda),\qquad
\rho(\lambda)=\underbrace{\sigma\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,B_\parallel'\,\xi_g}_{=:\rho_\infty}+\sigma\operatorname{Re}\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,e^{2i\sigma\omega\lambda}R_\sigma(p),
\tag{4.6}
$$

$$
R_\sigma:=-i\theta'\Bigl[B_\perp\,g^*g'+\tfrac12\bigl(\omega(B_\perp/\omega)'-\sigma\theta'\mathcal F\bigr)\lvert g\rvert^2\Bigr],
$$

with the coefficient $a_{k,\sigma}=g_\zeta\omega'^2+\theta'\omega'B_\perp-\sigma\omega''b_\sigma=\omega'B_\parallel'-\omega''B_\parallel+\sigma\mathcal F\omega''$. With $\omega\omega'=\tau^2\sin p\cos p$, $\omega'^2+\omega\omega''=\tau^2\cos2p$ and $\omega''=\tau^2(M^2\cos2p-\tau^2\sin^4p)/\omega^3$:

$$
\boxed{\;a_{k,\sigma}(p)=\frac{g_{\zeta,k}\tau_k^4\sin^2p-G_kM_k\tau_k^2\cos2p}{\omega_k^2}+\sigma\,\mathcal F_k\,\frac{\tau_k^2\bigl(M_k^2\cos2p-\tau_k^2\sin^4p\bigr)}{\omega_k^3}\;},\qquad\lvert a_{k,\sigma}(p)\rvert\le\frac{\tau_k^2}{M_k}\,\mathsf b_k .
\tag{4.7}
$$

**The rest $\rho$.** It is bounded for all $\lambda\ge0$: with $\lvert\theta'\rvert\le\tau/M$, $\lvert\omega'\rvert\le\tau$, $\lvert B_\perp\rvert\le\lvert G-g_\zeta M\rvert$, $\lvert\omega(B_\perp/\omega)'\rvert\le\frac\tau M\lvert G-g_\zeta M\rvert$, $\lvert\xi_g\rvert\le\lvert g\rvert\lvert g'\rvert$ and the Cauchy–Schwarz inequality,

$$
\lvert\rho(\lambda)\rvert\le\frac{\tau}{M}\,\mathsf b\,\Bigl(3X_g+\frac{\tau}{2M}\Bigr),\qquad\lvert\rho(\lambda)-\rho_\infty\rvert\le\frac{\tau}{M}\,\mathsf b\,\Bigl(X_g+\frac{\tau}{2M}\Bigr)\qquad(\text{exact, all }\lambda\ge0),
\tag{4.8}
$$

where $X_g\le(\sum_xx^2p_x(0))^{1/2}$ because $\lVert\hat x\psi\rVert^2=X_g^2+\tfrac14\langle\theta'^2\rangle_g$. It does not grow: no power of $\lambda$ multiplies the oscillating integral, since a branch start has no zeroth-order amplitude in the other branch. For $\lambda\to\infty$, $\rho(\lambda)\to\rho_\infty$ with $\rho(\lambda)-\rho_\infty=O(\lambda^{-1/2})$ (order estimate, constant depending on $g$): the phase $2\omega\lambda$ is stationary only at $p\in\{0,\pm\tfrac\pi2,\pi\}$, where $\omega''\neq0$, and labels with different $\omega(p)$ dephase. $\rho_\infty$ is the change of the velocity $\sigma\omega'$ due to the local parameters at the position of the start; it shifts by $\sigma x_c\langle B_\parallel'\rangle_g$ when the start is translated by $x_c$.

**Beyond first order (fixed chain).** On the chain $\lvert x\rvert\le L$, $\lVert C_1\rVert\le L\,\mathsf b'$, $\lVert V_0\rVert\le\tau$, $\lVert\hat v_1\rVert\le\tau\lvert g_\zeta\rvert L$ (open ends included) and $\lVert\psi-\psi_0\rVert\le\lambda\lVert C_1\rVert$, $\lVert\psi-\psi_0-\psi_1\rVert\le\tfrac12\lambda^2\lVert C_1\rVert^2$, so

$$
\bigl\lvert v_L(\lambda)-v_L^{(0)}(\lambda)-v_L^{(1)}(\lambda)\bigr\rvert\le2\tau L^2\,\mathsf b'\,\bigl(\lambda^2\mathsf b'+\lambda\lvert g_\zeta\rvert\bigr)\qquad(\text{exact, all }\lambda\ge0).
\tag{4.9}
$$

This bound grows with $\lambda$ and with $L$. An acceleration read off (4.6) over a window ending at $\lambda$ therefore carries the correction $O(\tau L^2\mathsf b'^2\lambda)$ of second order in the gradients, which is outside the scope; (4.6)–(4.8) are statements about the first-order coefficient in the stated order of limits.

### Step 4. Long window and sharp label (item 1b)

The first-order acceleration is $a^{(1)}(\lambda):=\mathrm dv^{(1)}/\mathrm d\lambda=\langle a_{k,\sigma}\rangle_g+\rho'(\lambda)$.

**Families of starts.** A *sharp-label family at $p_0$* is a sequence of branch starts $B_{k,\sigma}[g]$, all concentrated at $p_0$, with $\delta p\to0$; nothing else is assumed. For it $\lvert g\rvert^2\frac{\mathrm dp}{2\pi}\to\delta_{p_0}$ weakly (Chebyshev's inequality), so $\langle F\rangle_g\to F(p_0)$ for every continuous periodic $F$. The start enters (4.6) also through $\xi_g$, which $\delta p\to0$ does not fix. For smooth periodic $F$ let

$$
\Xi_g[F]:=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,\xi_g(p)\,F(p),\qquad
\mathcal K_1(x_c,\beta):\ \text{sharp-label families with}\ \ \Xi_g[F]\ \xrightarrow[\delta p\to0]{}\ x_c\,F(p_0)+\beta\,F'(p_0)\ \ \text{for every smooth periodic }F .
\tag{4.26}
$$

Here $x_c,\beta\in\mathbb R$, and necessarily $x_c=\lim\bar x_{\rm lab}(0)$, $\beta=\lim\Xi_g[\sin(p-p_0)]$. A family of $\mathcal K_1(x_c,0)$ is *centred at $x_c$*. The class is a property of the starts, not of the contract (M9). Examples: for $x_c\in\mathbb Z$, $\mu\in\mathbb R$ and real $\gamma$, $g=e^{-ipx_c}e^{-i\mu(1-\cos(p-p_0))}\gamma$ has $\xi_g=\bigl(x_c+\mu\sin(p-p_0)\bigr)\lvert g\rvert^2$. With $\mu=0$ and any real $\gamma$ of spread $\delta p\to0$ the family is centred at $x_c$. The family $E(x_c,\mu)$ has $\gamma(p_0+y)=(2\pi/\epsilon)^{1/2}\Gamma_0(y/\epsilon)$ for $\lvert y\rvert\le\pi$, with a fixed real, smooth, even $\Gamma_0$ supported in $[-1,1]$, $\int\Gamma_0^2=1$, and $\epsilon\to0$; then $\delta p\propto\epsilon$ and, by Taylor expansion, $\Xi_g[F]=x_c\langle F\rangle_g+\mu\,\delta p^2\bigl[F'(p_0)+O(\delta p^2)\bigr]$. So $E$ with $\mu\,\delta p^2\to\beta$ belongs to $\mathcal K_1(x_c,\beta)$, and for $E$ with $\mu\,\delta p^2\to\infty$, called $E_\infty$, $\Xi_g[F]$ diverges whenever $F'(p_0)\neq0$.

1. **Long window at fixed $g$.** $\overline{a^{(1)}}=\bigl[v^{(1)}(\lambda_1+\Lambda)-v^{(1)}(\lambda_1)\bigr]/\Lambda$, so by (4.6), (4.8), for all $\lambda_1\ge0$, $\Lambda>0$:
$$
\Bigl\lvert\overline{a^{(1)}}-\langle a_{k,\sigma}\rangle_g\Bigr\rvert\le\frac{2\tau}{M\Lambda}\,\mathsf b\,\Bigl(X_g+\frac{\tau}{2M}\Bigr)\ \ (\text{exact}),\qquad\lim_{\Lambda\to\infty}\overline{a^{(1)}}=\langle a_{k,\sigma}\rangle_g .
\tag{4.10}
$$
2. **Then sharp label (every sharp-label family; no class is needed).** On the arc $(p_0-\pi,p_0+\pi]$ the weight has mean $p_0$, so the linear Taylor term of $a_{k,\sigma}$ averages to zero:
$$
\bigl\lvert\langle a_{k,\sigma}\rangle_g-a_{k,\sigma}(p_0)\bigr\rvert\le\tfrac12\sup_p\lvert a_{k,\sigma}''\rvert\,\delta p^2\ \ (\text{exact}),\qquad\text{drift acceleration}=\lim_{\delta p\to0}\lim_{\Lambda\to\infty}\overline{a^{(1)}}=a_{k,\sigma}(p_0).
\tag{4.11}
$$
3. **Sharp label at fixed $\lambda$.** With $g^*g'=\tfrac12(\lvert g\rvert^2)'-i\xi_g$ and one integration by parts in (4.6), exactly,
$$
v^{(1)}(\lambda)=\Bigl\langle a_{k,\sigma}\lambda-\lambda\,n_k\cos(2\omega\lambda)-\tfrac12\bigl(\tfrac{u'}{\omega}B_\perp+\sigma\theta'^2\mathcal F\bigr)\sin(2\omega\lambda)\Bigr\rangle_g+\sigma\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,\xi_g\bigl[B_\parallel'-\theta'B_\perp\cos(2\omega\lambda)\bigr],
\tag{4.12}
$$
$$
n_k(p):=\omega'\theta'B_\perp=-\frac{\tau^4M\,(G-g_\zeta M)\,\sin^2(2p)}{4\,\omega^4}.
$$
Differentiating in $\lambda$, with $A_\sigma:=(u'+\theta'\omega')B_\perp+\sigma\omega\theta'^2\mathcal F$ and $w:=2\omega\theta'B_\perp=-\tau^2M(G-g_\zeta M)\sin2p/\omega^2$ (so $\omega'w=2\omega n_k$):
$$
\begin{aligned}
&a^{(1)}(\lambda)=\Bigl\langle a_{k,\sigma}-A_\sigma\cos(2\omega\lambda)+\omega'w\,\lambda\sin(2\omega\lambda)\Bigr\rangle_g+\sigma\,\Xi_g\bigl[w\sin(2\omega\lambda)\bigr]\qquad(\text{exact, every smooth }g,\ \text{all }\lambda\ge0);\\
&\text{every sharp-label family:}\quad\lim_{\delta p\to0}a^{(1)}(\lambda)\ \text{exists}\iff\lim_{\delta p\to0}\Xi_g\bigl[w\sin(2\omega\lambda)\bigr]\ \text{exists}.
\end{aligned}
\tag{4.27}
$$
The second line holds in both directions because the average $\langle\cdot\rangle_g$ converges to the value of its argument at $p_0$ for every sharp-label family; the limit is then that value plus $\sigma\lim\Xi_g[w\sin(2\omega\lambda)]$. Evaluating $\Xi_g$ by (4.26), and using $w\equiv0$, $A_\sigma=\sigma\omega\theta'^2\mathcal F$ for $G=g_\zeta M$:
$$
\begin{aligned}
\text{families of }\mathcal K_1(x_c,\beta):\quad a^{(1)}_{\rm sharp}(\lambda)&=\Bigl[a_{k,\sigma}-A_\sigma\cos(2\omega\lambda)+w\,(\omega'\lambda+\sigma x_c)\sin(2\omega\lambda)+\sigma\beta\bigl(w'\sin(2\omega\lambda)+2\omega'w\,\lambda\cos(2\omega\lambda)\bigr)\Bigr]_{p=p_0},\\
G_k=g_{\zeta,k}M_k,\ \text{every sharp-label family}:\quad a^{(1)}_{\rm sharp}(\lambda)&=\Bigl[a_{k,\sigma}-\sigma\,\omega\,\theta'^2\mathcal F\cos(2\omega\lambda)\Bigr]_{p=p_0}.
\end{aligned}
\tag{4.13}
$$
For centred families ($\beta=0$) the first line is $\bigl[a_{k,\sigma}-A_\sigma\cos(2\omega\lambda)+2\omega\theta'B_\perp(\omega'\lambda+\sigma x_c)\sin(2\omega\lambda)\bigr]_{p_0}$. Already within $\mathcal K_1$ the limit depends on the family through $\beta$, also at rest if $G_k\neq g_{\zeta,k}M_k$ (there $w=0$ and $w'\neq0$). Outside these classes it need not exist: for $G_k\neq g_{\zeta,k}M_k$ and the family $E_\infty$ it does not exist, at every $p_0$ and for all $\lambda$ outside a discrete set. Indeed $\partial_p\bigl(w\sin(2\omega\lambda)\bigr)_{p_0}=w'\sin(2\omega\lambda)+2\omega'w\,\lambda\cos(2\omega\lambda)$ is an entire function of $\lambda$ that is not identically zero, since $\omega'w=2\omega n_k\neq0$ for $\sin2p_0\neq0$, and $w'(p_0)=-2\tau^2M(G-g_\zeta M)\cos(2p_0)/\omega^2\neq0$ for $\sin2p_0=0$ (rest included). For other families outside $\mathcal K_1$ only (4.27) is established.
4. **The two orders.** Let $\lambda_2:=\lambda_1+\Lambda$, $[f]_{\lambda_1}^{\lambda_2}:=f(\lambda_2)-f(\lambda_1)$, $h:=\theta'B_\perp=w/2\omega$, $S_\sigma:=\tfrac{u'}{\omega}B_\perp+\sigma\theta'^2\mathcal F$. From (4.12), in which the term $\Xi_g[B_\parallel']$ cancels, exactly for every smooth $g$:
$$
\overline{a^{(1)}}=\Bigl\langle a_{k,\sigma}-\frac{n_k}{\Lambda}\bigl[\lambda\cos(2\omega\lambda)\bigr]_{\lambda_1}^{\lambda_2}-\frac{S_\sigma}{2\Lambda}\bigl[\sin(2\omega\lambda)\bigr]_{\lambda_1}^{\lambda_2}\Bigr\rangle_g-\frac{\sigma}{\Lambda}\,\Xi_g\Bigl[h\,\bigl[\cos(2\omega\lambda)\bigr]_{\lambda_1}^{\lambda_2}\Bigr].
\tag{4.28}
$$
For a family of $\mathcal K_1(x_c,\beta)$, with $\partial_p\bigl(h\cos(2\omega\lambda)\bigr)=h'\cos(2\omega\lambda)-2n_k\lambda\sin(2\omega\lambda)$, $\phi_j:=2\omega(p_0)\lambda_j$ and all functions taken at $p_0$:
$$
\lim_{\delta p\to0}\overline{a^{(1)}}=a_{k,\sigma}-n_k\bigl[\cos\phi_2-2\sigma\beta\sin\phi_2\bigr]+\frac{R}{\Lambda},\qquad\lvert R\rvert\le2\lvert n_k\rvert\lambda_1(1+2\lvert\beta\rvert)+\lvert S_\sigma\rvert+2\lvert x_ch+\beta h'\rvert\quad(\text{exact}).
$$
If $n_k(p_0)=0$ this tends to $a_{k,\sigma}(p_0)$ for $\Lambda\to\infty$. If $n_k(p_0)\neq0$ it has no limit, because $\cos\phi_2-2\sigma\beta\sin\phi_2=\sqrt{1+4\beta^2}\cos(\phi_2+\phi_\beta)$ and $\omega(p_0)>0$. Hence
$$
\text{families of }\mathcal K_1(x_c,\beta),\ \text{every }x_c,\beta:\qquad\lim_{\Lambda\to\infty}\lim_{\delta p\to0}\overline{a^{(1)}}\ \text{exists}\iff n_k(p_0)=0\iff(G_k-g_{\zeta,k}M_k)\sin^2(2p_0)=0,\ \text{and then equals }a_{k,\sigma}(p_0).
\tag{4.14}
$$
The limits commute when both orders exist and agree; the order "long window first" exists for every sharp-label family and gives $a_{k,\sigma}(p_0)$ (4.11). For all sharp-label families:
$$
\begin{array}{l|l|l|l|l}
\text{level data}&\text{label }p_0&\text{sharp-label family}&\lim_{\Lambda\to\infty}\lim_{\delta p\to0}\overline{a^{(1)}}&\text{commute}\\\hline
G_k=g_{\zeta,k}M_k&\text{every }p_0&\text{every family}&a_{k,\sigma}(p_0)&\text{yes}\\
G_k\neq g_{\zeta,k}M_k&\sin2p_0=0:\ \text{rest},\ \pm\tfrac\pi2,\ \pi&\mathcal K_1(x_c,\beta),\ \text{every }x_c,\beta&a_{k,\sigma}(p_0)&\text{yes}\\
G_k\neq g_{\zeta,k}M_k&\sin2p_0\neq0&\mathcal K_1(x_c,\beta),\ \text{every }x_c,\beta&\text{none: }\limsup,\liminf=a_{k,\sigma}(p_0)\pm\lvert n_k(p_0)\rvert\sqrt{1+4\beta^2}&\text{no}\\
G_k\neq g_{\zeta,k}M_k&\text{every }p_0,\ \text{rest included}&E_\infty&\text{none: the inner limit does not exist}&\text{no}\\
G_k\neq g_{\zeta,k}M_k&\text{every }p_0&\text{any other family}&\text{exists iff (4.28) converges for }\delta p\to0\text{, then for }\Lambda\to\infty&\text{not determined}
\end{array}
\tag{4.29}
$$
Proofs of the rows. Row 1: $n_k\equiv h\equiv0$, so (4.28) is an average $\langle\cdot\rangle_g$, which converges for every sharp-label family to $a_{k,\sigma}(p_0)-\frac{\sigma\theta'^2\mathcal F}{2\Lambda}[\sin(2\omega\lambda)]_{\lambda_1}^{\lambda_2}$ at $p_0$. Rows 2, 3: (4.14); in row 3 what converges is the window average minus the explicit term $-n_k[\cos\phi_2-2\sigma\beta\sin\phi_2]$. Row 4: at fixed $\lambda_1$, $\partial_p\bigl(h[\cos(2\omega\lambda)]_{\lambda_1}^{\lambda_2}\bigr)_{p_0}=\bigl[h'\cos(2\omega\lambda)-2n_k\lambda\sin(2\omega\lambda)\bigr]_{\lambda_1}^{\lambda_2}$ is an entire function of $\lambda_2$ that is not identically zero ($n_k(p_0)\neq0$, or $h'(p_0)=-\tau^2M(G-g_\zeta M)\cos(2p_0)/\omega^3\neq0$ for $\sin2p_0=0$), so for all $\Lambda$ outside a discrete set $\Xi_g[\cdot]$ in (4.28) diverges while the average converges. Row 5: nothing beyond (4.28) is proven.

The reason for the rows "no" in $\mathcal K_1$ is that the term $\lambda\,n_k\cos(2\omega\lambda)$ of (4.12) dephases only at fixed $g$; the bound (4.10) contains $X_g$, which diverges for $\delta p\to0$. Near rest $n_k=-(G/M-g_\zeta)v^2+O(v^4)$ in the variable $v$ of Step 6.

### Step 5. Comparison with the bare start $S_k$ (item 1c)

At $p_0=0$: $\omega'=B_\perp=B_\parallel'=h=n_k=0$, $\theta'=-\tau/M$, $A_\sigma=\sigma\tau^2\mathcal F/M$, $S_\sigma=\sigma\tau^2\mathcal F/M^2$, $h'=-\tau^2(G-g_\zeta M)/M^2$, $B_\parallel''=g_\zeta\omega''+h'=\tau^2(2g_\zeta M-G)/M^2$. Integrating (4.12) over $[0,\lambda]$ at fixed $g$ and then taking the limit with (4.26), for sharp-label families at rest of the class $\mathcal K_1(x_c,\beta)$, every $x_c$:

$$
\bar x^{(1)}_{\rm lab}(\lambda)-\bar x_{\rm lab}(0)\ \xrightarrow[\delta p\to0]{}\ \tfrac12\,a_{k,\sigma}(0)\,\lambda^2-\sigma\,\frac{\tau^2\mathcal F}{2M^3}\sin^2(M\lambda)+\sigma\beta\,\frac{\tau^2}{M^2}\Bigl[(2g_\zeta M-G)\,\lambda+(G-g_\zeta M)\,\frac{\sin(2M\lambda)}{2M}\Bigr].
\tag{4.15}
$$

For centred families ($\beta=0$) the right side is $\tfrac12a_{k,\sigma}(0)\lambda^2-\sigma\frac{\tau^2\mathcal F}{2M^3}\sin^2(M\lambda)$. Against (3.15) and (3.26), for $\sigma=+$:
- the secular term is unchanged at rest, $a_{k,+}(0)=a_{\rm sec}$ (every $g$); for all labels $a_{\rm sec}(p)$ of (3.26) is replaced by $a_{k,\sigma}(p)$, and $a_{\rm sec}(p)=\sum_\sigma w_\sigma a_{k,\sigma}(p)$ with the branch weights $w_\pm=\tfrac12(1\pm M/\omega)$ of the bare label (these are not the function $w$ of Step 4; checked term by term with $\sum_\sigma w_\sigma=1$, $\sum_\sigma\sigma w_\sigma=M/\omega$);
- the term $\frac{\tau^2G}{2M^2}\lambda\sin(2M\lambda)$ of (3.15) is absent for every family of $\mathcal K_1$ at rest: in (3.26) the term $\lambda^2A_2$ is absent and $A_1$ is replaced by $-n_k$ (every $g$, (4.12)), which vanishes at $p=0$, whereas $A_1(0)=\tau^2G/M$ for $S_k$;
- in the term $\sin^2(M\lambda)$ the coefficient $Mg_\zeta-\mathcal F-G$ is replaced by $-\mathcal F$ ($A_0$ at $p=0$ keeps only its $\mathcal F$ part), and $\langle\nu_0\rangle_W$ is replaced by $\rho_\infty$. For centred families no $G$ or $g_\zeta$ remains in the oscillation; for $\beta\neq0$ the last term of (4.15) adds a constant velocity and an oscillation $\sin(2M\lambda)$ that contain them;
- the second limit of (3.28), which does not exist for $S_k$ if $G_k\neq0$, exists and equals the first for every family of $\mathcal K_1$ at rest, and for every sharp-label family if $G_k=g_{\zeta,k}M_k$ ((4.14), (4.29)); for $E_\infty$ and $G_k\neq g_{\zeta,k}M_k$ it does not exist.

Why: the bare label is $\lvert1\rangle=\cos\tfrac\theta2\,\beta_+-\sin\tfrac\theta2\,\beta_-$, with the amplitude $-\sin\tfrac\theta2=\tfrac{\tau p}{2M}+O(p^3)$ in the lower branch. It vanishes at $p=0$, but $\hat x=i\partial_p$ in $C_1$ and $\hat v_1$ differentiates it, so the zeroth-order coherence between the branches contributes at order one for $W\to\infty$; with $G\neq0$ the beat frequency $2M_{x,k}$ varies along the start, which gives $\lambda\sin(2M\lambda)$. A branch start has no such coherence. Its branches are mixed only at first order by $C_1^{\rm od}$ (Step 2), whose $G$ and $g_\zeta$ part is proportional to $B_\perp$ and vanishes at $p=0$ (for $\beta\neq0$ its $p$-derivative at $p=0$ enters, with the weight $\beta$); the part $-\tfrac{\theta'}2\mathcal F\rho_2$ remains, because the start is not adapted to the gradient $\mathcal F$, and it is the same for both starts.

### Step 6. Rest and motion (item 2)

**(a)** The drift acceleration is (4.7) at $p=p_0$, for every $p_0$; it has the period $\pi$ in $p_0$. At rest:

$$
\boxed{\;a_{k,\sigma}(0)=\frac{\tau_k^2}{M_k}\,\bigl(\sigma\,\mathcal F_k-G_k\bigr)\;}
\tag{4.16}
$$

**(b)** $v:=\sigma\omega_k'(p_0)$ is the sharp-label limit of (4.5): the velocity of the branch start at zero gradients, in position labels per unit of $\lambda$, $\lvert v\rvert<\tau_k$. $a_{k,\sigma}$ is even and $v$ is odd in $p_0$, and $v=\sigma\frac{\tau^2}{M}p_0\,(1+O(p_0^2))$ is invertible near $p_0=0$; on this branch of $p_0(v)$, which contains rest, $a_{k,\sigma}$ is a smooth function of $v^2$ and $p_0^2=\frac{M^2}{\tau^4}v^2+O(v^4)$. Expanding (4.7) with $\cos2p=1-2p^2$, $\omega^{-2}=M^{-2}(1-\tau^2p^2/M^2)$, $\omega^{-3}=M^{-3}(1-\tfrac{3\tau^2p^2}{2M^2})$:

$$
a_{k,\sigma}=\frac{\tau^2}{M}(\sigma\mathcal F-G)+c_{2,k,\sigma}\,v^2+O(v^4),\qquad
c_{2,k,\sigma}=g_\zeta+\frac GM\Bigl(1+\frac{2M^2}{\tau^2}\Bigr)-\sigma\,\frac{\mathcal F}{M}\Bigl(\frac32+\frac{2M^2}{\tau^2}\Bigr).
\tag{4.17}
$$

| gradient | value at $v=0$ | coefficient of $v^2$ |
| --- | --- | --- |
| $g_{\zeta,k}$ | $0$ | $g_{\zeta,k}$ |
| $G_k$ | $-\tau_k^2G_k/M_k$ | $\frac{G_k}{M_k}\bigl(1+\frac{2M_k^2}{\tau_k^2}\bigr)$ |
| $\mathcal F_k$ | $\sigma\,\tau_k^2\mathcal F_k/M_k$ | $-\sigma\frac{\mathcal F_k}{M_k}\bigl(\frac32+\frac{2M_k^2}{\tau_k^2}\bigr)$ |

The two table columns are exact; $O(v^4)$ is an order estimate for $v\to0$ at fixed level data. The terms $2M^2/\tau^2$ come from $\cos2p=1-2p^2$, that is from the lattice.

**(c)** In the manner of (3.2), for the level $k$ of $C_{\rm eff}$: $E^\pm_{x,k}=V_{x,k}\pm M_{x,k}$ (modulo a common constant), gap $\Delta_{x,k}=2M_{x,k}$, norm of the hopping block $t_{x,k}=\tfrac12\tau_k\zeta_{x,k}$. For the linear profiles the differences $\delta E^\pm_k=\pm G_k-\mathcal F_k$, $\delta\Delta_k=2G_k$, $\delta t_k=\tfrac12\tau_kg_{\zeta,k}$ do not depend on $x$; the reference values are $\Delta_k:=\Delta_{0,k}=2M_k$ and $\bar t_k:=\tfrac12(t_{0,k}+t_{-1,k})=\tfrac12\tau_k$. Then

$$
a_{k,\sigma}(0)=-\sigma\,\frac{8\,\bar t_k^{\,2}}{\Delta_k}\,\delta E^{\sigma}_k,\qquad
c_{2,k,\sigma}=\frac{\delta t_k}{\bar t_k}+\frac{\delta\Delta_k}{\Delta_k}\Bigl(1+\frac{\Delta_k^2}{8\bar t_k^{\,2}}\Bigr)+\sigma\,\frac{\delta E^+_k+\delta E^-_k}{\Delta_k}\Bigl(\frac32+\frac{\Delta_k^2}{8\bar t_k^{\,2}}\Bigr).
\tag{4.18}
$$

These are functions of $C_{\rm eff}$ modulo $\mathbb 1$. They are invariant under every replacement (8.1), place-dependent $\alpha_h$ included (which leave $C_{\rm eff}$ unchanged), under $Z$ (which shifts $C_{\rm eff}$ by a multiple of $\mathbb 1$, (8.20)), and under the common phases of $\chi$ and of the state. The values $E^\pm_{x,k}$ and $V_{x,k}$ themselves are not invariant. The reference values belong to the origin $x=0$ of the linear profiles; moving the origin changes (4.18) at second order in the gradients.

### Step 7. Level-independent fall (item 3)

(4.16) and (4.17) hold for every member of $\mathcal C_{\rm lev}$ with linear level profiles, every $N\ge2$ and all admissible level data. Each statement below is an equality between these numbers, so both directions follow from (4.16), (4.17) alone; (b) also uses that a smooth function of $v^2$ fixes its Taylor coefficients.

**(a) From rest.**

$$
\begin{aligned}
a_{k,+}(0)=a_+\ \forall k&\iff\frac{\tau_k^2}{M_k}(\mathcal F_k-G_k)=a_+\ \forall k,\qquad
a_{k,-}(0)=a_-\ \forall k\iff\frac{\tau_k^2}{M_k}(\mathcal F_k+G_k)=-a_-\ \forall k,\\
\text{both, with }a_+=a_-=a&\iff\mathcal F_k=0\ \text{ and }\ \frac{\tau_k^2G_k}{M_k}=-a\quad\forall k .
\end{aligned}
\tag{4.19}
$$

Both without equal values: $\tau_k^2\mathcal F_k/M_k=\tfrac12(a_+-a_-)$ and $\tau_k^2G_k/M_k=-\tfrac12(a_++a_-)$ for all $k$. In well-defined terms the first condition is that $\bar t_k^{\,2}\,\delta E^+_k/\Delta_k$ does not depend on $k$, and $\mathcal F_k=0$ is $\delta E^+_k+\delta E^-_k=0$, the term $\kappa\langle X\rangle$ of the records included.

**(b) In motion (upper branch).** $a_{k,+}$ is the same function of $v$ through the order $v^2$ for all $k$ if and only if (i) $\frac{\tau_k^2}{M_k}(\mathcal F_k-G_k)=a_0$ for all $k$ and (ii) $c_{2,k,+}$ does not depend on $k$. With (i) inserted in (4.17), exactly,

$$
c_{2,k,+}=g_{\zeta,k}-\frac{\mathcal F_k}{2M_k}-\frac{a_0}{\tau_k^2}\Bigl(1+\frac{2M_k^2}{\tau_k^2}\Bigr)\quad\text{independent of }k .
\tag{4.20}
$$

In the regime $M_k\ll\tau_k$: $c_{2,k,+}=c^{(0)}_{2,k}+\frac{2M_k}{\tau_k^2}(G_k-\mathcal F_k)$ with $c^{(0)}_{2,k}:=g_{\zeta,k}+\frac{G_k}{M_k}-\frac{3\mathcal F_k}{2M_k}$. To leading order the condition is (i), which is unchanged, together with

$$
c^{(0)}_{2,k}=g_{\zeta,k}-\frac{\mathcal F_k}{2M_k}-\frac{a_0}{\tau_k^2}\quad\text{independent of }k;\qquad\text{then exactly}\quad c_{2,k,+}-c_{2,k',+}=-2a_0\Bigl(\frac{M_k^2}{\tau_k^4}-\frac{M_{k'}^2}{\tau_{k'}^4}\Bigr).
\tag{4.21}
$$

The dropped terms are the lattice terms $\frac{2M_k}{\tau_k^2}(G_k-\mathcal F_k)=-2a_0M_k^2/\tau_k^4$, of relative order $M_k^2/\tau_k^2$. For the lower branch replace $\mathcal F_k\to-\mathcal F_k$.

**(c) The class $\mathcal C$** ($\tau_k=\tau$, $M_k=m+d_k$, $G_k=mg_m+d_kg_\eta$, $\mathcal F_k=\mathcal F$, $g_{\zeta,k}=g_\zeta$). Here $a_{k,\pm}(0)=-\tau^2\bigl[g_\eta+\frac{m(g_m-g_\eta)\mp\mathcal F}{m+d_k}\bigr]$, the $d_k$ are distinct and $d\mapsto1/(m+d)$ is injective. For every $N\ge2$:

$$
a_{k,+}(0)\ \text{level-independent}\iff\mathcal F=m(g_m-g_\eta);\qquad a_{k,-}(0)\ \text{level-independent}\iff\mathcal F=-m(g_m-g_\eta);\qquad\text{value }-\tau^2g_\eta\text{ in each case};
\tag{4.22}
$$

both hold if and only if $\mathcal F=0$ and $g_m=g_\eta$, and then the two values are equal automatically. Through the order $v^2$ (upper branch): (i) gives $\mathcal F=m(g_m-g_\eta)$, $a_0=-\tau^2g_\eta$, and (4.20) requires $h(M_k):=-\frac{\mathcal F}{2M_k}+\frac{2g_\eta}{\tau^2}M_k^2$ to be the same for all $k$.

$$
\begin{aligned}
N=2:&\quad\mathcal F=m(g_m-g_\eta)=-\frac{4g_\eta}{\tau^2}\,m\Bigl(m+\frac{\pi}{\lambda_0}\Bigr)\Bigl(2m+\frac{\pi}{\lambda_0}\Bigr),\quad g_\eta,\ g_\zeta\ \text{free};\\
N\ge3:&\quad g_m=g_\eta=0,\ \mathcal F=0,\quad g_\zeta\ \text{free}\quad(\text{so }a_0=0);\\
M_k\ll\tau,\ \text{leading order, every }N\ge2:&\quad\mathcal F=0,\ g_m=g_\eta=:g,\quad g_\zeta\ \text{free}.
\end{aligned}
\tag{4.23}
$$

Proofs. $N=2$: $h(M_0)=h(M_1)$ is $\mathcal F=-\frac{4g_\eta}{\tau^2}M_0M_1(M_0+M_1)$. $N\ge3$: $h(M)=c$ is the cubic $\frac{2g_\eta}{\tau^2}M^3-cM-\tfrac12\mathcal F=0$, whose roots sum to zero, so it cannot have three distinct positive roots unless it vanishes identically. Leading order: $g_\zeta+g_\eta-\frac{\mathcal F}{2M_k}$ is level-independent if and only if $\mathcal F=0$. In the last case $a_{k,\pm}=-\tau^2g+\bigl[g_\zeta+g\bigl(1+\frac{2M_k^2}{\tau^2}\bigr)\bigr]v^2+O(v^4)$: the level enters only through the lattice term $2gM_k^2v^2/\tau^2$. Exactly, for $N\ge3$, a fall that is level-independent through the order $v^2$ exists in $\mathcal C$ only with $a_0=0$. (In this step $h$ is the function just defined, not $\theta'B_\perp$ of Step 4.)

### Step 8. Clock rate and fall in the class $\mathcal C$ (item 4)

The rate (3.23) in its wide-start form is $r(x)=\eta_x/\lambda_0=\frac N{2\pi}(E^+_{x,k+1}-E^+_{x,k})$ with $\eta_x=1+g_\eta x$; it depends on the place if and only if $g_\eta\neq0$.

**(a)** A place-dependent rate does not imply a level-independent fall: in the variant (b2) ($g_\eta=g\neq0$, the others zero) $a_{k,+}(0)=-\tau^2g\,d_k/(m+d_k)$, which is zero for $k=0$ and not for $k\ge1$. Conversely, by (4.22) a level-independent $a_{k,+}(0)$ has the value $-\tau^2g_\eta$. So the converse holds exactly when the common value is not zero; it fails when the value is zero, for example Model D, or $g_\eta=0$ with $\mathcal F=m\,g_m\neq0$ (level-independent, no fall from rest, uniform rate).

**(b)** Let $a_{k,+}(0)=a\neq0$ for all $k$. By (4.22), $a=-\tau^2g_\eta$, and $g_\eta=\frac{r(x+1)-r(x)}{r(x)}$ at $x=0$ (for other $x$ the ratio is $g_\eta+O(g_\eta^2x)$). In the background of (17.15) with $K_x=\kappa xX$: $u_x=\kappa x\,e$, the distance of neighbouring positions is $d_0:=\kappa\sigma_X$, $\bar x=d_0\,\bar x_{\rm lab}\,\hat e$ with $\hat e:=e/\sigma_X$, and (5.11) reads $v_{\max}=\tau d_0$. Hence, at first order in the gradients,

$$
\boxed{\;\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\Big\vert_{\rm drift}=d_0\,a_{k,+}(0)\,\hat e=-\,\frac{v_{\max}^2}{d_0}\;\frac{r(x+1)-r(x)}{r(x)}\;\hat e\;}
\tag{4.24}
$$

The relation contains $v_{\max}$, $d_0$ and the two rates only. $m$: no. $N$: no. $\lambda_0$: no (it cancels in the ratio). $\tau$: only inside $v_{\max}/d_0$. The level $k$: no. The plain gradient $\mathcal F$: no; it is fixed by the hypothesis, $\mathcal F=m(g_m-g_\eta)$, and $m$, $g_m$ enter this condition, not the relation. Accuracy: first order in the gradients with the ratio taken at the origin of the profiles; leading order in $\kappa$ (both sides are proportional to $d_0$; corrections as in Step 1(i)); $v_{\max}=\tau d_0$ is the bound (5.11) of the zeroth-order contract, and for $g_\zeta\neq0$ its form (17.7) gives $\tau d_0\max_x\tfrac12(\zeta_x+\zeta_{x-1})$, a difference of second order in (4.24).

**(c)** Lower branch: $a_{k,-}(0)=a\neq0$ for all $k$ if and only if $\mathcal F=-m(g_m-g_\eta)$ and $g_\eta\neq0$, with $a=-\tau^2g_\eta$. The relation (4.24) and the list of answers are the same, with the same rate (3.23). The level spacings of the lower on-site eigenvalues are $E^-_{x,k+1}-E^-_{x,k}=-\frac{2\pi}N\frac{\eta_x}{\lambda_0}$, of opposite sign and with the same ratio between neighbouring positions. The hypotheses of (b) and (c) differ in the sign of $\mathcal F$; they hold together only for $\mathcal F=0$, $g_m=g_\eta$.

**(d)** $r(x)$ and $r(x+1)$ are the rates (3.23) of two runs of the same description, with the starts $S_{\rm clock}(x)$ and $S_{\rm clock}(x+1)$. Each is the advance of the clock distribution per unit of $\lambda$, taken from readings of the clock label applied at stated values of $\lambda$. Comparing them relies on A8 (readings at stated $\lambda$, comparison at equal $\lambda$ through the order of $\lambda$) and on the contract being the same in both runs; $\lambda$ is not a time (M1). Only the ratio enters. It does not depend on the unit of $\lambda$ and equals a ratio of level spacings of $C_{\rm eff}$, $\frac{r(x+1)}{r(x)}=\frac{E^+_{x+1,k+1}-E^+_{x+1,k}}{E^+_{x,k+1}-E^+_{x,k}}$, which is well defined (M6). The acceleration is the second $\lambda$-derivative of $\bar x_{\rm lab}$, from readings of the position at stated $\lambda$ (A8). This is not a comparison of two clocks inside one description. No meaning beyond these definitions is attached to (4.24).

### Step 9. Variants (item 5)

From (4.16), (4.17) with $\tau_k=\tau$, $M_k=m+d_k$ and $\ell_k:=1+2M_k^2/\tau^2$. "Same for all levels" is meant for every $N\ge2$ and $g\neq0$ (in (f) also $c_{\rm f}\neq0$).

$$
\begin{array}{l|c|c|c|c|c|c}
\text{variant}&a_{k,+}(0)&a_{k,-}(0)&c_{2,k,\sigma}&\text{rest, upper}&\text{rest, lower}&\text{through }v^2\\\hline
\text{(a)}&-\tau^2g&-\tau^2g&g\,\ell_k&\text{yes}&\text{yes}&\text{exactly no; leading order in }M_k/\tau\text{ yes}\\
\text{(b1)}&-\tau^2\frac{mg}{M_k}&-\tau^2\frac{mg}{M_k}&\frac{mg}{M_k}\ell_k&\text{no}&\text{no}&\text{no}\\
\text{(b2)}&-\tau^2\frac{d_kg}{M_k}&-\tau^2\frac{d_kg}{M_k}&\frac{d_kg}{M_k}\ell_k&\text{no}&\text{no}&\text{no}\\
\text{(c)}&-\tau^2g&-\tau^2g&g\,(1+\ell_k)&\text{yes}&\text{yes}&\text{exactly no; leading order in }M_k/\tau\text{ yes}\\
\text{(d)}&+\frac{\tau^2\mathcal F}{M_k}&-\frac{\tau^2\mathcal F}{M_k}&-\sigma\frac{\mathcal F}{M_k}\bigl(\tfrac12+\ell_k\bigr)&\text{no}&\text{no}&\text{no}\\
\text{(f)}&-\tau^2g\bigl(1+\frac{c_{\rm f}}{M_k}\bigr)&-\tau^2g\bigl(1-\frac{c_{\rm f}}{M_k}\bigr)&g\,\ell_k+\sigma\frac{c_{\rm f}g}{M_k}\bigl(\tfrac12+\ell_k\bigr)&\text{no}&\text{no}&\text{no}
\end{array}
\tag{4.25}
$$

In (a) and (c) the only level dependence is the lattice term $2gM_k^2v^2/\tau^2$; at leading order in $M_k/\tau$, $a=-\tau^2g+g\,v^2$ in (a) and $a=-\tau^2g+2g\,v^2$ in (c). In (f) the number $c_{\rm f}$ enters only through the well-defined gradient $\delta E^+_k+\delta E^-_k=2c_{\rm f}g$: a factor $1+gx$ on the on-site block gives a level-independent fall only if it multiplies the traceless part alone, so the zero of the cost under the factor matters.

## Result

For every member of $\mathcal C_{\rm lev}$ with linear level profiles, every level $k$, both branches and every smooth $g$; first order in the gradients at fixed $\lambda$, start and chain, then a long chain; records at leading order ($O(\kappa^2\sigma_X^2)$ neglected, Step 1). Sharp-label limits are taken along sharp-label families, and the class of families is named with each statement (4.26).

1. **Drift.**
   - Every smooth $g$: $v^{(0)}=\sigma\langle\omega_k'\rangle_g$ (4.5). $v^{(1)}(\lambda)=\lambda\langle a_{k,\sigma}\rangle_g+\rho(\lambda)$ (4.6), with the coefficient (4.7) exact and the rest bounded for all $\lambda\ge0$ by (4.8), converging to $\rho_\infty$ like $\lambda^{-1/2}$. Long window: $\langle a_{k,\sigma}\rangle_g$ (4.10). Fixed-chain remainder: (4.9).
   - Every sharp-label family: the drift acceleration (long window, then sharp label) exists for every $p_0$ and is $a_{k,\sigma}(p_0)$ (4.11).
   - Sharp label at fixed $\lambda$: for every sharp-label family the limit exists if and only if $\Xi_g[w\sin(2\omega\lambda)]$ converges (4.27). It is (4.13) for the families of $\mathcal K_1(x_c,\beta)$, which contain the centred families ($\beta=0$), and for every sharp-label family if $G_k=g_{\zeta,k}M_k$. For $G_k\neq g_{\zeta,k}M_k$ it does not exist for the family $E_\infty$, at rest included.
   - Commutation, (4.14) and table (4.29): for the families of $\mathcal K_1(x_c,\beta)$ the limits commute if and only if $(G_k-g_{\zeta,k}M_k)\sin^2(2p_0)=0$, in particular at rest. If $G_k=g_{\zeta,k}M_k$ they commute for every sharp-label family and every $p_0$. If $G_k\neq g_{\zeta,k}M_k$ they do not commute for $E_\infty$ at any $p_0$, rest included, and for other families outside $\mathcal K_1$ this is not determined beyond (4.28).
   - Against $S_k$: (4.15), for sharp-label families of $\mathcal K_1(x_c,\beta)$ at rest, and Step 5.
2. **Rest and motion.** Closed form (4.7); at rest (4.16); expansion in $v$ and its coefficients (4.17); well-defined form (4.18).
3. **Level-independent fall.** From rest (4.19); through $v^2$ exactly (4.20) and for $M_k\ll\tau_k$ (4.21); in the class $\mathcal C$ (4.22), (4.23).
4. **Clock rate and fall in $\mathcal C$.** A place-dependent rate does not imply a level-independent fall; the converse holds exactly when the common acceleration is not zero. Then (4.24), for both branches, without $m$, $N$, $\lambda_0$, $k$, $\mathcal F$ and with $\tau$ only inside $v_{\max}/d_0$. The comparison of the rates relies on A8 (Step 8(d)).
5. **Variants.** Table (4.25).

Controls (M9): the lower branch ((4.16): sign of $\mathcal F_k$); the bare start (Step 5); the family $E_\infty$, a class of starts for which the sharp-label limit at fixed $\lambda$ and the commutation at rest fail; the variants (b1), (b2), (d), (f), where the fall depends on the level.

## Consistency checks

1. **Dimensions and zero gradients.** With $\tau,M,G,\mathcal F$ in units of $1/\lambda$ and $g_\zeta$, $x$, $p$ pure numbers: $a_{k,\sigma}\sim\tau^2\mathsf b/M\sim\lambda^{-2}$, $c_2v^2\sim\lambda^{-2}$, and (4.24) is a distance per $\lambda^2$ on both sides. With all gradients zero, $v^{(1)}=0$ and $v=\sigma\langle\omega'\rangle_g$ is constant, as it must be for eigenvectors of $C_0$. Passed.
2. **Upper branch at rest against 03-clock-and-fall, and (4.13) against (4.15).** $a_{k,+}(0)=\frac{\tau^2}{M_k}(\mathcal F-G_k)$ equals $a_{\rm sec}$ of (3.15) and (3.28); for all $p$, $\sum_\sigma w_\sigma a_{k,\sigma}(p)$ reproduces $a_{\rm sec}(p)$ of (3.26), which was derived there for another start; the $\mathcal F$ oscillation of (4.15) equals that of (3.15). The second $\lambda$-derivative of (4.15) is $a_{k,\sigma}(0)-\sigma\frac{\tau^2\mathcal F}{M}\cos(2M\lambda)-2\sigma\beta\frac{\tau^2}{M}(G-g_\zeta M)\sin(2M\lambda)$, which is (4.13) at $p_0=0$ with $w'(0)=-2\tau^2(G-g_\zeta M)/M$; its first derivative at $\lambda=0$, $\sigma\beta g_\zeta\tau^2/M$, equals $\langle\psi_0\rvert\hat v_1\lvert\psi_0\rangle=\sigma g_\zeta\Xi_g[\omega']$ of Step 3. Passed.
3. **Symmetry and $\tau_k\to0$.** $\sigma_2C^{(k)}[G,\mathcal F,g_\zeta]\sigma_2=-C^{(k)}[G,-\mathcal F,g_\zeta]$ and $\sigma_2\beta_{k,+}=i\beta_{k,-}$, so the upper branch at $\lambda$ is the lower branch with $\mathcal F\to-\mathcal F$ at $-\lambda$; the coefficient of $\lambda^2$ is even in $\lambda$, hence $a_{k,+}[G,\mathcal F,g_\zeta]=a_{k,-}[G,-\mathcal F,g_\zeta]$, as in (4.7). For $\tau_k\to0$, $a_{k,\sigma}\to0$ and $v\to0$: no hopping, no motion. Passed.

## Open issues

- The second order in the gradients is not derived; the bound (4.9) is not uniform in $L$ and grows with $\lambda$, so (4.7) is an acceleration of the first-order coefficient, not a statement about long falls.
- For $G_k\neq g_{\zeta,k}M_k$ and sharp-label families outside $\mathcal K_1(x_c,\beta)$, other than $E_\infty$, the sharp-label limit at fixed $\lambda$ is known only through the criterion (4.27), and whether the two limits commute is not determined (last row of (4.29)). The drift acceleration (4.11) is not affected.
- The constants in $O(v^4)$ of (4.17), in $\sup\lvert a_{k,\sigma}''\rvert$ of (4.11) and in the $\lambda^{-1/2}$ decay of $\rho-\rho_\infty$ are not made explicit.
- Item 4 uses the rate (3.23) in its wide-start form, defined at zeroth order in the gradients; the comparison of two clocks inside one description is left to 05-rate-comparison.
- Records beyond leading order in $\kappa$, and starts in a superposition of levels or branches, are outside the scope.

## Methods used

- First-order perturbation theory in a term of the contract (Duhamel formula)
- Plane-wave representation, $\hat x=i\partial_p$, branch (eigenvector) basis of a two-level block
- Separation of secular and oscillating terms; integration by parts
- Riemann–Lebesgue lemma and stationary phase
- Weak convergence of the label weight; limit functionals of families of starts; zeros of entire functions
- Operator-norm and Cauchy–Schwarz bounds; finite-propagation estimate
- Spectral decomposition of the record operator; effective body contract
- Taylor expansion in the drift velocity; injectivity and root-sum arguments for the level conditions

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