Version 1 · earlier version; the current one is v3 · Review of v1

# Fall of a branch start and level-independent fall

- **Subproject:** 05-ilang-free-fall
- **Package:** 04-branch-fall
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-10

## 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.
- **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 $\partial_p$.
- **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.

## 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 $a^{(1)}(\lambda):=\mathrm dv^{(1)}/\mathrm d\lambda=\langle a_{k,\sigma}\rangle_g+\rho'(\lambda)$.

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.** 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,
$$
v^{(1)}(\lambda)=\Bigl\langle a_{k,\sigma}\lambda-\lambda\,n_k\cos(2\omega\lambda)-\tfrac12\bigl(\tfrac{u'}{\omega}B_\perp+\sigma\theta'^2\mathcal F\bigr)\sin(2\omega\lambda)\Bigr\rangle_g+\sigma\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,\xi_g\bigl[B_\parallel'-\theta'B_\perp\cos(2\omega\lambda)\bigr],
\tag{4.12}
$$
$$
n_k(p):=\omega'\theta'B_\perp=-\frac{\tau^4M\,(G-g_\zeta M)\,\sin^2(2p)}{4\,\omega^4}.
$$
The limit needs one more piece of data of the family of starts (an assumption on the family, named here): the family is *centred at $x_c$* if $\xi_g\frac{\mathrm dp}{2\pi}\to x_c\,\delta_{p_0}$ weakly, as for $g=e^{-ipx_c}\gamma$ with real $\gamma\ge0$, where $\xi_g=x_c\lvert g\rvert^2$. For such a family with $\delta p\to0$ ($\lvert g\rvert^2\frac{\mathrm dp}{2\pi}\to\delta_{p_0}$ by Chebyshev's inequality), at fixed $\lambda$,
$$
a^{(1)}_{\rm sharp}(\lambda)=\Bigl[a_{k,\sigma}-A_\sigma\cos(2\omega\lambda)+2\omega\,\theta'B_\perp\,(\omega'\lambda+\sigma x_c)\sin(2\omega\lambda)\Bigr]_{p=p_0},\qquad A_\sigma:=(u'+\theta'\omega')B_\perp+\sigma\omega\theta'^2\mathcal F .
\tag{4.13}
$$
For a family that is not centred, the term with $\xi_g$ need not converge unless $\theta'B_\perp(p_0)=0$.
4. **The two orders.** The window average of (4.13) is $a_{k,\sigma}(p_0)-n_k(p_0)\cos\bigl(2\omega(p_0)(\lambda_1+\Lambda)\bigr)+O(1/\Lambda)$. Hence
$$
\lim_{\Lambda\to\infty}\lim_{\delta p\to0}\overline{a^{(1)}}\ \text{exists}\iff n_k(p_0)=0\iff(G_k-g_{\zeta,k}M_k)\sin^2(2p_0)=0,\ \text{and then equals }a_{k,\sigma}(p_0).
\tag{4.14}
$$
The limits commute at rest, at $p_0\in\{\pm\tfrac\pi2,\pi\}$, and for all $p_0$ if $G_k=g_{\zeta,k}M_k$. Otherwise they do not: in the order "sharp label first" the window average oscillates between $a_{k,\sigma}(p_0)\pm\lvert n_k(p_0)\rvert$, and what converges is the average minus the explicit term $-n_k\cos(2\omega(\lambda_1+\Lambda))$. The reason 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 $p_0=0$: $\omega'=B_\perp=B_\parallel'=0$, $\theta'=-\tau/M$, $A_\sigma=\sigma\tau^2\mathcal F/M$. Integrating (4.13) (or from (4.12)), for a branch start at rest in the sharp-label limit and for every $x_c$:

$$
\bar x^{(1)}_{\rm lab}(\lambda)-\bar x_{\rm lab}(0)=\tfrac12\,a_{k,\sigma}(0)\,\lambda^2-\sigma\,\frac{\tau^2\mathcal F}{2M^3}\sin^2(M\lambda).
\tag{4.15}
$$

Against (3.15) and (3.26), for $\sigma=+$:
- the secular term is unchanged at rest, $a_{k,+}(0)=a_{\rm sec}$; 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 (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: in (3.26) the term $\lambda^2A_2$ is absent and $A_1$ is replaced by $-n_k$, 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 $\rho_\infty$;
- the second limit of (3.28), which does not exist for $S_k$ if $G_k\neq0$, exists for a branch start at rest and equals the first (4.14).

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 $p=0$; 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 $a_0=0$.

### 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).

1. **Drift.** $v^{(0)}=\sigma\langle\omega_k'\rangle_g$ (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). Sharp label at fixed $\lambda$: (4.13), for a centred family. The limits commute if and only if $(G_k-g_{\zeta,k}M_k)\sin^2(2p_0)=0$ (4.14), in particular at rest; the drift acceleration exists for every $p_0$ and is $a_{k,\sigma}(p_0)$ (4.11). Against $S_k$: (4.15) and Step 5. Fixed-chain remainder: (4.9).
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 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.** $a_{k,+}(0)=\frac{\tau^2}{M_k}(\mathcal F-G_k)$ 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). 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.
- The sharp-label limit at fixed $\lambda$ (4.13) is derived for centred families; families that are not centred are not classified.
- 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
- 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