Version 2 · earlier version; the current one is v6 · External review, round 1: major errors

# Field with its own contract: seen factor, second-order terms, fall and clock

- **Subproject:** 05-ilang-free-fall
- **Package:** 10-field-own-contract
- **Version:** v2
- **Mode:** internal regeneration
- **Date:** 2026-10-11

## Changes from previous version

Only the three issues of review.v1.md are answered; every other sentence and display of v1 is unchanged.

- **I1 (phase of the hopping).** Step 2(b): the sentence on the unitary that removes the phase is corrected (it acts on the start as well). Step 3(a): the sentence on the uniform phase $\theta_k$ is reworded. Step 3(b): new paragraph "Velocity from the phase" with the new displays (10.24), (10.25): a velocity of a branch start at rest, derived as a velocity, with its limits and its status; it is not a drift acceleration. Repeated in: Step 4(c) (it vanishes for the cells), the table of Step 5 (new row 5), (10.21), Result, check 1, open issues.
- **I2 (plain gradient of the records).** New bullet "Records at leading order in $\kappa$" in the Setup, with the convention of 04-branch-fall; the sentence of Step 3(a) that dropped the mean record is corrected. The term is kept: new displays (10.22) (drift acceleration with the record term), (10.23) (conditions for $\langle X\rangle\neq0$), (10.26) (defect of (4.24) with the record term). The condition $\langle X\rangle=0$ is added to every statement on the independence of the level and on (4.24): Step 3(b), 3(d), 4(c), rows 2 and 4 of the table, Result, check 2.
- **I3 (third small number).** The Setup and the Result state that $\epsilon_3\ll1$ is a condition of this package in addition to the two small numbers of the question, and that members with $\epsilon_3$ not small are covered by R0 only. The first open issue of v1 is removed and the last one reworded.
- **Tags.** (10.1)–(10.21) are kept and none is withdrawn. Restated with the same formulas and the added condition $\langle X\rangle=0$: (10.13), (10.14), (10.17), (10.20). Restated with a second clause for the new row 5: (10.21). New: (10.22)–(10.26).

## Setup and assumptions

- **Setting.** The class $\mathcal D$, the start $\Psi_0=\phi\otimes\varphi_0\otimes\chi$ with $C_f\varphi_0=e_0\varphi_0$ ($e_0$ simple, gap $\Delta_f$), and $N(x)$, $\tilde\Phi_x$, $\varepsilon_x$, $\varepsilon$, $d_k$, $M_k$, $M_{\max}$, $N_{\max}$, $\epsilon_1$, $\epsilon_2$, $\mathcal C$, $\mathcal C_{\rm lev}$ are those of question.md; $\lambda\ge0$. Class, field start, linear profiles and cells are assumptions of this package (M8, M9, A11). Inputs, as quoted in question.md: (2.1), (2.3), (2.20), (2.22) of 02-field-object; (3.23), (3.36) of 03-clock-and-fall; (4.7), (4.16), (4.19), (4.22), (4.24) of 04-branch-fall; (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.
- **Notation.** $P_x:=\lvert x\rangle\langle x\rvert$, $h_x:=\lvert x+1\rangle\langle x\rvert$, $\mathcal M:=m\mathbb 1_N+D$ (so $\mathcal M\lvert f_k\rangle=M_k\lvert f_k\rangle$, $G=\sigma_3\otimes\mathcal M$, $\lVert G\rVert=M_{\max}$), $B_x:=P_x\otimes G$, $\langle O\rangle:=\langle\varphi_0\vert O\vert\varphi_0\rangle$, $Q:=\mathbb 1_f-\lvert\varphi_0\rangle\langle\varphi_0\rvert$, $Q_0:=\mathbb 1-\Pi_0$, and the reduced resolvent $S:=Q\,(C_f-e_0)^{-1}Q$, $\lVert S\rVert=1/\Delta_f$. Tensor factors are written in the order $(b,c),f$ where convenient. The $M_k$ are positive and pairwise different.
- **A third number of the class.** $\nu:=\max_x\lVert Q\tilde\Phi_xQ\rVert$ and $\epsilon_3:=\nu M_{\max}/\Delta_f$ (the pair term inside the excited states of the field against the gap). In $\mathcal D$ it is bounded neither by $\epsilon_1$ nor by $\epsilon_2$; for the cells of item 4, $\epsilon_3=\epsilon_1$. $\epsilon_3\ll1$ is a condition of this package, in addition to the two small numbers of the question: without it the two diagonal blocks of the contract are not separated ("Perturbation theory used"). Every order estimate below is derived in the regime R1, which contains this condition, and this is stated with each result; the members of $\mathcal D$ in which $\epsilon_3$ is not small are covered by the statements of R0 only: (10.1), (10.2), (10.4) and the first part of (10.7).
- **Regimes.** *R0:* no condition, every member of $\mathcal D$, every start $\phi$, all $\lambda\ge0$ (exact statements and bounds). *R1:* $2\epsilon_1+2\epsilon_2+\epsilon_3<1$, in the weak-record regime ($\kappa\to0$ at fixed $\lambda$, chain and start, so that $\lVert\hat C-e_0\rVert\to\tau+N_{\max}M_{\max}=\epsilon_2\Delta_f$); order estimates are expansions in $\epsilon_1$ whose coefficients are expanded in $\epsilon_2$ and $\epsilon_3$.
- **Records at leading order in $\kappa$.** The convention is that of 04-branch-fall, whose (4.7), (4.16) and (4.24) are used: at leading order in $\kappa$ the body evolves by the effective body contract (8.19) of 04-ilang-time/08-maximal-speed, which is its own contract plus $\kappa\langle X\rangle\hat x$, $\hat x:=\sum_xx\,P_x$, $\langle X\rangle=\langle\chi\vert X\vert\chi\rangle$. The plain gradient $-\kappa\langle X\rangle$ of this contract is one of the first-order gradients and is kept. $\langle X\rangle=0$ is a condition, named with every statement that needs it.
- **Perturbation theory used (standard).** In R1 the diagonal blocks $\Pi_0C\Pi_0$ and $Q_0CQ_0$ have spectra separated by at least $\Delta_f(1-2\epsilon_2-\epsilon_3)$, more than twice the norm $\le\varepsilon M_{\max}$ of the off-diagonal block. Then a unitary $e^{\mathcal S}$ with $\mathcal S$ anti-self-adjoint and block-off-diagonal with respect to $\Pi_0$ block-diagonalises $C$ exactly (Schrieffer–Wolff form of nondegenerate perturbation theory), and $\mathcal S$ is given by its series in the off-diagonal block.
- **Status words.** *Exact*: an identity. *Bound*: a proven inequality. *Order estimate*: leading term computed, remainder given by its order under the stated conditions.

## Derivation

### Step 1. First order (item 1)

**(a)** Because $\langle\Phi_x\rangle=N(x)$ and $\langle C_f\rangle=e_0$, the compression of $C$ is

$$
\hat C=\hat C_b\otimes\mathbb 1_c+\sum_xP_x\otimes\kappa\,x\,X,\qquad
\hat C_b=A+\sum_xP_x\otimes\Bigl[N(x)\,\sigma_3\otimes(m\mathbb 1_N+D)+e_0\,\mathbb 1_2\otimes\mathbb 1_N\Bigr].
\tag{10.1}
$$

*Class and status:* every member of $\mathcal D$ and every $\varphi_0$ as stated, $\varepsilon=0$ or $\varepsilon>0$; exact. For $\varepsilon=0$ it is (2.22) with $n_x=N(x)$, $e_f=e_0$. The body part $\hat C_b$ is the member of $\mathcal C$ with $\tau$, $\zeta_x=1$, $m_x=m\,N(x)$, $\eta_x=N(x)$, $V_x=e_0$ (it is a member because $N(x)>0$); for $N(x)=1+gx$ it is the variant (a), $g_m=g_\eta=g$, $g_\zeta=0$, $\mathcal F=0$, plus the constant $e_0$. $\hat C$ is self-adjoint, independent of $\lambda$ and of the state of $b$ and $c$, and is one pair term between $b$ and $c$ (A5): **yes**, it is the contract of an admissible description of $b$ and $c$. That this description reproduces the one with $C$ is a separate statement, (10.7).

Under (2.20) with $j=x$, $B_j=B_x$ (the $B_x$ and $\mathbb 1_b$ are linearly independent, so these are all replacements):

$$
N(x)\to N(x)+a_x,\qquad e_0\to e_0+c,\qquad \varphi_0,\ \tilde\Phi_x,\ S,\ \varepsilon_x,\ \Delta_f,\ \nu\ \text{unchanged},\qquad \hat C\to\hat C .
\tag{10.2}
$$

$\hat C$ is invariant because it is built from $C$ and $\varphi_0$ only. $N(x)$ alone is not well defined (M6): after the replacement $C_{br}$ contains $-a_xB_x$, and only the sum of $N(x)$ and the coefficient of $B_x$ in $C_{br}$ is invariant. In the form of $\mathcal D$ that coefficient is zero, which fixes $a_x=0$; $\epsilon_1$, $\epsilon_3$ are invariant, $\epsilon_2$ is defined in this form.

**(b)** Split the contract:

$$
C=H_0+W,\qquad H_0:=(\hat C-e_0)\otimes\mathbb 1_f+\mathbb 1\otimes C_f,\qquad W:=\sum_xB_x\otimes\tilde\Phi_x,\qquad [H_0,\Pi_0]=0,\quad \Pi_0W\Pi_0=0,\quad \lVert W\Pi_0\rVert\le\varepsilon M_{\max}.
\tag{10.3}
$$

The last relation: $W(\Psi\otimes\varphi_0)=\sum_xP_xG\Psi\otimes\tilde\Phi_x\varphi_0$ has orthogonal summands, so its squared norm is $\sum_x\lVert P_xG\Psi\rVert^2\varepsilon_x^2\le\varepsilon^2M_{\max}^2\lVert\Psi\rVert^2$. Since $e^{-iH_0\lambda}\Psi_0=\bigl(e^{-i\hat C\lambda}(\phi\otimes\chi)\bigr)\otimes\varphi_0$ stays in $\Pi_0\mathcal H$, Duhamel's formula gives, in R0,

$$
\lVert Q_0\Psi(\lambda)\rVert\le\bigl\lVert\Psi(\lambda)-e^{-iH_0\lambda}\Psi_0\bigr\rVert\le\int_0^\lambda\lVert W\Pi_0\rVert\,\mathrm ds\le\varepsilon\,M_{\max}\,\lambda\qquad(\text{bound, all }\lambda\ge0).
\tag{10.4}
$$

In R1, with the block-diagonalising unitary of "Setup",

$$
e^{\mathcal S}Ce^{-\mathcal S}=H_{\rm eff}\otimes\lvert\varphi_0\rangle\langle\varphi_0\rvert+Q_0\,e^{\mathcal S}Ce^{-\mathcal S}Q_0,\qquad \lVert\mathcal S\rVert\le\epsilon_1\bigl[1+O(\epsilon_1^2+\epsilon_2+\epsilon_3)\bigr],
\tag{10.5}
$$

where the leading term of $Q_0\mathcal S\Pi_0$ is $(\mathbb 1\otimes S)W\Pi_0$, of norm $\le\varepsilon M_{\max}/\Delta_f$ by the argument of (10.3). For an off-diagonal anti-self-adjoint $\mathcal S$, $\lVert Q_0e^{\pm\mathcal S}\Pi_0\rVert\le\sin\lVert\mathcal S\rVert$ and $\lVert\Pi_0e^{\pm\mathcal S}\Pi_0-\Pi_0\rVert\le1-\cos\lVert\mathcal S\rVert$. Writing $\Psi(\lambda)=e^{-\mathcal S}e^{-i(e^{\mathcal S}Ce^{-\mathcal S})\lambda}e^{\mathcal S}\Psi_0$ and using that the middle factor commutes with $\Pi_0$:

$$
\lVert Q_0\Psi(\lambda)\rVert\le2\lVert\mathcal S\rVert\le2\,\epsilon_1\bigl[1+O(\epsilon_1^2+\epsilon_2+\epsilon_3)\bigr]\qquad(\text{R1, order estimate of the constant, all }\lambda\ge0).
\tag{10.6}
$$

For $\varepsilon=0$ both (10.4) and (10.6) give $Q_0\Psi(\lambda)=0$.

**(c)** Let $p$ be the outcome distribution of a schedule (readings $\Pi_i\otimes\mathbb 1_f$ at $\lambda_1\le\dots\le\lambda_k$) in the description with $C$, and $\hat p$ that in the description of $b$, $c$ with $\hat C$ and the start $\phi\otimes\chi$. The readings commute with $\Pi_0$, so the vectors of the $\hat C$-description stay in $\Pi_0\mathcal H$ and (10.4) applies on every interval; the outcome-labelled vectors of the two descriptions then differ by at most $\varepsilon M_{\max}\lambda_k$ in norm (telescoping). In R1, $\Pi_0\Psi$ follows $H_{\rm eff}$ up to $O(\epsilon_1^2)$ by (10.8), the $Q_0$ components have squared norm $O(\epsilon_1^2)$ by (10.6) and are orthogonal to the $\Pi_0$ components, and $\lVert H_{\rm eff}-\hat C\rVert\le\epsilon_1^2\Delta_f[1+O(\epsilon_1^2+\epsilon_2+\epsilon_3)]$ by (10.9). Hence

$$
\sum_{r_1,\dots,r_k}\lvert p-\hat p\rvert\le2\,\varepsilon M_{\max}\lambda_k\ \ (\text{R0, bound}),\qquad
\sum_{r_1,\dots,r_k}\lvert p-\hat p\rvert=O(\epsilon_1^2)+O\bigl(\epsilon_1^2\Delta_f\lambda_k\bigr)\ \ (\text{R1, order estimate, fixed schedule}).
\tag{10.7}
$$

*Answer:* in R1 the statistics of every schedule on $b$ and $c$ are those of the body in a sharp field with the profile $N(x)$ through first order in $\epsilon_1$ (there is no term linear in $\epsilon_1$), on the range $\lambda\ll1/(\epsilon_1^2\Delta_f)=\Delta_f/(\varepsilon M_{\max})^2$; the deviation starts at $\epsilon_1^2$, with a part that does not grow with $\lambda$ and a part that grows linearly. Without any condition on $\epsilon_2$, $\epsilon_3$ they agree at zeroth order on $\lambda\ll1/(\varepsilon M_{\max})$. Control: for $\varepsilon=0$, $p=\hat p$ for all $\lambda$.

### Step 2. Second order (item 2)

**(a)** Write $W_1:=Q_0W\Pi_0$, $W_{\rm d}:=Q_0WQ_0$, $\mathcal S_1:=Q_0\mathcal S\Pi_0$. Expanding $e^{\mathcal S}Ce^{-\mathcal S}$ and requiring the terms of first order in $W_1$ to be block-diagonal gives the Sylvester equation for the leading part of $\mathcal S_1$ and the effective operator:

$$
(\mathbb 1\otimes(C_f-e_0))\,\mathcal S_1+[\hat C\otimes\mathbb 1_f,\mathcal S_1]+W_{\rm d}\,\mathcal S_1=W_1,\qquad
H_{\rm eff}=\hat C-\tfrac12\bigl(\mathcal S_1^{\dagger}W_1+W_1^{\dagger}\mathcal S_1\bigr)+O(\epsilon_1^4\Delta_f),
\tag{10.8}
$$

read on $\mathcal H_b\otimes\mathcal H_c$ (sandwiched with $\varphi_0$). $H_{\rm eff}$ is self-adjoint. By the two norm relations after (10.5), for the exact $H_{\rm eff}$ of (10.5): $\bigl\lVert\Pi_0\Psi(\lambda)-\bigl(e^{-iH_{\rm eff}\lambda}(\phi\otimes\chi)\bigr)\otimes\varphi_0\bigr\rVert\le2\lVert\mathcal S\rVert^2\le2\epsilon_1^2[1+\dots]$ for all $\lambda\ge0$; a truncation $H'$ of $H_{\rm eff}$ adds at most $\lambda\,\lVert H_{\rm eff}-H'\rVert$.

Solve the Sylvester equation by iteration in $[\hat C,\cdot\,]$ and $W_{\rm d}$ against $C_f-e_0$: $\mathcal S_1=(\mathbb 1\otimes S)W_1-(\mathbb 1\otimes S)[\hat C,(\mathbb 1\otimes S)W_1]-(\mathbb 1\otimes S)W_{\rm d}(\mathbb 1\otimes S)W_1+\dots$. With $B_xB_y=\delta_{xy}P_x\otimes\mathbb 1_2\otimes\mathcal M^2$, the leading term is

$$
\boxed{\;H^{(2)}=-\sum_x\gamma_x\,P_x\otimes\mathbb 1_2\otimes(m\mathbb 1_N+D)^2\otimes\mathbb 1_c,\qquad \gamma_x:=\langle\tilde\Phi_x\,S\,\tilde\Phi_x\rangle=\sum_{e\neq e_0}\frac{\lVert\Pi_e\tilde\Phi_x\varphi_0\rVert^2}{e-e_0},\qquad\lvert\gamma_x\rvert\le\frac{\varepsilon_x^2}{\Delta_f}\;}
\tag{10.9}
$$

($\Pi_e$ the eigenprojectors of $C_f$), so $\lVert H^{(2)}\rVert\le\epsilon_1^2\Delta_f$. *Class:* all of $\mathcal D$; *regime:* R1; *status:* exact coefficient of the second order in $\epsilon_1$ at leading order in $\epsilon_2$, $\epsilon_3$. $\gamma_x>0$ ($<0$) whenever $\varepsilon_x>0$ and $e_0$ is the lowest (highest) eigenvalue; for an inner $e_0$, $\gamma_x=0$ with $\varepsilon_x>0$ is possible (exceptional).

First correction. Only the motion term fails to commute with $B_x$ (the on-site terms and the record term commute with $P_x\otimes G$): $[\hat C,B_x]=[A,B_x]=\tfrac\tau2\,\sigma_2\otimes\mathcal M\,\bigl[(h_x^{\dagger}-h_x)+(h_{x-1}^{\dagger}-h_{x-1})\bigr]$. Inserting the second and third terms of $\mathcal S_1$ into (10.8), with $\rho_{xy}:=\langle\tilde\Phi_xS^2\tilde\Phi_y\rangle$ ($\lvert\rho_{xy}\rvert\le\varepsilon_x\varepsilon_y/\Delta_f^2$):

$$
\begin{aligned}
H^{(2,1)}&=-\frac\tau2\sum_x\Bigl(c_x\,h_x\otimes(-i\sigma_1)+c_x^{*}\,h_x^{\dagger}\otimes(i\sigma_1)\Bigr)\otimes\mathcal M^2,\qquad c_x:=\tfrac12(\rho_{xx}+\rho_{x+1,x+1})+\rho_{x+1,x},\\
H^{(3)}&=\sum_x\gamma^{(3)}_x\,P_x\otimes\sigma_3\otimes\mathcal M^3,\qquad \gamma^{(3)}_x:=\langle\tilde\Phi_xS\tilde\Phi_xS\tilde\Phi_x\rangle,\qquad \lVert H^{(2,1)}\rVert\le2\tau\epsilon_1^2,\quad\lVert H^{(3)}\rVert\le\epsilon_1^2\epsilon_3\Delta_f .
\end{aligned}
\tag{10.10}
$$

So $H_{\rm eff}=\hat C+H^{(2)}+H^{(2,1)}+H^{(3)}+O\bigl(\epsilon_1^2(\epsilon_2+\epsilon_3)^2\Delta_f\bigr)+O(\epsilon_1^4\Delta_f)$ (R1, order estimate of the remainder): the first correction has relative order $\epsilon_2$ in the motion term (form: a factor on the hopping) and relative order $\epsilon_3$ on site (form: a term in $\sigma_3\otimes\mathcal M^3$). Through this order $H_{\rm eff}-\hat C$ acts as the identity on $c$ and does not contain $\kappa$. *Range:* by (10.8) and the argument of (10.7), the description of $b$, $c$ with $\hat C+H^{(2)}$ reproduces the statistics of a fixed schedule to an accuracy $\eta\gg\epsilon_1^2$ for $\lambda\lesssim\eta/[\epsilon_1^2\Delta_f(\epsilon_2+\epsilon_3+\epsilon_1^2)]$, and with $H^{(2,1)}+H^{(3)}$ added for $\lambda\lesssim\eta/[\epsilon_1^2\Delta_f((\epsilon_2+\epsilon_3)^2+\epsilon_1^2)]$; the part $O(\epsilon_1^2)$ of the error that does not grow with $\lambda$ is not removed by these terms.

**(b)** Every term is a function of $D$ on the clock label, so the level is conserved. On the level $k$ the on-site block at $x$ and the hopping block from $x$ to $x+1$ of the body part are

$$
\bigl(N(x)M_k+\gamma^{(3)}_xM_k^3\bigr)\sigma_3+\bigl(e_0-\gamma_xM_k^2\bigr)\mathbb 1_2,\qquad T_{x,k}=-\frac{i\tau}2\bigl(1-M_k^2c_x\bigr)\sigma_1 .
\tag{10.11}
$$

At second order and leading order in $\epsilon_2$, $\epsilon_3$ the body part is a member of $\mathcal C_{\rm lev}$: **yes**, with $\tau_k=\tau$, $\zeta_{x,k}=1$, $M_{x,k}=N(x)M_k$, $V_{x,k}=e_0-\gamma_xM_k^2$; it is a member of $\mathcal C$ if and only if $\gamma_x=0$ for all $x$ (the $M_k^2$ are pairwise different). With the first corrections, $M_{x,k}=N(x)M_k+\gamma_x^{(3)}M_k^3$ and $\tau_k\zeta_{x,k}=\tau\lvert1-M_k^2c_x\rvert$ are of the form of $\mathcal C_{\rm lev}$. The one term not of that form is the phase $\arg(1-M_k^2c_x)=-M_k^2\operatorname{Im}\rho_{x+1,x}+O(\epsilon_1^4)$ of the hopping, present iff $\operatorname{Im}\rho_{x+1,x}\neq0$. On the open chain the unitary $U:=\sum_{x,k}e^{i\vartheta_{x,k}}P_x\otimes\mathbb 1_2\otimes\lvert f_k\rangle\langle f_k\rvert$ with $\vartheta_{x+1,k}-\vartheta_{x,k}=\arg(1-M_k^2c_x)$ removes it from the contract: $U^{\dagger}H_{\rm eff}U$ has the hopping factors $\lvert1-M_k^2c_x\rvert$ and is otherwise unchanged. The unitary acts on the start as well: $e^{-iH_{\rm eff}\lambda}(\phi\otimes\chi)=U\,e^{-iU^{\dagger}H_{\rm eff}U\lambda}\,(U^{\dagger}\phi\otimes\chi)$. $U$ is diagonal in $(x,s,k)$ and is the identity on $c$, so the statistics of the readings of the position and of the branch label, and of every reading on $c$, are for the contract with the phase and the start $\phi$ those for the contract without the phase and the start $U^{\dagger}\phi$; for readings of the clock label, which are not diagonal in $k$, this does not hold. $U^{\dagger}\phi\neq\phi$ in general, also for a start in one level: there $U^{\dagger}$ multiplies the weight at $x$ by $e^{-i\vartheta_{x,k}}$, which changes the plane-wave labels of the start. The phase is therefore not without effect: it is equivalent to a change of the start that depends on the level. For the linear profiles of the field its effect on a branch start at rest is the velocity (10.25). (Beyond the order kept, at relative order $\epsilon_2^2$, hops over two sites and record-dependent terms appear.)

**(c)** By (10.2), $\tilde\Phi_x$, $S$ and $\varphi_0$ are unchanged under (2.20), and all coefficients are sesquilinear in $\varphi_0$. Hence $\gamma_x$, $\rho_{xy}$, $c_x$, $\gamma^{(3)}_x$, and with them $H^{(2)}$, $H^{(2,1)}$, $H^{(3)}$, each separately, are well defined: **yes**. In (10.11) the differences $V_{x,k}-V_{x',k'}$ are well defined, the common constant $e_0$ is not (it shifts by $c$), and $N(x)$ only in the sense of (10.2). $H_{\rm eff}$ is fixed by the off-diagonal choice of $\mathcal S$; a conjugation by a unitary $\mathbb 1+O(\epsilon_1^2)$ on $\mathcal H_b\otimes\mathcal H_c$ satisfies (10.8) equally and leaves the spectrum unchanged.

### Step 3. Fall, clock and relation (item 3)

*Class:* $\Phi_x=N_0+xY$, every $N_0$, $Y$, $C_f$, $\varphi_0$ as stated. *Regime:* R1, second order in $\epsilon_1$ with the first corrections in $\epsilon_2$, $\epsilon_3$, first order in $Y$, leading order in $\kappa$ in the convention of "Setup" (the plain gradient $-\kappa\langle X\rangle$ is kept). *Status:* closed forms of the body part of $\hat C+H^{(2)}+H^{(2,1)}+H^{(3)}$; remainders as in (10.10).

**(a)** Put $\tilde N_0:=N_0-\mathbb 1_f$, $\tilde Y:=Y-g\,\mathbb 1_f$, so $\tilde\Phi_x=\tilde N_0+x\tilde Y$, and $\varepsilon_0:=\lVert\tilde N_0\varphi_0\rVert$, $\varepsilon_Y:=\lVert\tilde Y\varphi_0\rVert$. To first order in $Y$: $\gamma_x=\gamma_0+\gamma_1x$, $\rho_{xy}=\rho_0+x\rho_Y^{*}+y\rho_Y$, $\gamma^{(3)}_x=\gamma^{(3)}_0+\gamma^{(3)}_1x$, $c_x=2\rho_0+4\operatorname{Re}\rho_Y\,(x+\tfrac12)-i\operatorname{Im}\rho_Y$, with

$$
\begin{aligned}
&\gamma_0:=\langle\tilde N_0S\tilde N_0\rangle,\quad \gamma_1:=2\operatorname{Re}\langle\tilde N_0S\tilde Y\rangle,\quad \rho_0:=\langle\tilde N_0S^2\tilde N_0\rangle\ge0,\quad \rho_Y:=\langle\tilde N_0S^2\tilde Y\rangle,\\
&\gamma^{(3)}_0:=\langle\tilde N_0S\tilde N_0S\tilde N_0\rangle,\quad \gamma^{(3)}_1:=\langle\tilde N_0S\tilde YS\tilde N_0\rangle+2\operatorname{Re}\langle\tilde N_0S\tilde N_0S\tilde Y\rangle;\\
&\tau_k=\tau\,(1-2\rho_0M_k^2),\quad \hat M_k=M_k\,(1+\gamma^{(3)}_0M_k^2),\quad G_k=g\,M_k+\gamma^{(3)}_1M_k^3,\quad \mathcal F_k=\gamma_1M_k^2,\quad g_{\zeta,k}=-4M_k^2\operatorname{Re}\rho_Y,\quad V_k=e_0-\gamma_0M_k^2 .
\end{aligned}
\tag{10.12}
$$

These follow from (10.11) by matching $M_{x,k}=\hat M_k+G_kx$, $V_{x,k}=V_k-\mathcal F_kx$, $\tau_k\zeta_{x,k}=\tau(1-M_k^2\operatorname{Re}c_x)$; $\hat M_k$ is the rest term of the level $k$ (the "$M_k$" of the level data). At leading order in $\epsilon_2$, $\epsilon_3$: $\tau_k=\tau$, $\hat M_k=M_k$, $G_k=gM_k$, $g_{\zeta,k}=0$, and $\mathcal F_k$, $V_k$ as above. The phase of (10.11) is uniform, $\theta_k=M_k^2\operatorname{Im}\rho_Y$ (from here on $\theta_k$ is this phase; the angle of (5.6) of 04-ilang-time/05-moving-clock is not used by name); it shifts the plane-wave label of the level block, $p\to p-\theta_k$, and is of first order in $Y$ and independent of $\lambda$. It changes the drift acceleration $a_{k,\sigma}$ only at second order in $Y$, and at first order in $Y$ it gives a branch start at rest a velocity; both statements are derived in (b), (10.24), (10.25). The mean record $\kappa\langle X\rangle x$ of (8.19) of 04-ilang-time/08-maximal-speed adds $-\kappa\langle X\rangle$ to every $\mathcal F_k$ of the effective body contract. It is one of the first-order gradients and is kept ("Setup"): in (4.7) and (4.16) the $\mathcal F_k$ of (10.12) is replaced by $\mathcal F_k-\kappa\langle X\rangle$.

*Level offset.* The level is conserved, and on the level $k$ the offset is $V_k$ times the identity. For a start in one level it multiplies the state by the common phase $e^{-iV_k\lambda}$ (1.2), so no view and no reading changes: (4.7) and (4.16), derived with $V_k=0$, hold unchanged.

**(b)** Inserting (10.12) into (4.16), $a_{k,\sigma}(0)=\frac{\tau_k^2}{\hat M_k}(\sigma\mathcal F_k-G_k)$, and keeping the second order in $\varepsilon$, for $\langle X\rangle=0$ (for $\langle X\rangle\neq0$ see (10.22)):

$$
\boxed{\;\langle X\rangle=0:\quad a_{k,\sigma}(0)=\tau^2\Bigl[-g+\sigma\,\gamma_1M_k+\beta\,M_k^2\Bigr],\qquad \beta:=g\,\bigl(4\rho_0+\gamma^{(3)}_0\bigr)-\gamma^{(3)}_1,\qquad \lvert\gamma_1\rvert M_k\le\frac{2\varepsilon_0\varepsilon_YM_{\max}}{\Delta_f},\quad \rho_0M_k^2\le\epsilon_1^2\;}
\tag{10.13}
$$

The term in $\gamma_1$ is the leading order in $\epsilon_2$, $\epsilon_3$; the term in $\beta$ is their first correction (relative to the $\gamma_1$ term it carries $M_{\max}/\Delta_f\le\epsilon_2$, respectively $\epsilon_3$). These are drift accelerations of the description of $b$, $c$ with the contract $H_{\rm eff}$; the description with $C$ has the same position statistics within (10.10) on its range, and no rate in the description with $C$ is inferred beyond that. Since the $M_k$ are pairwise different:

$$
\begin{aligned}
&\langle X\rangle=0,\ \text{leading order in }\epsilon_2,\epsilon_3\ (\text{every }N\ge2):\quad a_{k,+}(0)\ \text{level-independent}\iff\gamma_1=0\iff a_{k,-}(0)\ \text{level-independent}\iff\text{both, with the one value }-\tau^2g;\\
&\langle X\rangle=0,\ \text{with the first corrections},\ N\ge3:\quad \text{each of the three}\iff\gamma_1=0\ \text{and}\ \beta=0;\\
&\langle X\rangle=0,\ \text{with the first corrections},\ N=2:\quad \text{upper}\iff\gamma_1+\beta(M_0+M_1)=0,\quad \text{lower}\iff\gamma_1-\beta(M_0+M_1)=0,\quad \text{both (then one value)}\iff\gamma_1=\beta=0 .
\end{aligned}
\tag{10.14}
$$

(For $N\ge3$ a polynomial $\sigma\gamma_1M+\beta M^2$ with equal values at three points is constant.) Sufficient for $\gamma_1=0$: $\varepsilon_0=0$ or $\varepsilon_Y=0$; it also holds when $\operatorname{Re}\langle\tilde N_0S\tilde Y\rangle$ vanishes otherwise. *Order,* for $\langle X\rangle=0$: at first order in $\epsilon_1$ the fall from rest is level-independent in both branches, with the value $-\tau^2g$ ((10.1) and (4.22) with $\mathcal F=m(g_m-g_\eta)=0$). The independence is lost at second order in $\epsilon_1$, already at the leading order of the coefficient in $\epsilon_2$, by the branch-odd term $\sigma\tau^2\gamma_1M_k$, bilinear in $\varepsilon_0$ and $\varepsilon_Y$; if $\gamma_1=0$, it is lost at second order in $\epsilon_1$ and first order in $\epsilon_2$, $\epsilon_3$ by $\tau^2\beta M_k^2$ unless $\beta=0$.

*Records, every $\langle X\rangle$.* Through the order kept $H_{\rm eff}-\hat C$ acts as the identity on $c$, so the effective body contract of the description with $H_{\rm eff}$ is the body part of $H_{\rm eff}$ plus $\kappa\langle X\rangle\hat x$. Then (4.16) with $\mathcal F_k-\kappa\langle X\rangle$ gives

$$
a_{k,\sigma}(0)=\tau^2\Bigl[-g+\sigma\,\gamma_1M_k+\beta\,M_k^2\Bigr]+a^{\rm rec}_{k,\sigma},\qquad a^{\rm rec}_{k,\sigma}:=-\sigma\,\frac{\tau_k^2}{\hat M_k}\,\kappa\langle X\rangle=-\sigma\,\tau^2\,\kappa\langle X\rangle\Bigl[\frac1{M_k}-\bigl(4\rho_0+\gamma^{(3)}_0\bigr)M_k\Bigr].
\tag{10.22}
$$

*Class, regime, status:* as (10.13), for every $\langle X\rangle$; closed form. The record term is of zeroth order in $\epsilon_1$ (it is there for a sharp field), odd in the branch, and different for every two levels when $\langle X\rangle\neq0$ (the $1/M_k$ are pairwise different). *Size:* against $-\tau^2g$ it is $\kappa\lvert\langle X\rangle\rvert/(\lvert g\rvert M_k)$; against $\sigma\tau^2\gamma_1M_k$ it is $\kappa\lvert\langle X\rangle\rvert/(\lvert\gamma_1\rvert M_k^2)$, where $\lvert\gamma_1\rvert M_k^2\le2\varepsilon_0\varepsilon_YM_k^2/\Delta_f$ is of second order in $\epsilon_1$. With $\gamma_1':=\gamma_1+\kappa\langle X\rangle\,(4\rho_0+\gamma^{(3)}_0)$, $a_{k,\sigma}(0)=a$ for all $k$ if and only if the cubic polynomial $\beta M^3+\sigma\gamma_1'M^2-(g+a/\tau^2)M-\sigma\kappa\langle X\rangle$ vanishes at all $M_k$. Its constant term is the record term, so that for $\langle X\rangle\neq0$:

$$
\begin{aligned}
&\text{leading order in }\epsilon_2,\epsilon_3,\ N\ge3:\quad a_{k,\sigma}(0)\ \text{level-dependent in each branch};\\
&\text{leading order in }\epsilon_2,\epsilon_3,\ N=2:\quad \text{upper}\iff\gamma_1M_0M_1=-\kappa\langle X\rangle\iff\text{lower},\quad\text{with the two values }\tau^2\bigl[-g+\sigma\gamma_1(M_0+M_1)\bigr];\\
&\text{with the first corrections}:\quad N\ge4:\ \text{level-dependent in each branch};\qquad N=3:\ \text{branch }\sigma\iff\sigma\kappa\langle X\rangle=\beta M_0M_1M_2\ \text{and}\ \sigma\gamma_1'=-\beta(M_0+M_1+M_2);\\
&\phantom{\text{with the first corrections}:\quad}N=2:\ \text{branch }\sigma\iff\gamma_1'+\frac{\kappa\langle X\rangle}{M_0M_1}+\sigma\beta(M_0+M_1)=0;\\
&\text{every }N\ge2:\quad\text{both branches with one value}\ \Longrightarrow\ \langle X\rangle=0\ \ (\text{then (10.14)}).
\end{aligned}
\tag{10.23}
$$

(A polynomial of degree three, at leading order two, with more zeros than its degree vanishes identically, which needs $\langle X\rangle=0$; for both branches with one value, the sum and the difference of the two polynomials give $\beta M_k^2=g+a/\tau^2$ and $\gamma_1'M_k^2=\kappa\langle X\rangle$ for all $k$.) So for $\langle X\rangle\neq0$ the statement "level-independent at first order in $\epsilon_1$" fails: already for $\varepsilon=0$, $a_{k,\sigma}(0)=-\tau^2g-\sigma\tau^2\kappa\langle X\rangle/M_k$ depends on the level for every $N\ge2$, as (4.22) requires ($\mathcal F=-\kappa\langle X\rangle\neq m(g_m-g_\eta)=0$). The equivalences (10.14) are replaced by (10.23), whose cases with independence are exceptional members in which the record term cancels against terms of second order in $\epsilon_1$. The condition $\langle X\rangle=0$ is read in the form of $\mathcal D$, in which the body has no plain term of its own; under (8.1) of 04-ilang-time/08-maximal-speed only the plain gradient of the effective body contract, here $-\kappa\langle X\rangle$, is well defined ((8.20) there).

*Velocity from the phase.* Take the level $k$ with all gradients set to zero and the uniform phase $\theta_k$ kept, on the infinite chain (the limit of a long chain at fixed $\lambda$ and start, as in 04-branch-fall). The computation of (5.5) of 04-ilang-time/05-moving-clock with the hopping factor $e^{i\theta_k}$ gives the level block $-\tau_k\sin(p-\theta_k)\,\sigma_1+\hat M_k\sigma_3$, with the eigenvalues $\pm\omega_k(p-\theta_k)$ and the eigenvectors $\beta_{k,\pm}(p-\theta_k)$. The position label acts on the plane-wave weights as $i\,\partial_p$, so $\mathrm d\bar x_{\rm lab}/\mathrm d\lambda$ is the mean of $i[C^{(k)},\hat x]=-\tau_k\cos(p-\theta_k)\,\sigma_1$: the rate itself is computed. The part of this operator that is diagonal in the eigenvectors is $\pm\omega_k'(p-\theta_k)$. The branch start $B_{k,\sigma}[g]$ is built with $\beta_{k,\sigma}(p)$, which contains the other eigenvector with an amplitude of first order in $\theta_k$, so the off-diagonal part contributes at first order as well, with the frequency $2\omega_k$. Evaluating the mean of the $2\times2$ block to first order in $\theta_k$ (where $\tau_k\to\tau$, $\hat M_k\to M_k$):

$$
\frac{\mathrm d\bar x_{\rm lab}}{\mathrm d\lambda}=\sigma\langle\omega_k'\rangle_g+v^{\rm ph}_{k,\sigma}(\lambda),\qquad v^{\rm ph}_{k,\sigma}(\lambda)=-\sigma\,\theta_k\Bigl\langle\omega_k''-\frac{\tau^2M_k^2\cos^2p}{\omega_k^3}\cos(2\omega_k\lambda)\Bigr\rangle_g,\qquad \theta_k=M_k^2\operatorname{Im}\rho_Y,\quad\lvert\rho_Y\rvert\le\frac{\varepsilon_0\varepsilon_Y}{\Delta_f^2},
\tag{10.24}
$$

with $\omega_k''=\tau^2(M_k^2\cos2p-\tau^2\sin^4p)/\omega_k^3$. With the gradients, the terms of (4.7) are added; the cross terms are of second order in the gradients. $v^{\rm ph}_{k,\sigma}$ is bounded for all $\lambda\ge0$, so it does not change the coefficient of $\lambda$: the drift accelerations (10.13), (10.22) are unchanged at first order in $Y$. For a start at rest the first term of (10.24) vanishes in the limit of a sharp label, and

$$
\boxed{\;\begin{aligned}
&\text{sharp label at fixed }\lambda:\quad v^{\rm ph}_{k,\sigma}(\lambda)=-\sigma\,\tau^2M_k\operatorname{Im}\rho_Y\,\bigl[1-\cos(2M_k\lambda)\bigr];\\
&\text{long window, before or after the sharp label}:\quad \bar v^{\rm ph}_{k,\sigma}=-\sigma\,\tau^2M_k\operatorname{Im}\rho_Y,\qquad \lvert\bar v^{\rm ph}_{k,\sigma}\rvert\le\tau^2M_k\,\frac{\varepsilon_0\varepsilon_Y}{\Delta_f^2}.
\end{aligned}\;}
\tag{10.25}
$$

*Order of the limits.* The growth of $\lambda$ and the limit of a sharp label do not commute for the velocity itself. With the sharp label first, the velocity is zero at $\lambda=0$, oscillates between $0$ and $2\bar v^{\rm ph}_{k,\sigma}$ and has no limit. With $\lambda\to\infty$ first, at a fixed smooth $g$, the oscillating term of (10.24) decays (stationary phase; $\omega_k$ is not constant because $\tau>0$), the limit is $-\sigma\theta_k\langle\omega_k''\rangle_g$, and the sharp label then gives $\bar v^{\rm ph}_{k,\sigma}$. The average over a window of length $\Lambda$ removes the oscillating term up to $\tau^2\lvert\theta_k\rvert/(M_k^2\Lambda)$ uniformly in $g$, so it commutes with the limit of a sharp label. (A start built with the eigenvectors $\beta_{k,\sigma}(p-\theta_k)$ of the block with the phase has the velocity $\sigma\langle\omega_k'(p-\theta_k)\rangle_g$ at every $\lambda$, at the label $p_0=0$ equal to $\sigma\omega_k'(-\theta_k)$, which is $\bar v^{\rm ph}_{k,\sigma}$ at the order kept.)

*Class, regime, status:* linear profiles of the field, branch starts at rest, every $\langle X\rangle$; R1, second order in $\epsilon_1$ (bilinear in $\varepsilon_0$, $\varepsilon_Y$) and first order in $\epsilon_2$ (the term comes from $H^{(2,1)}$; there is no term in $\epsilon_3$ at this order), first order in $Y$ (in $\tilde Y$; it does not contain $g$); exact coefficient at these orders, in the description with $H_{\rm eff}$, as stated after (10.13). It is a velocity and not a drift acceleration. It is odd in the branch and proportional to $M_k$, hence different for every two levels, and it vanishes if and only if $\operatorname{Im}\rho_Y=0$. Sufficient for that: $\varepsilon_0=0$, or $\varepsilon_Y=0$, or real matrices of $C_f$, $N_0$, $Y$ in one basis. It does not need $\gamma_1\neq0$: a field with two places, $C_f=\tfrac12\Delta(\lvert1\rangle\langle1\rvert-\lvert0\rangle\langle0\rvert)$, $\varphi_0=\lvert0\rangle$, $N_0=\mathbb 1+\varepsilon_0(\lvert0\rangle\langle1\rvert+\lvert1\rangle\langle0\rvert)$, $Y=g\,\mathbb 1+i\varepsilon_Y(\lvert1\rangle\langle0\rvert-\lvert0\rangle\langle1\rvert)$, has $\gamma_1=0$ and $\rho_Y=i\varepsilon_0\varepsilon_Y/\Delta^2$. The level block of the level $k$ has zero group velocity at the label $p=\theta_k$, which depends on the level: a start with one position factor for two levels $k\neq k'$ is not at rest in both, and their window-averaged velocities differ by $-\sigma\tau^2\operatorname{Im}\rho_Y\,(M_k-M_{k'})$.

**(c)** With $E^+_{x,k}=e_0+N(x)M_k-\gamma_xM_k^2+\gamma^{(3)}_xM_k^3$:

$$
E^+_{x,k+1}-E^+_{x,k}=\frac{2\pi}{N\lambda_0}\Bigl[N(x)-\gamma_x\,(M_k+M_{k+1})+\gamma^{(3)}_x\bigl(M_k^2+M_kM_{k+1}+M_{k+1}^2\bigr)\Bigr].
\tag{10.15}
$$

At leading order in $\epsilon_2$, $\epsilon_3$ the differences are independent of $k$ if and only if $\gamma_x=0$ or $N=2$ (one difference only): for $N\ge3$ and $\gamma_x\neq0$, **no**; the second difference is $-2\gamma_x(2\pi/N\lambda_0)^2$. Clock of $S_{\rm clock}(x_0)$, at the orders of (3.36) (zeroth order in the gradients): the body part is then $\hat C_b-\gamma_{x_0}\mathcal M^2$, the two terms commute, and $\mathcal M^2=m^2+\Sigma\,D-D(d_{N-1}-D)$ with $\Sigma:=M_0+M_{N-1}$. The factor $e^{i\gamma_{x_0}\Sigma D\lambda}$ shifts every advance of the mixture (3.36) (taken with $\eta^0=N(x_0)$, $m^0=mN(x_0)$, $\tau^0=\tau$) by $-\gamma_{x_0}\Sigma\lambda/\lambda_0$, in both branch components alike; the factor $e^{-i\gamma_{x_0}D(d_{N-1}-D)\lambda}$ has phases in $[0,\tfrac14\lvert\gamma_{x_0}\rvert d_{N-1}^2\lambda]$ and is the identity for $N=2$. Hence

$$
\text{clock statistics}=\text{ring-clock mixture (3.36) with rate }\bar r(x_0)\text{ in the wide-start limit, up to total variation }\le\tfrac12\lvert\gamma_{x_0}\rvert\,d_{N-1}^2\,\lambda;\qquad \text{accuracy }\eta\ \text{for}\ \lambda\le\frac{2\eta}{\lvert\gamma_{x_0}\rvert\,d_{N-1}^2}\ (N\ge3),\ \ \text{all }\lambda\ (N=2\text{ or }\gamma_{x_0}=0).
\tag{10.16}
$$

*Status:* order estimate; $\lvert\gamma_{x_0}\rvert d_{N-1}^2\le\epsilon_1^2\Delta_f\,(d_{N-1}/M_{\max})^2$. The level dependence of $\tau_k$ changes $\varsigma_W$ of (3.23) only at relative order $\epsilon_1^2$.

**(d)** From (10.15), at leading order in $\epsilon_2$, $\epsilon_3$, $\bar r(x)=\lambda_0^{-1}\bigl[N(x)-\gamma_x\Sigma\bigr]$, so $\frac{\bar r(x+1)-\bar r(x)}{\bar r(x)}\big\vert_{x=0}=g\,(1+\gamma_0\Sigma)-\gamma_1\Sigma$ at first order in $Y$ and second order in $\varepsilon$. With $v_{\max}^2/d_0^2=\tau^2$, (4.24) for the body part of $H_{\rm eff}$ would read $a_{k,+}(0)=-\tau^2\,(\bar r(x+1)-\bar r(x))/\bar r(x)$. Its defect is, for $\langle X\rangle=0$,

$$
\boxed{\;\mathcal R_k:=a_{k,+}(0)+\tau^2\,\frac{\bar r(x+1)-\bar r(x)}{\bar r(x)}\Big\vert_{x=0}=\tau^2\Bigl[\gamma_1\,(M_k-\Sigma)+g\,\gamma_0\,\Sigma\Bigr]\ \ (\langle X\rangle=0),\qquad \Sigma=M_0+M_{N-1}=2m+d_{N-1}\;}
\tag{10.17}
$$

up to the first corrections in $\epsilon_2$, $\epsilon_3$ and $O(\varepsilon^4)$. **No**: for $\langle X\rangle=0$, (4.24) holds for all levels if and only if $\gamma_1=0$ and $g\gamma_0=0$. Under the hypothesis of (4.24) (a level-independent $a\neq0$, which for $\langle X\rangle=0$ needs $\gamma_1=0$ and $g\neq0$) it holds iff $\gamma_0=0$; otherwise $a=-\tau^2\,\frac{\bar r(x+1)-\bar r(x)}{\bar r(x)}\,(1-\gamma_0\Sigma)$, and the leading deviation $d_0\mathcal R_k\hat e=d_0\tau^2g\gamma_0\Sigma\,\hat e$ contains the data $m$, $d_{N-1}$ of the body. At first order in $\epsilon_1$ and for $\langle X\rangle=0$, (4.24) holds with $r(x)=N(x)/\lambda_0$.

For every $\langle X\rangle$: the plain term $\kappa\langle X\rangle x$ is the same in all levels and drops out of (10.15), so $\bar r$ is unchanged, and the record term $a^{\rm rec}_{k,+}$ of (10.22) is added to the defect. At leading order in $\epsilon_2$, $\epsilon_3$:

$$
\mathcal R_k=\tau^2\Bigl[\gamma_1\,(M_k-\Sigma)+g\,\gamma_0\,\Sigma-\frac{\kappa\langle X\rangle}{M_k}\Bigr]\qquad(\text{every }\langle X\rangle).
\tag{10.26}
$$

*Class, regime, status:* as (10.17); closed form of the leading defect. For $\langle X\rangle\neq0$, (4.24) fails already at zeroth order in $\epsilon_1$ (for a sharp field), with the level-dependent defect $-\tau^2\kappa\langle X\rangle/M_k$. It holds for all levels if and only if $\gamma_1=0$, $g\gamma_0=0$ and $\langle X\rangle=0$ for $N\ge3$ (a quadratic polynomial in $M_k$ with three zeros), and if and only if $g\gamma_0=0$ and $\gamma_1M_0M_1=-\kappa\langle X\rangle$ for $N=2$ (the exceptional members of (10.23)).

### Step 4. Example and control: independent cells (item 4)

*Class:* the cells, every $\Delta_x>0$, $n_x>0$, $\varepsilon_x\ge0$. *Regime:* R1 for (a), (c); (b) is exact on site.

**(a)** $\varphi_0=\bigotimes_x\lvert0\rangle$ has $e_0=-\tfrac12\sum_x\Delta_x$, the lowest and simple eigenvalue of $C_f$; an excited eigenvector has a set of flipped cells. With $\sigma^{(x)}_1:=\lvert0\rangle\langle1\rvert+\lvert1\rangle\langle0\rvert$ on the cell $x$, $\tilde\Phi_x=\varepsilon_x\sigma_1^{(x)}$ and $\tilde\Phi_x\varphi_0$ is the eigenvector with the cell $x$ flipped, at $e_0+\Delta_x$:

$$
N(x)=n_x,\quad \lVert\tilde\Phi_x\varphi_0\rVert=\varepsilon_x,\quad \varepsilon=\max_x\varepsilon_x,\quad \Delta_f=\min_x\Delta_x,\quad \nu=\varepsilon,\quad
\gamma_x=\frac{\varepsilon_x^2}{\Delta_x},\quad \rho_{xy}=\delta_{xy}\frac{\varepsilon_x^2}{\Delta_x^2},\quad c_x=\frac12\Bigl(\frac{\varepsilon_x^2}{\Delta_x^2}+\frac{\varepsilon_{x+1}^2}{\Delta_{x+1}^2}\Bigr),\quad \gamma^{(3)}_x=0 .
\tag{10.18}
$$

So $H^{(2)}=-\sum_x\frac{\varepsilon_x^2}{\Delta_x}P_x\otimes\mathbb 1_2\otimes\mathcal M^2$, $H^{(3)}=0$, the hopping has the real factor $1-M_k^2c_x$, and the body part is a member of $\mathcal C_{\rm lev}$ with $M_{x,k}=n_xM_k$, $V_{x,k}=e_0-\varepsilon_x^2M_k^2/\Delta_x$, $\tau_k\zeta_{x,k}=\tau(1-M_k^2c_x)$; no term is outside the form.

**(b)** In the basis $(\lvert0\rangle,\lvert1\rangle)$, $C_{f_x}\pm M_k\Phi_x=\pm M_kn_x\mathbb 1-\Omega_{x,k}\bigl(\cos\alpha_{x,k}\,\sigma_3\mp\sin\alpha_{x,k}\,\sigma_1\bigr)$ with $\Omega_{x,k}\cos\alpha_{x,k}=\Delta_x/2$, $\Omega_{x,k}\sin\alpha_{x,k}=M_k\varepsilon_x$. Exactly:

$$
\text{eigenvalues }\pm M_kn_x-\Omega_{x,k}\ (\text{lower}),\ \ \pm M_kn_x+\Omega_{x,k}\ (\text{upper}),\quad \Omega_{x,k}=\sqrt{\tfrac14\Delta_x^2+M_k^2\varepsilon_x^2};\qquad \text{lower eigenvector }\cos\tfrac{\alpha_{x,k}}2\lvert0\rangle\mp\sin\tfrac{\alpha_{x,k}}2\lvert1\rangle;\qquad \cos\tfrac{\alpha_{x,k}}2=\Bigl[\tfrac12\Bigl(1+\tfrac{\Delta_x}{2\Omega_{x,k}}\Bigr)\Bigr]^{1/2}.
\tag{10.19}
$$

*Comparison.* With the other cells in $\lvert0\rangle$ the field costs $e_0+\tfrac12\Delta_x\pm M_kn_x-\Omega_{x,k}=e_0\pm M_kn_x-\frac{\varepsilon_x^2M_k^2}{\Delta_x}+\frac{\varepsilon_x^4M_k^4}{\Delta_x^3}-\dots$: the part odd in the branch is $n_xM_k$ exactly (so $\gamma_x^{(3)}=0$), the even part reproduces $-\gamma_xM_k^2$ of (10.11), and the remainder has the relative size $(M_k\varepsilon_x/\Delta_x)^2\le\epsilon_1^2$. *Overlap:* $\langle0\vert\text{lower}\rangle=\cos\tfrac{\alpha_{x,k}}2=1-\frac{M_k^2\varepsilon_x^2}{2\Delta_x^2}+O(\epsilon_1^4)$, the same for both signs. *What it multiplies:* at leading order in $\tau/\Delta_x$ the cells follow the body; a hop from $x$ to $x+1$ returns the cell $x$ to $\lvert0\rangle$ and takes the cell $x+1$ to its lower eigenvector, so the hopping block of the level $k$, that is the motion term $\tau$, is multiplied by $\cos\tfrac{\alpha_{x,k}}2\cos\tfrac{\alpha_{x+1,k}}2=1-M_k^2c_x+O(\epsilon_1^4)$, in agreement with (10.18).

**(c)** The cells are not of the form $N_0+xY$ (the $\sigma_1^{(x)}$ of more than two cells are linearly independent), so (10.11) is used directly, with $n_x=1+gx$ and, to first order in the gradients of the cell data, $\gamma_x=\gamma_0+\gamma_1x$, $\rho_{xx}=\rho_0+\rho_1x$. Then $\tau_k=\tau(1-\rho_0M_k^2)$, $g_{\zeta,k}=-\rho_1M_k^2$, $\hat M_k=M_k$, $G_k=gM_k$, $\mathcal F_k=\gamma_1M_k^2$, $V_k=e_0-\gamma_0M_k^2$, and by (4.16), for $\langle X\rangle=0$,

$$
a_{k,\sigma}(0)=\tau^2\Bigl[-g+\sigma\,\gamma_1M_k+2g\,\rho_0M_k^2\Bigr]\ \ (\langle X\rangle=0),\qquad \gamma_1=\frac{\varepsilon_{x+1}^2}{\Delta_{x+1}}-\frac{\varepsilon_x^2}{\Delta_x},\quad \rho_0=\frac{\varepsilon_x^2}{\Delta_x^2}\Big\vert_{x=0}\ (\text{the cell at }x=0),
\tag{10.20}
$$

that is (10.13), (10.14) with $\beta=2g\rho_0$ (here $\rho_{x+1,x}=0$, whence $\rho_0$ in place of the $2\rho_0$ of (10.12)). For $\langle X\rangle\neq0$ the record term of (10.22) is added with the cell data, $a^{\rm rec}_{k,\sigma}=-\sigma\tau^2\kappa\langle X\rangle\,[1/M_k-2\rho_0M_k]$, and (10.23) applies with $\gamma_1'=\gamma_1+2\rho_0\,\kappa\langle X\rangle$. The hopping factors of the cells are real, so $\theta_k=0$ and the velocity (10.25) vanishes. *Control,* for $\langle X\rangle=0$, $\varepsilon_x^2/\Delta_x=\gamma$ for all $x$ against cells in which it depends on $x$: for constant $\gamma$, $\gamma_1=0$, so $\mathcal F_k$, the branch-odd term $\sigma\tau^2\gamma_1M_k$ and the $\gamma_1$ part of $\mathcal R_k$ vanish. What remains: the level offsets $V_k=e_0-\gamma M_k^2$, hence the unequal spacings (10.15) and the range (10.16); the defect $\mathcal R_k=\tau^2g\gamma\Sigma$ of (4.24); and the level-dependent hopping, $a_{k,\sigma}(0)=-\tau^2g\,(1-2\rho_0M_k^2)$ with $\rho_0=\gamma/\Delta_0>0$, level-dependent iff $g\neq0$. If $\gamma_x$ depends on $x$, $\gamma_1\neq0$ and the fall is level-dependent at the leading order, with opposite sign in the two branches. For $\langle X\rangle\neq0$ the record term $a^{\rm rec}_{k,\sigma}$ remains in both cases. For $\varepsilon_x=0$ for all $x$ every new term vanishes.

### Step 5. Orders (item 5)

All rows: class $\mathcal D$ (rows 2–5: linear profiles of the field), regime R1, the ranges of $\lambda$ of (10.7), (10.10), (10.16). The entries state what a field with its own contract changes. Rows 2 and 4 are stated for $\langle X\rangle=0$: for $\langle X\rangle\neq0$ the fall from rest depends on the level and (4.24) has a defect already at zeroth order in $\epsilon_1$, also for a sharp field, by the record term of (10.22), (10.26), and the conditions of row 2 are those of (10.23).

| Quantity | Order $\epsilon_1$ | First change: order $\epsilon_1^2$, leading order in $\epsilon_2$, $\epsilon_3$ | Order $\epsilon_1^2$, first order in $\epsilon_2$, $\epsilon_3$ | Tags |
| --- | --- | --- | --- | --- |
| Factor that the body sees | $N(x)$ on $G$; no term linear in $\epsilon_1$ | factor unchanged; new on-site term $-\gamma_xM_k^2\,\mathbb 1_2$ | factor $N(x)+\gamma^{(3)}_xM_k^2$ ($\epsilon_3$); motion term times $1-M_k^2c_x$ ($\epsilon_2$) | (10.1), (10.7), (10.9), (10.10), (10.11) |
| Fall from rest independent of the level, for $\langle X\rangle=0$ | holds, both branches, $-\tau^2g$ | lost iff $\gamma_1\neq0$: $\sigma\tau^2\gamma_1M_k$ | lost iff $\beta\neq0$ ($N\ge3$): $\tau^2\beta M_k^2$ | (10.13), (10.14), (10.22), (10.23), (4.22) |
| Clock a ring clock with one rate | holds, rate $N(x)/\lambda_0$ | lost iff $\gamma_{x_0}\neq0$ and $N\ge3$; range (10.16); rate $\bar r$ shifted by $-\gamma_x\Sigma/\lambda_0$ | cubic term $\gamma^{(3)}_xM_k^3$ in $E^+_{x,k}$ | (10.15), (10.16), (3.23), (3.36) |
| Relation (4.24), for $\langle X\rangle=0$ | holds | defect $\mathcal R_k=\tau^2[\gamma_1(M_k-\Sigma)+g\gamma_0\Sigma]$; holds iff $\gamma_1=0$ and $g\gamma_0=0$ | further defect from $\beta$ and $\gamma^{(3)}_x$ | (10.17), (10.26), (4.24) |
| Velocity of a branch start at rest from the phase of the hopping (a velocity, not a drift acceleration), every $\langle X\rangle$ | none | none | $\bar v^{\rm ph}_{k,\sigma}=-\sigma\tau^2M_k\operatorname{Im}\rho_Y$ ($\epsilon_2$): odd in the branch, different for every two levels, present iff $\operatorname{Im}\rho_Y\neq0$ | (10.24), (10.25) |

$$
\text{rows 1–4: unchanged at order }\epsilon_1;\quad\text{first change at order }\epsilon_1^2\text{ with the coefficient at its leading order in }\epsilon_2,\ \epsilon_3,\ \text{of relative size }\lvert\gamma_x\rvert M_{\max}\le\epsilon_1\varepsilon=\epsilon_1^2\,\Delta_f/M_{\max};\quad\text{row 5: first change at order }\epsilon_1^2\text{ and first order in }\epsilon_2 .
\tag{10.21}
$$

## Result

- **Regime of the order estimates.** Every order estimate of this package holds in R1, $2\epsilon_1+2\epsilon_2+\epsilon_3<1$ with weak records. The smallness of $\epsilon_3=\nu M_{\max}/\Delta_f$ is a condition of this package in addition to the two small numbers of the question; members of $\mathcal D$ with $\epsilon_3$ not small are covered by the R0 statements only ((10.1), (10.2), (10.4), first part of (10.7)).
- **Item 1.** (10.1): $\hat C$, exact, whole class; body part in $\mathcal C$ with $m_x=mN(x)$, $\eta_x=N(x)$, $\zeta_x=1$, $V_x=e_0$; admissible. (10.2): rules under (2.20). (10.4): bound $\varepsilon M_{\max}\lambda$, R0, all $\lambda$; (10.6): $2\epsilon_1$, R1, all $\lambda$, order estimate. (10.7): statistics as in a sharp field with $N(x)$ through first order in $\epsilon_1$ for $\lambda\ll1/(\epsilon_1^2\Delta_f)$.
- **Item 2.** (10.8)–(10.10): $H_{\rm eff}=\hat C+H^{(2)}+H^{(2,1)}+H^{(3)}$ with remainder and range, R1. (10.11): blocks; member of $\mathcal C_{\rm lev}$ up to the hopping phase, which a unitary moves from the contract into the start (a change of the start that depends on the level, not the identity for a start in one level); all new terms well defined under (2.20).
- **Item 3.** (10.12): level data. (10.13), (10.14): fall from rest and its level-independence, for $\langle X\rangle=0$. (10.22), (10.23): the same for every $\langle X\rangle$, with the record term $a^{\rm rec}_{k,\sigma}$ of zeroth order in $\epsilon_1$; linear profiles, R1, closed form and conditions exact at the stated order. (10.24), (10.25): the velocity of a branch start at rest from the hopping phase, $\bar v^{\rm ph}_{k,\sigma}=-\sigma\tau^2M_k\operatorname{Im}\rho_Y$; linear profiles, R1, second order in $\epsilon_1$, first order in $\epsilon_2$ and in $Y$, exact coefficient in the description with $H_{\rm eff}$; a velocity, not a drift acceleration. (10.15), (10.16): spacings and clock range. (10.17): defect of (4.24) for $\langle X\rangle=0$; (10.26): for every $\langle X\rangle$.
- **Item 4.** (10.18)–(10.20): cells, exact on-site solution, fall (for $\langle X\rangle=0$; record term by (10.22); no velocity (10.25)) and control.
- **Item 5.** The table of Step 5 and (10.21).

## Consistency checks

1. **Dimensions.** $\Phi_x$, $N$, $\varepsilon$, $g$ are pure numbers; $\tau$, $M_k$, $\Delta_f$, $e_0$ are costs. Then $\gamma_x$ is an inverse cost and $\gamma_xM_k^2$ a cost; $\rho_{xy}M_k^2$, $\gamma^{(3)}_xM_k^2$, $\gamma_1M_k$, $\beta M_k^2$ are pure numbers like $g$; $a_{k,\sigma}(0)$ and $\mathcal R_k$ are $\tau^2$ times a number, as in (4.16). $\kappa\langle X\rangle$ is a cost, so $\kappa\langle X\rangle/M_k$ in (10.22), (10.26) is a pure number; $\theta_k$ is a pure number and $\bar v^{\rm ph}_{k,\sigma}=\tau^2M_k\rho_Y$ a cost, like $\omega_k'$, the dimension of $\mathrm d\bar x_{\rm lab}/\mathrm d\lambda$. Passed.
2. **Sharp field and commuting operators.** For $\varepsilon=0$: $W\Pi_0=0$, so $\gamma_x=\rho_{xy}=\gamma^{(3)}_x=0$, $H_{\rm eff}=\hat C$, $Q_0\Psi=0$ and $p=\hat p$ exactly, which is (2.3) with $n_x=N(x)$, $e_f=e_0$. If every $\Phi_x$ commutes with $C_f$, the simple eigenvector $\varphi_0$ is an eigenvector of every $\Phi_x$, so $\varepsilon=0$: all new terms need $[\Phi_x,C_f]\neq0$. For $\varepsilon=0$, (10.22) is $a_{k,\sigma}(0)=-\tau^2g-\sigma\tau^2\kappa\langle X\rangle/M_k$, which is (4.16) for the variant (a) with the plain gradient $\mathcal F=-\kappa\langle X\rangle$; it is level-independent iff $\langle X\rangle=0$, with the value $-\tau^2g$, as (4.22) states; and $\theta_k=0$, so (10.25) vanishes. Passed.
3. **Cell, exact against second order.** The exact lower eigenvalue and overlap (10.19) reproduce $-\gamma_xM_k^2$, $\gamma^{(3)}_x=0$ and the hopping factor $1-M_k^2c_x$ of (10.18), derived independently from (10.9), (10.10). Passed.

## Open issues

- (10.6), the R1 part of (10.7) and the remainders of (10.10) are order estimates: their constants, and the dependence of (10.7) on the number of readings, are not bounded explicitly.
- The accelerations (10.13), (10.20), (10.22) and the velocity (10.24), (10.25) are those of the description with $H_{\rm eff}$; for the description with $C$ only the agreement of the statistics within (10.10) is derived, not a statement about its acceleration or its velocity.
- (10.16) has the orders of (3.36) (zeroth order in the gradients). The loss of clock coherence by the level-dependent force $\mathcal F_k$ and by the level-dependent velocity (10.25), which separate the levels in position at first order in the gradients, is not treated.
- For profiles of the field that are not linear the phase of the hopping depends on $x$; its effect is given only as the change of the start of Step 2(b), and the velocity (10.24), (10.25) is derived for the linear profiles.
- Where $2\epsilon_1+2\epsilon_2+\epsilon_3\ge1$ (a slow field, or $\epsilon_3$ not small) the expansion fails and only the R0 statements (10.1), (10.2), (10.4) and the first part of (10.7) remain.

## Methods used

- Compression of the contract onto an eigenvector of the field
- Duhamel formula and norm bounds
- Schrieffer–Wolff block diagonalisation (nondegenerate perturbation theory with a gap)
- Sylvester equation, expansion in commutators with the body contract
- Rayleigh–Schrödinger second and third order with the reduced resolvent
- Exact diagonalisation of a two-place cell
- Level blocks and linear level profiles of the class $\mathcal C_{\rm lev}$
- Unitary removal of a hopping phase, with its action on the start
- Velocity operator of a level block (commutator with the position label), stationary phase and window average
- Effective body contract with the mean record as a plain gradient

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