Version 1 · earlier version; the current one is v6 · Review of v1

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 10-field-own-contract
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## 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, $\epsilon_3=\epsilon_1$. Every order estimate below needs $\epsilon_3\ll1$ besides $\epsilon_1,\epsilon_2\ll1$; this is stated with each result.
- **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 $\epsilon_3$.
- **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 $\sum_{x,k}e^{i\vartheta_{x,k}}P_x\otimes\mathbb 1_2\otimes\lvert f_k\rangle\langle f_k\rvert$ with $\vartheta_{x+1,k}-\vartheta_{x,k}=\arg(1-M_k^2c_x)$ removes it; this unitary is diagonal in $(x,s,k)$, so it changes no position or branch statistics and nothing for a start in one level. (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 $\kappa$. *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, $\theta_k=M_k^2\operatorname{Im}\rho_Y$; it shifts the plane-wave label of the level block, $p\to p-\theta_k$, is of first order in $Y$ and independent of $\lambda$, and therefore changes $a_{k,\sigma}$ only at second order in $Y$. The mean record $\kappa\langle X\rangle x$ of (8.19) adds $-\kappa\langle X\rangle$ to every $\mathcal F_k$ and vanishes at leading order in $\kappa$.

*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 $\varepsilon$:

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

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:

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

(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:* 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 $\beta=0$.

**(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

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

up to the first corrections in $\epsilon_2$, $\epsilon_3$ and $O(\varepsilon^4)$. **No**: (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 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 $\epsilon_1$, (4.24) holds with $r(x)=N(x)/\lambda_0$.

### 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)

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

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)). *Control,* $\varepsilon_x^2/\Delta_x=\gamma$ 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 $\varepsilon_x=0$ for all $x$ every new term vanishes.

### Step 5. Orders (item 5)

All rows: class $\mathcal D$ (rows 2–4: linear profiles of the field), regime R1, the ranges of $\lambda$ of (10.7), (10.10), (10.16).

| 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 | 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), (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) | 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), (4.24) |

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

## Result

- **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; 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. (10.15), (10.16): spacings and clock range. (10.17): defect of (4.24).
- **Item 4.** (10.18)–(10.20): cells, exact on-site solution, fall 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). 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 $[\Phi_x,C_f]\neq0$. 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

- The two small numbers of the question do not control the class $\mathcal D$: a third one, $\epsilon_3=\nu M_{\max}/\Delta_f$, enters the conditions and the first corrections. It is stated with every result; whether it should be part of the regime of the question is for the main model.
- (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) 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.
- (10.16) has the orders of (3.36) (zeroth order in the gradients). The loss of clock coherence by the level-dependent force $\mathcal F_k$, which separates the levels in position at first order in the gradients, is not treated.
- Where $2\epsilon_1+2\epsilon_2+\epsilon_3\ge1$ (a slow field) 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 $\mathcal C_{\rm lev}$

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