Version 1 · earlier version; the current one is v4 · External review, round 1: major errors

# A body in a changing field: gradient seen, fall from rest, level dependence

- **Subproject:** 05-ilang-free-fall
- **Package:** 12-changing-field
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## Setup and assumptions

**Model (assumption of the package, M8).** One body $b$ (Model D), one field $f$, one medium $c$, of pairwise different types (no swap condition). The contract of the question is (2.1) of 02-field-object with $r=\{c\}$ and one index:

$$C=C_{br}\otimes\mathbb 1_f+B\otimes Y+\mathbb 1_{br}\otimes C_f,\qquad C_{br}:=C_{\rm D}\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\kappa\,x\,X,\qquad B:=\hat x\otimes G\otimes\mathbb 1_c ,$$

with $G=\sigma_3\otimes(m\mathbb 1_N+D)$, $Y$ and $C_f$ self-adjoint on a finite-dimensional $\mathcal H_f$, $\lVert Y\rVert(L+1)<1$, all contracts independent of $\lambda$ (A5), $\lambda\ge0$. The start is $\Psi_0=\Psi_{br}\otimes\varphi$ with $\Psi_{br}=\phi\otimes\chi$ and $\varphi$ any unit vector (A11).

**Gradient seen.** With $P_e$ the eigenprojectors of $C_f$ and $\nu$ running over the field frequencies ($\nu=0$ included),

$$\gamma(\lambda)=\langle\varphi\vert e^{iC_f\lambda}Ye^{-iC_f\lambda}\vert\varphi\rangle=\sum_\nu\hat\gamma_\nu\,e^{i\nu\lambda},\qquad\hat\gamma_\nu:=\sum_{e-e'=\nu}\langle\varphi\vert P_eYP_{e'}\vert\varphi\rangle,\qquad\hat\gamma_{-\nu}=\hat\gamma_\nu^{*} ,\tag{12.1}$$

a real trigonometric polynomial. It is constant on $\lambda\ge0$ iff $\hat\gamma_\nu=0$ for every $\nu\neq0$ (always so for the exceptional member $Y=y\,\mathbb 1_f$).

**Level notation.** Inside one level $M:=M_k$, $\omega:=\omega_k$, $\theta:=\theta_k$, a prime is $\partial_p$, $\Delta:=\lambda-s$, $u:=1+\tau^2/M^2$. From (5.6) of 04-ilang-time/05-moving-clock: $\cos\theta=M/\omega$, $\sin\theta=-\tau\sin p/\omega$, $\omega'=\tau^2\sin p\cos p/\omega$, $\theta'=-\tau M\cos p/\omega^2$, $\omega\omega''+\omega'^2=\tau^2\cos2p$.

**Functionals of a branch start $B_{k,\sigma}[g]$** (all integrals over $(-\pi,\pi]$):

$$j_g(p):=\operatorname{Re}\bigl(g^{*}\,i\,g'\bigr),\qquad\chi_g:=2\int\frac{\mathrm dp}{2\pi}\,\sin p\;j_g=\langle\phi\vert\{\sin\hat p,\hat x\}\vert\phi\rangle,\qquad\rho_g:=\int\frac{\mathrm dp}{2\pi}\,\lvert\sin p\rvert\,\sin^2\tfrac p2\,\lvert j_g\rvert .\tag{12.2}$$

$j_g$ is the position density of the start ($\int\frac{\mathrm dp}{2\pi}j_g=\bar x_{\rm lab}(0)$, because $\beta_{k,\sigma}^{T}\beta_{k,\sigma}'=0$); $\chi_g$ is its **position–label correlation**. The **class $\mathcal S_\chi$** ($\chi\in\mathbb R$) consists of the sequences of branch starts at rest with $\delta p\to0$, $\rho_g\to0$ and $\chi_g\to\chi$. Members: $g=e^{-ipx_0}\times(\text{real function})$ with fixed $x_0\in\mathbb Z$ gives $j_g=x_0\lvert g\rvert^2$, hence $\mathcal S_0$; a chirped start $g\propto\exp[-(1+i\chi)p^2/4\delta p^2]$ (smoothly cut off at $\pm\pi$) gives $\mathcal S_\chi$; a member of $\mathcal S_0$ evolved freely over $\lambda_e$ has $\chi_g$ increased by $2\sigma\lambda_e\langle\omega'\sin p\rangle_g$.

**Orders and regimes (A13).**
- First order in $Y$ at fixed $\lambda$, start and chain, then a long chain; for smooth $g$ the position amplitudes decay faster than any power, so the first-order coefficients of the chain $\lvert x\rvert\le L$ converge to those of the infinite chain.
- Records (items 2–4): $\kappa\to0$ at fixed $\lambda$, chain and start, $\langle X\rangle=0$. The medium has no contract of its own, so the $O(\kappa)$ term of the view of $b$ contains one factor $\langle X\rangle$ and vanishes ((8.19) of 04-ilang-time/08-maximal-speed); the place weights are those of $\kappa=0$ up to $O(\kappa^2)$.
- Two orders of limits for starts at rest: **(I)** $\delta p\to0$ at fixed $\lambda$; **(II)** $\lambda\to\infty$ at a fixed start, then $\delta p\to0$. They do not commute (Step 9).
- Inputs: as quoted in the question; (5.11) of 04-ilang-time/05-moving-clock and (8.1), (8.20) of 04-ilang-time/08-maximal-speed are not needed.

## Derivation

### Step 1. First order in the field (item 1a)

$C_{br}\otimes\mathbb 1_f$ and $\mathbb 1_{br}\otimes C_f$ commute. With $B(s):=e^{iC_{br}s}Be^{-iC_{br}s}$, $Y(s):=e^{iC_fs}Ye^{-iC_fs}$, the Dyson expansion in the term $B\otimes Y$ (convergent in finite dimension) gives

$$\Psi(\lambda)=e^{-i(C_{br}+C_f)\lambda}\Bigl[\Psi_{br}\otimes\varphi-i\int_0^\lambda\mathrm ds\,B(s)\Psi_{br}\otimes Y(s)\varphi\Bigr]+O(\lVert Y\rVert^2).$$

Taking the partial trace over $f$ with $\operatorname{Tr}_f\bigl[Y(s)\lvert\varphi\rangle\langle\varphi\rvert\bigr]=\gamma(s)\in\mathbb R$, and $\rho_0:=\lvert\Psi_{br}\rangle\langle\Psi_{br}\rvert$:

$$V(\lambda)=e^{-iC_{br}\lambda}\Bigl\{\rho_0-i\int_0^\lambda\mathrm ds\,\gamma(s)\,\bigl[B(s),\rho_0\bigr]\Bigr\}e^{iC_{br}\lambda}+O(\lVert Y\rVert^2).\tag{12.3}$$

For a schedule, $p(r_1,\dots,r_k)=\lVert\Pi_kU(\lambda_k-\lambda_{k-1})\cdots\Pi_1U(\lambda_1)\Psi_0\rVert^2$ with projectors acting on $\mathcal H_{br}$ only; every first-order term contains exactly one factor $Y(s)$, $s\in[0,\lambda_k]$, between $\langle\varphi\rvert$ and $\lvert\varphi\rangle$, hence again $\gamma(s)$.

**Result 1(a).** *Class:* every $C_f$, $Y$, $\varphi$, every $\Psi_{br}$ (any $\kappa$). *Status:* exact first-order coefficient. The first-order term of the view of $\{b,c\}$ at $\lambda$, and of every schedule ending at $\lambda_k$, depends on $(\varphi,C_f,Y)$ only through the one real function $\gamma$ on $[0,\lambda]$ (on $[0,\lambda_k]$), i.e. through the finitely many numbers $\hat\gamma_\nu$ of (12.1). *Control:* the second-order term contains $\langle\varphi\vert Y(s)Y(s')\vert\varphi\rangle$, which is not a function of $\gamma$ (Step 3).

### Step 2. The generator and the two special cases (item 1b)

$$C_{\rm eff}(\lambda):=C_{br}+\gamma(\lambda)\,B=C_{\rm D}\otimes\mathbb 1_c+\gamma(\lambda)\,\hat x\otimes G\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\kappa\,x\,X,\qquad i\,\frac{\mathrm d}{\mathrm d\lambda}\Psi_{\rm eff}=C_{\rm eff}(\lambda)\Psi_{\rm eff},\quad\Psi_{\rm eff}(0)=\Psi_{br}.\tag{12.4}$$

Its Dyson expansion is that of Step 1 with $Y(s)\to\gamma(s)$, so $V(\lambda)=\lvert\Psi_{\rm eff}(\lambda)\rangle\langle\Psi_{\rm eff}(\lambda)\rvert+O(\lVert Y\rVert^2)$, and the same for every schedule: the body sees the factor $1+\gamma(\lambda)x$ on its rest and clock term.

- *Admissible?* By A5 the operators of a contract do not depend on $\lambda$. $C_{\rm eff}(\lambda)$ is the contract of an admissible description iff $\gamma$ is constant, i.e. iff $\hat\gamma_\nu=0$ for all $\nu\neq0$; it is then $C_{\rm red}$ of (2.3) of 02-field-object with $n_1=\gamma$ (a body with a place-dependent own term, A10, and its medium). Otherwise: **no**; it is a derived generator, not a description.
- *Sharp and stationary* ($Y\varphi=\gamma\varphi$, $C_f\varphi=e_f\varphi$): $\gamma(\lambda)=\gamma$, and by (2.3) $\Psi(\lambda)=e^{-ie_f\lambda}e^{-iC_{\rm red}\lambda}\Psi_{br}\otimes\varphi$: the statement of (a) holds at all orders in $Y$.
- *Field with (2.11)*: $e^{-iC_f\lambda}\varphi$ lies in the span of the $C_f^{\,k}\varphi$, so $Ye^{-iC_f\lambda}\varphi=\gamma e^{-iC_f\lambda}\varphi$ and $\gamma(\lambda)=\gamma$; by (2.12) $V(\lambda)=\lvert\Psi_{\rm red}(\lambda)\rangle\langle\Psi_{\rm red}(\lambda)\rvert$ exactly. The field changes with $\lambda$ ($V_f$ of (2.12)), the body does not see it.

### Step 3. Where the generator fails (item 1c)

Let $\varphi$ be sharp, $Y\varphi=\gamma\varphi$, with (2.11) violated. The view produced by (12.4) is pure at all orders, and by Step 2 it agrees with $V(\lambda)$ at the orders $Y^0$ and $Y^1$. By (2.14) of 02-field-object with $J=\{1\}$, $B_1=B$, $n_1=\gamma$:

$$1-\operatorname{Tr}V(\lambda)^2=\frac{\lambda^4}{2}\,\Gamma\,\bigl\lVert(Y-\gamma)\,C_f\varphi\bigr\rVert^2+O(\lambda^5),\qquad\Gamma=\langle B^2\rangle_{\Psi_{br}}-\langle B\rangle_{\Psi_{br}}^2 ,\tag{12.5}$$

which is homogeneous of degree two in $Y$. **Result 1(c):** the view first differs from the one of the generator at order $Y^2$ and at order $\lambda^4$, if $(Y-\gamma)C_f\varphi\neq0$ and $\Gamma\neq0$. Exceptional members: if (2.11) first fails at $k_0>1$, then $(Y-\gamma(s))e^{-iC_fs}\varphi=O(s^{k_0})$; the second-order difference of the views is bilinear in this vector at two parameters $s,s'\le\lambda$ and is integrated twice, so the order is $Y^2$ and $\lambda^{2k_0+2}$; if $\Psi_{br}$ is an eigenvector of $B$ ($\Gamma=0$) the coefficient of $\lambda^4$ vanishes.

### Step 4. First-order velocity in a gradient that changes with $\lambda$ (item 2a)

*Reduction.* By Step 2 and the record regime, the first-order place weights of a branch start are those of the body alone under $C_{\rm D}+\gamma(\lambda)\hat x\otimes G$. $D$ is conserved, and on the level $k$ the generator is, by (5.5) of 04-ilang-time/05-moving-clock ((4.2) of 04-branch-fall with $B(p)=\gamma M\sigma_3$, now with $\gamma$ depending on $\lambda$),

$$C^{(k)}(\lambda)=C_0+\gamma(\lambda)\,M\,\hat x\,\sigma_3,\qquad C_0=-\tau\sin\hat p\,\sigma_1+M\sigma_3 .$$

*Velocity.* $\frac{\mathrm d}{\mathrm d\lambda}\bar x_{\rm lab}=\langle i[C^{(k)}(\lambda),\hat x]\rangle=\langle\hat V\rangle$ with $\hat V=\partial_pC_0=-\tau\cos\hat p\,\sigma_1$, because $\hat x\sigma_3$ commutes with $\hat x=i\partial_p$. With $\phi_0(\lambda)=e^{-iC_0\lambda}\phi$ (amplitude $g\,e^{-i\sigma\omega\lambda}\beta_\sigma$) and the first-order term $\phi_1(\lambda)=-iM\int_0^\lambda\mathrm ds\,\gamma(s)\,e^{-iC_0\Delta}\,\hat x\sigma_3\,\phi_0(s)$, the first-order velocity is $v^{(1)}(\lambda)=2\operatorname{Re}\langle\phi_0(\lambda)\vert\hat V\vert\phi_1(\lambda)\rangle$.

*Branch basis.* From (5.6) of 04-ilang-time/05-moving-clock: $\sigma_3\beta_\sigma=\sigma\cos\theta\,\beta_\sigma-\sin\theta\,\beta_{-\sigma}$, $\hat V\beta_\sigma=\sigma\omega'\beta_\sigma+\omega\theta'\beta_{-\sigma}$, $\beta_\sigma'=\sigma\tfrac{\theta'}2\beta_{-\sigma}$. Writing $e^{i\sigma\omega s}\partial_p\bigl[g\,e^{-i\sigma\omega s}\sigma_3\beta_\sigma\bigr]=u_\sigma\beta_\sigma+u_{-\sigma}\beta_{-\sigma}$,

$$u_\sigma=\sigma\bigl[(g'-i\sigma\omega'sg)\cos\theta-\tfrac12\theta'g\sin\theta\bigr],\qquad u_{-\sigma}=-(g'-i\sigma\omega'sg)\sin\theta-\tfrac12\theta'g\cos\theta ,$$

$$\langle\phi_0(\lambda)\vert\hat V\vert\phi_1(\lambda)\rangle=M\int_0^\lambda\mathrm ds\,\gamma(s)\int\frac{\mathrm dp}{2\pi}\,g^{*}\Bigl[\sigma\omega'\,u_\sigma+\omega\theta'\,e^{2i\sigma\omega\Delta}\,u_{-\sigma}\Bigr].$$

*Real part.* Use $\operatorname{Re}(g^{*}g')=\tfrac12(\lvert g\rvert^2)'$, $\operatorname{Im}(g^{*}g')=-j_g$, integrate the terms with $(\lvert g\rvert^2)'$ by parts ($g$ smooth and periodic), and use $\omega\theta'\sin\theta=M\omega'/\omega$, $(\omega'\cos\theta)'+\omega'\theta'\sin\theta=\omega''\cos\theta$, $(\omega\theta'\sin\theta)'-\omega\theta'^2\cos\theta=(\omega\theta')'\sin\theta$. The two terms that carry $s$ and $\Delta$ as factors combine to $s+\Delta=\lambda$. The result is

$$\boxed{\;v^{(1)}_g(\lambda)=\int_0^\lambda\mathrm ds\;\gamma(s)\,\mathcal K_g(\lambda,s)\;}\tag{12.6}$$

$$\mathcal K_g(\lambda,s)=\Bigl\langle k_0+k_1\cos\bigl(2\omega(\lambda-s)\bigr)\Bigr\rangle_g-2\sigma M^2\int\frac{\mathrm dp}{2\pi}\,\frac{\omega'}{\omega}\,\sin\bigl(2\omega(\lambda-s)\bigr)\,\Bigl[j_g+\sigma\lambda\,\omega'\lvert g\rvert^2\Bigr],\tag{12.7}$$

$$k_0(p)=-\frac{M^2\omega''}{\omega}=-\frac{M^2\tau^2\bigl(M^2\cos2p-\tau^2\sin^4p\bigr)}{\omega^4},\qquad k_1(p)=M(\omega\theta')'\sin\theta=-\frac{M^2\tau^2(M^2+\tau^2)\sin^2p}{\omega^4}.$$

**Result 2(a).** *Class:* every level, both branches, every smooth $g$, every continuous $\gamma$. *Status:* exact first-order coefficient at $\kappa\to0$, $\langle X\rangle=0$. The bracket $j_g+\sigma\lambda\omega'\lvert g\rvert^2$ is the position density of the free motion at $\lambda$. Case distinction: if $j_g=x_0\lvert g\rvert^2$ the kernel is the average $\langle\cdot\rangle_g$ of an explicit function of $(p,\lambda,s)$, the bracket being $(x_0+\sigma\lambda\omega')\lvert g\rvert^2$; in general the start enters through $j_g$ as well, and no formula through $\lvert g\rvert^2$ alone exists (*control:* two starts with equal $\lvert g\rvert^2$ and different chirp have different kernels, Step 6).

### Step 5. Check against (4.7) for a constant $\gamma$

For $\gamma(s)=\gamma$, $\int_0^\lambda\mathrm ds\,\sin(2\omega\Delta)=(1-\cos2\omega\lambda)/2\omega$ and $\int_0^\lambda\mathrm ds\,\cos(2\omega\Delta)=\sin(2\omega\lambda)/2\omega$. The terms that grow are $\gamma\lambda\langle k_0-M^2\omega'^2/\omega^2\rangle_g$; the term $\gamma\lambda M^2\langle\omega'^2\omega^{-2}\cos2\omega\lambda\rangle_g$ is bounded for $\lambda\ge0$ after one integration by parts in $p$ ($\cos2\omega\lambda=\partial_p\sin2\omega\lambda/2\omega'\lambda$), and all others are bounded. With $\omega\omega''+\omega'^2=\tau^2\cos2p$:

$$v^{(1)}_g(\lambda)=\gamma\,\lambda\,\langle\alpha_0\rangle_g+(\text{bounded}),\qquad\alpha_0(p):=-\frac{M_k^2\tau^2\cos2p}{\omega_k^2},$$

which is (4.7) of 04-branch-fall with $G_k=\gamma M_k$, $\mathcal F_k=0$, $g_{\zeta,k}=0$. The check passes.

### Step 6. Starts at rest, sharp label at fixed $\lambda$: kernel, velocity, acceleration (item 2b)

*Elementary bounds, all $p$, $0\le s\le\lambda$* (from the explicit forms; the last one uses $1-\cos p=2\sin^2\frac p2$, $0\le\omega-M\le2\tau^2\sin^2\frac p2/M$ and the mean value theorem for $w\mapsto\sin(2w\Delta)/w^2$ on $w\ge M$):

$$0\le k_0+\tau^2\le2\tau^2u\sin^2p,\quad\lvert k_1\rvert\le\tau^2u\sin^2p,\quad2M^2\lambda\frac{\omega'^2}{\omega}\le\frac{2\tau^4\lambda}{M}\sin^2p,\quad\Bigl\lvert\frac{\omega'}{\omega}\sin(2\omega\Delta)-\frac{\tau^2}{M^2}\sin(2M\Delta)\sin p\Bigr\rvert\le\frac{4\tau^2}{M^2}\Bigl(u+\frac{\tau^2\lambda}{M}\Bigr)\lvert\sin p\rvert\sin^2\tfrac p2 .$$

At rest $\langle\sin^2p\rangle_g\le\delta p^2$. Inserting into (12.7), with (12.2): for **every** smooth $g$ at rest,

$$\Bigl\lvert\mathcal K_g(\lambda,s)-K_{\chi_g}(\lambda-s)\Bigr\rvert\le4\tau^2\Bigl(u+\frac{\tau^2\lambda}{M}\Bigr)\bigl(\delta p^2+2\rho_g\bigr),\qquad\boxed{\;K_\chi(\Delta):=-\tau^2\bigl[1+\sigma\,\chi\,\sin(2M_k\Delta)\bigr]\;}\tag{12.8}$$

**Kernel in closed form.** *Class:* sequences of $\mathcal S_\chi$, every level, both branches. *Order:* (I), $\delta p\to0$ at fixed $\lambda$, uniformly in $s\in[0,\lambda]$. *Status:* limit, with the exact bound (12.8) for each member. $\mathcal K_g\to K_\chi$; for a sequence with $\rho_g\to0$ the limit exists iff $\chi_g$ converges (for $\sin2M_k\Delta\neq0$). The level enters only through the term $\sigma\chi\sin(2M_k\Delta)$: for $\chi=0$ the kernel is $-\tau^2$, without memory.

**Velocity and acceleration, any continuous $\gamma$.** From (12.6) and (12.8), with the error at most $4\tau^2\lambda\sup_{[0,\lambda]}\lvert\gamma\rvert\,(u+\tau^2\lambda/M)(\delta p^2+2\rho_g)$ for each member, the limit of the velocity is the first formula of (12.9). The acceleration is derived on its own: by the Leibniz rule $a^{(1)}_g(\lambda)=\gamma(\lambda)\mathcal K_g(\lambda,\lambda)+\int_0^\lambda\mathrm ds\,\gamma(s)\,\partial_\lambda\mathcal K_g(\lambda,s)$, where

$$\mathcal K_g(\lambda,\lambda)=\langle k_0+k_1\rangle_g=-\Bigl\langle\frac{M^2\tau^2\cos^2p}{\omega^2}\Bigr\rangle_g,\quad\partial_\lambda\mathcal K_g=-\Bigl\langle\Bigl(2\omega k_1+\frac{2M^2\omega'^2}{\omega}\Bigr)\sin(2\omega\Delta)+4M^2\lambda\,\omega'^2\cos(2\omega\Delta)\Bigr\rangle_g-4\sigma M^2\!\int\!\frac{\mathrm dp}{2\pi}\,\omega'\cos(2\omega\Delta)\,j_g ,$$

and the same bounds give $\lvert\mathcal K_g(\lambda,\lambda)+\tau^2\rvert\le\tau^2u\,\delta p^2$ and $\lvert\partial_\lambda\mathcal K_g+2\sigma\chi_gM\tau^2\cos(2M\Delta)\rvert\le4M\tau^2(u+2\tau^2\lambda/M)(\delta p^2+2\rho_g)$. Hence, in the limit (I) of the class $\mathcal S_\chi$,

$$\boxed{\;v^{(1)}_{k,\sigma}(\lambda)=-\tau^2\!\int_0^\lambda\!\mathrm ds\,\gamma(s)\bigl[1+\sigma\chi\sin\bigl(2M_k(\lambda-s)\bigr)\bigr],\qquad a^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma(\lambda)-2\sigma\chi M_k\tau^2\!\int_0^\lambda\!\mathrm ds\,\gamma(s)\cos\bigl(2M_k(\lambda-s)\bigr)\;}\tag{12.9}$$

and for every smooth $g$ at rest, with $a_{\chi_g}$ the second expression of (12.9) at $\chi=\chi_g$ (exact bound):

$$\bigl\lvert a^{(1)}_g(\lambda)-a_{\chi_g}(\lambda)\bigr\rvert\le\tau^2\sup_{[0,\lambda]}\lvert\gamma\rvert\,(1+4M\lambda)\Bigl(u+\frac{2\tau^2\lambda}{M}\Bigr)\bigl(\delta p^2+2\rho_g\bigr).\tag{12.10}$$

(12.9) describes a fixed start as long as $\lambda\ll\lambda_g:=M/\bigl[\tau^2(\delta p^2+2\rho_g)\bigr]$.

### Step 7. One field frequency, $\gamma(\lambda)=\gamma_0+\gamma_1\cos(\Omega\lambda+\vartheta)$ (item 2b)

$y(\lambda)=\int_0^\lambda\mathrm ds\,f(s)\sin(2M(\lambda-s))$ is the solution of $y''+4M^2y=2Mf$, $y(0)=y'(0)=0$. For $f=\cos(\Omega s+\vartheta)$ and $\Omega\neq2M$ this gives $y=\bigl[2M(\cos(\Omega\lambda+\vartheta)-\cos\vartheta\cos2M\lambda)+\Omega\sin\vartheta\sin2M\lambda\bigr]/(4M^2-\Omega^2)$. Inserting into (12.9), for $\Omega\neq2M_k$ ($M=M_k$):

$$\begin{aligned}v^{(1)}_{k,\sigma}(\lambda)={}&\underbrace{-\tau^2\gamma_0\,\lambda}_{\text{grows}}\;\underbrace{-\,\tau^2\gamma_1\Bigl[\frac{\sin(\Omega\lambda+\vartheta)}{\Omega}+\sigma\chi\,\frac{2M\cos(\Omega\lambda+\vartheta)}{4M^2-\Omega^2}\Bigr]}_{\text{frequency }\Omega}\\&\underbrace{+\,\sigma\chi\tau^2\Bigl[\frac{\gamma_0}{2M}\cos2M\lambda+\gamma_1\frac{2M\cos\vartheta\cos2M\lambda-\Omega\sin\vartheta\sin2M\lambda}{4M^2-\Omega^2}\Bigr]}_{\text{frequency }2M_k}\;\underbrace{+\,\tau^2\Bigl[\frac{\gamma_1\sin\vartheta}{\Omega}-\frac{\sigma\chi\gamma_0}{2M}\Bigr]}_{\text{constant}}\end{aligned}\tag{12.11}$$

$$\begin{aligned}a^{(1)}_{k,\sigma}(\lambda)={}&-\tau^2\gamma_0\;\underbrace{-\,\tau^2\gamma_1\Bigl[\cos(\Omega\lambda+\vartheta)-\sigma\chi\,\frac{2M\Omega\,\sin(\Omega\lambda+\vartheta)}{4M^2-\Omega^2}\Bigr]}_{\text{frequency }\Omega}\\&\underbrace{-\,\sigma\chi\tau^2\Bigl[\gamma_0\sin2M\lambda+\gamma_1\frac{4M^2\cos\vartheta\sin2M\lambda+2M\Omega\sin\vartheta\cos2M\lambda}{4M^2-\Omega^2}\Bigr]}_{\text{frequency }2M_k}\end{aligned}\tag{12.12}$$

(12.12) follows from the second formula of (12.9) (equivalently, $\mathrm d/\mathrm d\lambda$ of (12.11)); $v^{(1)}(0)=0$ and $a^{(1)}(0)=-\tau^2\gamma(0)$ hold. **Result 2(b).** *Class:* $\mathcal S_\chi$, every level, both branches, all $\gamma_0,\gamma_1,\vartheta$, $\Omega>0$, $\Omega\neq2M_k$. *Order:* (I). *Status:* exact evaluation of the limit (12.9). The only part that grows is the drift $-\tau^2\gamma_0\lambda$ of the velocity; the acceleration has no growing part. The parts of frequency $2M_k$ are proportional to $\chi$ and are present already for a constant $\gamma$ (they are the bounded rest of (4.7)).

### Step 8. The response at the field frequency (item 3a)

Since $\operatorname{Re}[(1+i\,b)e^{i\phi}]=\cos\phi-b\sin\phi$, the part of (12.12) with the frequency $\Omega$ is $\operatorname{Re}\bigl[\mathcal A_{k,\sigma}(\Omega)\gamma_1e^{i(\Omega\lambda+\vartheta)}\bigr]$ with

$$\boxed{\;\mathcal A_{k,\sigma}(\Omega)=-\tau^2\Bigl[1+i\,\sigma\,\chi\,\frac{2M_k\Omega}{4M_k^2-\Omega^2}\Bigr]\;},\qquad\Omega\neq2M_k .\tag{12.13}$$

*Class, order, status:* as Result 2(b); exact.
- **$\Omega\to0$:** $\mathcal A_{k,\sigma}\to-\tau^2$ for every $\chi$, level and branch. This is (4.16) of 04-branch-fall per unit gradient: $a_{k,\sigma}(0)=\frac{\tau^2}{M_k}(-G_k)=-\tau^2\gamma$ for $G_k=\gamma M_k$, $\mathcal F_k=0$.
- **Same for all levels.** For two levels, $\frac{M_k}{4M_k^2-\Omega^2}=\frac{M_{k'}}{4M_{k'}^2-\Omega^2}$ is equivalent to $(M_{k'}-M_k)(4M_kM_{k'}+\Omega^2)=0$, which fails because the $M_k=m+d_k$ of Model D are positive and pairwise different ($N\ge2$). Hence, for starts of one class $\mathcal S_\chi$ on all levels and any $\Omega>0$ different from all $2M_k$: $\mathcal A_{k,\sigma}(\Omega)$ is the same for all levels **iff $\chi=0$**, and then also for both branches and all $\Omega$, with the value $-\tau^2$. If the classes differ between the levels ($\chi_k$), the condition is that $\sigma_k\chi_kM_k/(4M_k^2-\Omega^2)$ is the same for all levels; for all $\Omega$ of an interval this again means $\chi_k=0$ for all $k$.
- **Leading level dependence, $\Omega\ll2M_k$:** $\mathcal A_{k,\sigma}(\Omega)=-\tau^2\bigl[1+i\sigma\chi\frac{\Omega}{2M_k}\bigl(1+O(\Omega^2/4M_k^2)\bigr)\bigr]$: a phase shift of the response of relative size $\lvert\chi\rvert\Omega/2M_k$, opposite for the two branches.

### Step 9. Resonance $\Omega=2M_k$ and the other order of limits (item 3b)

*Order (I).* For $f=\cos(2Ms+\vartheta)$ the equation of Step 7 is resonant: $y=\frac\lambda2\sin(2M\lambda+\vartheta)-\frac{\sin\vartheta\sin2M\lambda}{4M}$. With (12.9), for the class $\mathcal S_\chi$ and $\Omega=2M_k$:

$$v^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma_0\lambda-\sigma\chi\,\tau^2\gamma_1\,\frac\lambda2\,\sin(2M_k\lambda+\vartheta)+(\text{bounded}),\qquad a^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma(\lambda)-\sigma\chi\tau^2\Bigl[\gamma_1M_k\lambda\cos(2M_k\lambda+\vartheta)+\bigl(\gamma_0+\tfrac{\gamma_1}2\cos\vartheta\bigr)\sin2M_k\lambda\Bigr].\tag{12.14}$$

The velocity carries an oscillation of frequency $2M_k$ whose amplitude grows linearly, $\tfrac12\lvert\chi\gamma_1\rvert\tau^2\lambda$ (the acceleration: $\lvert\chi\gamma_1\rvert\tau^2M_k\lambda$). In every other level the oscillating parts are bounded, by (12.11). For $\chi=0$ nothing grows besides the drift, and no level is distinguished.

*Order (II): fixed smooth $g$ (any $p_0$), $\lambda\to\infty$.* In (12.7) write $\int_0^\lambda\mathrm ds\,e^{i\Omega s}\sin(2\omega\Delta)=e^{i\Omega\lambda}\int_0^\lambda\mathrm du\,e^{-i\Omega u}\sin2\omega u$. By stationary phase, $\langle W\sin2\omega u\rangle_g$ decays like $u^{-3/2}$ for the weight $W=\omega'^2/\omega$, which vanishes to second order at all critical points of $\omega$ ($\sin p\cos p=0$); so the term of (12.7) with the factor $\lambda$ gives $\lambda$ times a convergent integral. The terms with $k_1$ and with $j_g$ have weights decaying like $u^{-1/2}$ and $u^{-1}$ and contribute $O(\lambda^{1/2})$. With the $\gamma_0$ part as in Step 5, for **every** $\Omega>0$ (leading order, rest $O(\lambda^{1/2})$):

$$v^{(1)}_g(\lambda)=\lambda\Bigl\{\gamma_0\langle\alpha_0\rangle_g+\gamma_1\operatorname{Re}\bigl[c_g(\Omega)\,e^{i(\Omega\lambda+\vartheta)}\bigr]\Bigr\}+o(\lambda),\qquad c_g(\Omega):=-\lim_{\epsilon\to0^+}\Bigl\langle\frac{4M^2\omega'^2}{4\omega^2-(\Omega-i\epsilon)^2}\Bigr\rangle_g .\tag{12.15}$$

Then $\delta p\to0$ at rest: $\langle\alpha_0\rangle_g\to-\tau^2$; for $\Omega<2M_k$, $\lvert c_g\rvert\le4\tau^4\delta p^2/(4M^2-\Omega^2)\to0$; for $\Omega=2M_k$, $4\omega^2-4M^2=4\tau^2\sin^2p$ gives $c_g=-\langle M^2\tau^2\cos^2p/\omega^2\rangle_g\to-\tau^2$ for **every** sequence at rest, whatever $\chi_g$; for $\Omega>2M_k$, $c_g\to0$ along sequences that vanish on a fixed neighbourhood of $\{p:2\omega_k(p)=\Omega\}$ (an empty set for $\Omega>2\sqrt{M_k^2+\tau^2}$).

**Result 3(b), the two orders.** (I) defines the response of a start for $\lambda\ll\lambda_g$: at $\Omega=2M_k$ the growing part is $-\sigma\chi\tau^2\gamma_1\frac\lambda2\sin(2M_k\lambda+\vartheta)$. (II) defines the long-run growth rate of the velocity of a fixed start: at $\Omega=2M_k$ it is $-\tau^2\gamma_1\lambda\cos(2M_k\lambda+\vartheta)$, independent of $\chi$. The two differ, so the limits do not commute at resonance; both agree on the drift $-\tau^2\gamma_0\lambda$ and, for $\Omega<2M_k$, on the absence of any other growth. What distinguishes the level $k$ with $2M_k=\Omega$: the field frequency equals the gap $2\omega_k(0)$ between its two branches at rest, and only there an oscillation of the velocity grows linearly.

### Step 10. Accuracy $\delta$ and the general statement (item 3c)

By (12.13), $\lvert\mathcal A_{k,\sigma}(\Omega)+\tau^2\rvert\le\delta\,\tau^2$ iff $2M_k\Omega\lvert\chi\rvert\le\delta\,\lvert4M_k^2-\Omega^2\rvert$, a quadratic inequality in $\Omega$:

$$\bigl\lvert\mathcal A_{k,\sigma}(\Omega)+\tau^2\bigr\rvert\le\delta\,\tau^2\ \text{ for all levels}\iff\Omega\notin\bigcup_k\bigl(\Omega^{(k)}_-,\Omega^{(k)}_+\bigr),\qquad\Omega^{(k)}_\mp:=\frac{M_k}{\delta}\Bigl(\sqrt{\chi^2+4\delta^2}\mp\lvert\chi\rvert\Bigr),\quad\Omega^{(k)}_-\Omega^{(k)}_+=4M_k^2 .\tag{12.16}$$

*Class, order, status:* $\mathcal S_\chi$, order (I), exact (iff). For $\chi=0$ the intervals are empty: every field frequency. For $\delta\ll\lvert\chi\rvert$: $\Omega^{(k)}_-\approx2\delta M_k/\lvert\chi\rvert$; the low-frequency condition for all levels is $\Omega\le\Omega^{(0)}_-$ ($M_0=m$ is the smallest). The parts of frequency $2M_k$ in (12.12) depend on the level with the relative size $\lvert\chi\rvert(1+2M_k/\lvert2M_k-\Omega\rvert)$ at most; lowering $\Omega$ does not remove them, an average over a long window does.

**General $\gamma(\lambda)$.** By (12.9), linear in $\gamma$, and (12.1): $a^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma(\lambda)+\sigma\chi\,R_k[\gamma](\lambda)$ with $R_k[\gamma](\lambda)=-2M_k\tau^2\int_0^\lambda\mathrm ds\,\gamma(s)\cos(2M_k(\lambda-s))$; if no $\lvert\nu\rvert$ equals $2M_k$, $a^{(1)}_{k,\sigma}(\lambda)=\sum_\nu\mathcal A_{k,\sigma}(\nu)\hat\gamma_\nu e^{i\nu\lambda}+(\text{terms of frequency }2M_k)$, with $\mathcal A_{k,\sigma}(-\nu)=\mathcal A_{k,\sigma}(\nu)^{*}$. The statement that replaces "the fall is the same for all levels": *in the limit (I), the first-order acceleration from rest at $\lambda$ is $-\tau^2$ times the gradient seen at the same $\lambda$, for all levels and both branches, plus a term proportional to the position–label correlation $\chi$ of the start, which filters $\gamma$ at the frequency $2M_k$ of the level.* For $\chi=0$ the first term is all, for every $\gamma(\cdot)$; in order (II) the growth rate of the velocity is $-\tau^2\gamma_0$ ($\gamma_0=\hat\gamma_0$) for all levels if every field frequency in $\gamma$ is below $2m$.

### Step 11. Example and controls (item 4)

**(a) The field (2.13) coupled as a gradient.** $e^{-iC_f\lambda}\lvert0\rangle=\cos(\gamma_f\lambda)\lvert0\rangle-i\sin(\gamma_f\lambda)\lvert1\rangle$, hence

$$\gamma(\lambda)=y\sin^2(\gamma_f\lambda)=\tfrac y2-\tfrac y2\cos(2\gamma_f\lambda):\qquad\gamma_0=\tfrac y2,\quad\gamma_1=-\tfrac y2,\quad\Omega=2\lvert\gamma_f\rvert,\quad\vartheta=0 .\tag{12.17}$$

- *Item 1.* $\varphi$ is sharp ($n_1=0$) and not stationary; $C_{\rm eff}(\lambda)$ is not a contract; by (12.5) $1-\operatorname{Tr}V^2=\tfrac12\lambda^4\,\Gamma\,y^2\gamma_f^2+O(\lambda^5)$.
- *Item 2(b)*, class $\mathcal S_\chi$, order (I), $\lvert\gamma_f\rvert\neq M_k$, from (12.11), (12.12):
$$v^{(1)}_{k,\sigma}=-\frac{\tau^2y}2\Bigl[\lambda-\frac{\sin2\gamma_f\lambda}{2\gamma_f}+\frac{\sigma\chi}{2M_k}\Bigl(1+\frac{\gamma_f^2\cos2M_k\lambda-M_k^2\cos2\gamma_f\lambda}{M_k^2-\gamma_f^2}\Bigr)\Bigr],\qquad a^{(1)}_{k,\sigma}=-\tau^2y\Bigl[\sin^2\gamma_f\lambda+\sigma\chi\,\frac{\gamma_f\bigl(M_k\sin2\gamma_f\lambda-\gamma_f\sin2M_k\lambda\bigr)}{2(M_k^2-\gamma_f^2)}\Bigr].$$
  The body starts with zero acceleration and drifts with $-\tfrac12\tau^2y\,\lambda$, the fall in the mean gradient $y/2$.
- *Item 3.* $\mathcal A_{k,\sigma}(2\lvert\gamma_f\rvert)=-\tau^2\bigl[1+i\sigma\chi\,M_k\lvert\gamma_f\rvert/(M_k^2-\gamma_f^2)\bigr]$; the same for all levels iff $\chi=0$, and then $a^{(1)}_{k,\sigma}=-\tau^2y\sin^2\gamma_f\lambda$ for every $\gamma_f$; to the accuracy $\delta$ iff $\lvert\chi\rvert M_k\lvert\gamma_f\rvert\le\delta\lvert M_k^2-\gamma_f^2\rvert$ for all $k$. At $\lvert\gamma_f\rvert=M_k$ the growing part of the velocity of the level $k$ is $+\sigma\chi\tfrac14\tau^2y\,\lambda\sin2M_k\lambda$ in order (I) and $+\tfrac12\tau^2y\,\lambda\cos2M_k\lambda$ in order (II), by (12.14), (12.15).

**(b) Controls.**
- *Sharp and stationary field:* $\gamma(\lambda)=\gamma$, $\gamma_1=0$. (12.6) reduces to (4.7) (Step 5); (12.9) gives $a^{(1)}_{k,\sigma}=-\tau^2\gamma\,[1+\sigma\chi\sin2M_k\lambda]$, whose window average and drift are (4.10), (4.11), (4.16) of 04-branch-fall, $-\tau^2\gamma$. Of item 3 only $\mathcal A_{k,\sigma}(0)=-\tau^2$ remains; there is no field frequency and no resonance. By (2.3) the body sees the contract $C_{\rm red}$ at all orders in $Y$.
- *$[Y,C_f]=0$, not sharp:* $Y(s)=Y$, so $\gamma(\lambda)=\langle\varphi\vert Y\vert\varphi\rangle=:\bar\gamma$ is constant although $\varphi$ need not be stationary. Results 1(a), 1(b) and items 2, 3 with $\gamma_0=\bar\gamma$, $\gamma_1=0$ describe it: at first order it acts as a sharp stationary field with the gradient $\bar\gamma$, and $C_{\rm eff}$ is an admissible contract. Result 1(c), (12.5) and the two special cases of Step 2 do not: $Y$ is conserved, $V(\lambda)=\sum_yw_y\lvert\Psi_y(\lambda)\rangle\langle\Psi_y(\lambda)\rvert$ with $w_y$ the weights of $\varphi$ on the eigenspaces of $Y$ and $\Psi_y=e^{-i(C_{br}+yB)\lambda}\Psi_{br}$, and $1-\operatorname{Tr}V^2=2\lambda^2\,\Gamma\,\bigl(\langle Y^2\rangle_\varphi-\bar\gamma^2\bigr)+O(\lambda^3)$: the difference from the generator is of order $Y^2$ already at $\lambda^2$ (the static field that is not sharp, outside this package).

### Step 12. Summary (item 5)

| Quantity | Stationary sharp field | What a field that is not stationary changes | Order in $Y$ | Field frequencies | Tags |
|---|---|---|---|---|---|
| Factor that the body sees | the number $1+\gamma x$; contract $C_{\rm red}$, exact | the function $1+\gamma(\lambda)x$; generator $C_{\rm eff}(\lambda)$, not a contract; beyond first order the view is mixed | $Y^1$: only $\gamma(\cdot)$; difference at $Y^2$ ($\lambda^4$ for a sharp start) | those with $\hat\gamma_\nu\neq0$; none under (2.11) or $[Y,C_f]=0$ | (12.1), (12.3)–(12.5) |
| Drift acceleration from rest | $-\tau^2\gamma$, (4.16) | (I): $a^{(1)}=-\tau^2\gamma(\lambda)$ plus a memory term $\propto\chi$; drift $-\tau^2\gamma_0$. (II): growth rate $-\tau^2\gamma_0$ plus $\gamma_1\operatorname{Re}[c_ge^{i(\Omega\lambda+\vartheta)}]$ | $Y^1$ | all; an oscillation grows only at $\nu=2M_k$ | (12.8)–(12.12), (12.14), (12.15) |
| Independence of the clock level | yes, $-\tau^2$ | $\chi=0$: kept at every frequency (I). $\chi\neq0$: relative deviation $\lvert\chi\rvert\,2M_k\Omega/\lvert4M_k^2-\Omega^2\rvert$. Lost at $\Omega=2M_k$ (in (II) for every $\chi$) | $Y^1$ | to accuracy $\delta$: $\Omega\notin\bigcup_k(\Omega^{(k)}_-,\Omega^{(k)}_+)$ | (12.13), (12.14), (12.16) |

## Result

Each result with its class, order of limits and status.

- **(12.3), (12.4)** — every $C_f$, $Y$, $\varphi$, $\Psi_{br}$; fixed $\lambda$; exact first-order coefficient. The view of $\{b,c\}$ and every schedule depend on the field only through $\gamma(\cdot)$ of (12.1), and are reproduced by the generator $C_{\rm eff}(\lambda)=C_{br}+\gamma(\lambda)B$, which is an admissible contract iff $\gamma$ is constant. Exact at all orders for a sharp stationary field (2.3) and under (2.11), by (2.12).
- **(12.5)** — sharp, not stationary; expansion in $\lambda$ at fixed operators; exact leading term. First difference from the generator at order $Y^2$, $\lambda^4$ ($\lambda^{2k_0+2}$ if (2.11) first fails at $k_0$).
- **(12.6), (12.7)** — every level, branch, smooth $g$, continuous $\gamma$; fixed start and $\lambda$, $\kappa\to0$, $\langle X\rangle=0$; exact first-order coefficient. $v^{(1)}_g=\int_0^\lambda\gamma\,\mathcal K_g$; the kernel involves $\lvert g\rvert^2$ and the position density $j_g$.
- **(12.8)–(12.10)** — starts at rest; exact bounds for each $g$; limit (I) for the class $\mathcal S_\chi$: $K_\chi(\Delta)=-\tau^2[1+\sigma\chi\sin2M_k\Delta]$, velocity and acceleration (12.9), the acceleration with its own bound (12.10).
- **(12.11), (12.12)** — $\mathcal S_\chi$, (I), $\Omega\neq2M_k$; exact. Parts of frequency $\Omega$, of frequency $2M_k$ ($\propto\chi$), and the drift $-\tau^2\gamma_0\lambda$.
- **(12.13)** — $\mathcal A_{k,\sigma}(\Omega)=-\tau^2[1+i\sigma\chi\,2M_k\Omega/(4M_k^2-\Omega^2)]$; $\to-\tau^2$, i.e. (4.16), for $\Omega\to0$; the same for all levels iff $\chi=0$; leading level dependence $-i\sigma\chi\tau^2\Omega/2M_k$.
- **(12.14), (12.15)** — $\Omega=2M_k$: linear growth of an oscillation of frequency $2M_k$ in the level $k$ only; coefficient $-\sigma\chi\tau^2\gamma_1/2$ (sine) in order (I), $-\tau^2\gamma_1$ (cosine) in order (II), leading order with rest $O(\lambda^{1/2})$; the limits do not commute.
- **(12.16)** — $\mathcal S_\chi$, (I), exact iff: the response at $\Omega$ is level-independent to the relative accuracy $\delta$ iff $\Omega\notin\bigcup_k(\Omega^{(k)}_-,\Omega^{(k)}_+)$; for a general $\gamma$: $a^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma(\lambda)+\sigma\chi R_k[\gamma](\lambda)$.
- **(12.17)** — the two-place field: $\gamma(\lambda)=y\sin^2\gamma_f\lambda$, with the evaluations of Step 11 and the two controls.

## Consistency checks

1. **Constant $\gamma$.** (12.6), (12.7) give the growing part $\gamma\lambda\langle\alpha_0\rangle_g$ with $\alpha_0=-M_k^2\tau^2\cos2p/\omega_k^2$, which is (4.7) with $G_k=\gamma M_k$ (Step 5); at rest (12.9) gives $-\tau^2\gamma[1+\sigma\chi\sin2M_k\lambda]$ with the window average $-\tau^2\gamma$, which is (4.16). Passed.
2. **A gradient pulse, by an independent route.** $\gamma(s)=\epsilon\,\delta(s-s_0)$ acts as $e^{-i\epsilon M\hat x\sigma_3}$, a shift of the label by $\mp\epsilon M$ on the upper and lower component. For the branch $+$ at rest and a narrow $g$: the shift inside the branch changes the velocity by $-\epsilon M\omega''(0)=-\epsilon\tau^2$; the amplitude on $\beta_-$ is $\tfrac{\epsilon\tau}2(2pg'+g)$ to leading order, its overlap with $g$ is $\int\frac{\mathrm dp}{2\pi}g^{*}(2pg'+g)=-i\chi_g$, and with the off-diagonal element $-\tau$ of $\hat V$ and the phase $e^{2iM\Delta}$ it adds $-\epsilon\tau^2\chi_g\sin2M\Delta$. The sum is $\epsilon K_{\chi_g}(\Delta)$ of (12.8). Passed.
3. **Dimensions.** $x$, $p$, $\gamma$, $\chi$, $\delta$ are pure numbers; $\tau$, $M$, $\omega$, $\Omega$, $\gamma_f$ are costs, of dimension $1/\lambda$. A velocity is a cost: $\tau^2\int\mathrm ds\,\gamma$ in (12.9); an acceleration is a cost squared: $\tau^2\gamma$, $\mathcal A_{k,\sigma}$; the kernel (12.7) is a cost squared in every term ($M^2\lambda\omega'^2/\omega$ included); (12.5) is a pure number ($\Gamma$ and $\lVert(Y-\gamma)C_f\varphi\rVert^2$ are costs squared). Passed.

## Open issues

- The level-dependent part of the response at fixed $\lambda$ is carried entirely by the position–label correlation $\chi$ of the start; $\chi_g$ grows under free motion by $2\sigma\lambda\langle\omega'\sin p\rangle_g$, which is why the orders (I) and (II) differ. A formula uniform in $\lambda$ between $\lambda\ll\lambda_g$ and $\lambda\gg\lambda_g$ is not derived; the exact (12.6), (12.7) cover it implicitly.
- Order (II) for a field frequency inside the band, $2M_k<\Omega\le2\sqrt{M_k^2+\tau^2}$: $c_g(\Omega)$ of (12.15) is defined for every start, but its limit $\delta p\to0$ depends on the weight of the starts near the labels with $2\omega_k(p)=\Omega$, which $\delta p\to0$ does not control; it is evaluated only for sequences that vanish near these labels.
- Sequences at rest with $\rho_g\not\to0$ belong to no class $\mathcal S_\chi$; for them only (12.6), (12.7) and the bounds (12.8), (12.10) are available.
- The growth at $\Omega=2M_k$ is a statement on the first-order coefficient at each fixed $\lambda$; the range of $\lambda$ in which it is the leading effect needs the second order in $Y$, which is outside the scope (as is the mixing behind (12.5)).

## Methods used

- Dyson expansion (first-order perturbation theory) in a pair term; interaction picture
- Partial trace and views; purity of the view
- Plane-wave representation with $\hat x=i\partial_p$; two-branch decomposition of the level block
- Integration by parts on periodic functions; elementary bounds with explicit constants
- Duhamel formula and the driven oscillator equation (resonance)
- Stationary phase for the long-$\lambda$ behaviour; order of limits

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