Version 3 · earlier version; the current one is v4 · External review, round 2: minor issues

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 12-changing-field
- **Version:** v3
- **Mode:** internal regeneration
- **Date:** 2026-10-11

## Changes from previous version

- **Unchanged, word for word:** Setup; Steps 1–5; Step 6 except the three places listed under I1; Steps 7 and 8; Step 9 except the first item of Result 3(b); Step 10 except the last sentence of item 4(iii) of (A2); Step 11; the three consistency checks; Methods used. All displays and the tags (12.1)–(12.22) are unchanged; no tag is renumbered, withdrawn or added, and no result is changed. Only sentences that quote a display are changed. The section "Response to verification" of v2 belonged to the external round and is not repeated.
- **I1 (ranges quoted without all their conditions).**
  - Step 6, paragraph after the three ranges of (12.18): the sentence "the range $\lambda\ll\lambda_g$ of the drift of the velocity covers neither the acceleration nor the oscillating parts of the velocity" is replaced. The range of the drift is quoted with both conditions, and its comparison with $\lambda_a$ carries the factor $\lvert\gamma_0\rvert/\Gamma_\lambda$: an exact criterion, a sufficient one, the case of a factor of order one named as such, and the opposite case of a gradient that mostly oscillates.
  - Step 6, second range: "a shorter range than that of the drift" is replaced by the condition under which it is the stronger one, $\Omega\lambda>\lvert\gamma_1\rvert/\lvert\gamma_0\rvert$; the amplitude of the part of frequency $\Omega$ is now "at least" $\tau^2\lvert\gamma_1\rvert/\Omega$ (equal for $\chi=0$), as (12.11) shows.
  - Step 6, lead sentence of the three ranges: "holds for every continuous $\gamma$" now refers to the bounds $E_v$, $E_a$; the first two ranges name their $\gamma$ themselves.
  - Step 9, Result 3(b), first item: the range of the growing parts is quoted with both conditions and, for the acceleration, with $M\lambda\ge1$.
  - Step 12, row "Drift acceleration from rest", third column; Result, items (12.18) and (12.14), (12.15), (12.19): every range with all its conditions, factors and reference quantities.
- **I2 ("only" for the bounded parts at resonance).** Step 9, Result 3(b), first item: the sentence "The bounded parts of (12.14) are described only for $\lambda\ll\lambda_a$" is replaced by two sufficient ranges, one for the velocity (from $E_v$, against the amplitude of its bounded oscillation) and one for the acceleration (from $E_a$), and by the statement of what is not derived beyond them. The last sentence of the item now says "before dephasing" and names (12.15) for the long run. The second item of Open issues is extended accordingly.
- **Other quotations made complete** (every sentence that quotes a range of $\lambda$, an "only", an "iff" or a "sufficient" was read against its display):
  - Step 10, (A2), item 4(iii), last sentence: "for a fixed start the window must lie in the range $\lambda\ll\lambda_a$" stated a sufficient range as a necessary one; it now quotes the third range of (12.18) with both conditions, as sufficient.
  - Step 12, row "Drift acceleration from rest", last column: the growth of a fixed start in order (II) is quoted with its amplitude and with the sign statement of (12.15).
  - Step 12, row "Independence of the clock level": the window average with its $O(1/\Lambda)$; the two criteria with their classes, which tolerance is sufficient and which necessary; the sufficient condition of (12.22) with the quantity it bounds.
  - Result: (12.5) with its two conditions; (12.13) with its class, the conditions of the iff and of the leading term; (12.16), (12.20), (12.21) with "one class and one branch" for the relation between the two criteria; (12.22) with the class and the quantity of the sufficient condition and the $O(1/\Lambda)$ of the window average; (12.19) with the amplitudes against which the relative size is taken.

## 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):

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

**Ranges of a fixed start.** Let $w_g:=\delta p^2+2\rho_g$, $\Gamma_\lambda:=\sup_{[0,\lambda]}\lvert\gamma\rvert$, and $v_{\chi_g}$ the first expression of (12.9) at $\chi=\chi_g$. The velocity bound stated above and (12.10) are, for every smooth $g$ at rest and every continuous $\gamma$ (exact),

$$\bigl\lvert v^{(1)}_g-v_{\chi_g}\bigr\rvert\le E_v:=4\tau^2\Gamma_\lambda\,\lambda\Bigl(u+\frac{\tau^2\lambda}{M}\Bigr)w_g,\qquad\bigl\lvert a^{(1)}_g-a_{\chi_g}\bigr\rvert\le E_a:=\tau^2\Gamma_\lambda(1+4M\lambda)\Bigl(u+\frac{2\tau^2\lambda}{M}\Bigr)w_g,\qquad\lambda_g:=\frac{M}{\tau^2w_g},\quad\lambda_a:=\min\Bigl\{\frac{1}{Mu\,w_g},\frac{1}{\tau\sqrt{w_g}}\Bigr\}.\tag{12.18}$$

A range in which (12.9) describes a fixed start is a statement on one quantity and is taken from the bound of that quantity. Each of the following is *sufficient*; the bounds $E_v$, $E_a$ behind them hold for every continuous $\gamma$:
- *Velocity, against its growing drift* $\tau^2\lvert\gamma_0\rvert\lambda$ ($\gamma$ of Step 7, $\gamma_0\neq0$): the relative error is at most $E_v/(\tau^2\lvert\gamma_0\rvert\lambda)=4(\Gamma_\lambda/\lvert\gamma_0\rvert)(u+\tau^2\lambda/M)\,w_g$, small if $u\,w_g\ll\lvert\gamma_0\rvert/\Gamma_\lambda$ and $\lambda\ll(\lvert\gamma_0\rvert/\Gamma_\lambda)\,\lambda_g$.
- *Velocity, against a bounded oscillating part*, e.g. the part of frequency $\Omega$ of (12.11), whose amplitude is at least $\tau^2\lvert\gamma_1\rvert/\Omega$ (equal for $\chi=0$): $E_v\ll\tau^2\lvert\gamma_1\rvert/\Omega$, i.e. $4\Omega\lambda\,(u+\tau^2\lambda/M)\,w_g\ll\lvert\gamma_1\rvert/\Gamma_\lambda$; at a given $\lambda$ this is a stronger condition than that of the drift iff $\Omega\lambda>\lvert\gamma_1\rvert/\lvert\gamma_0\rvert$.
- *Acceleration, against its amplitude* $\tau^2\Gamma_\lambda$ (off resonance (12.12) is bounded and oscillating and has no growing part): the relative error is at most $(1+4M\lambda)(u+2\tau^2\lambda/M)\,w_g$, small if $u\,w_g\ll1$ and $\lambda\ll\lambda_a$.

$\lambda_a\le\lambda_g$, with $\lambda_a/\lambda_g=\min\{\tau^2/(M^2+\tau^2),\,\tau\sqrt{w_g}/M\}\le\tau\sqrt{w_g}/M$, which tends to zero for $w_g\to0$. The range of the drift of the velocity is $u\,w_g\ll\lvert\gamma_0\rvert/\Gamma_\lambda$ and $\lambda\ll(\lvert\gamma_0\rvert/\Gamma_\lambda)\,\lambda_g$. At a given $\lambda$ the bound $(\lvert\gamma_0\rvert/\Gamma_\lambda)\,\lambda_g$ of its second condition exceeds $\lambda_a$ iff $\lvert\gamma_0\rvert/\Gamma_\lambda>\lambda_a/\lambda_g$, for which $\lvert\gamma_0\rvert/\Gamma_\lambda>\tau\sqrt{w_g}/M$ is sufficient. In that case (in particular for $w_g\to0$ at a ratio $\lvert\gamma_0\rvert/\Gamma_\lambda$ of order one) the conditions of the drift do not imply $\lambda\ll\lambda_a$, so the range of the drift does not cover the acceleration. By the second item it does not cover the part of frequency $\Omega$ of the velocity where $\Omega\lambda>\lvert\gamma_1\rvert/\lvert\gamma_0\rvert$. For a gradient that mostly oscillates, $\lvert\gamma_0\rvert/\Gamma_\lambda<\lambda_a/\lambda_g$, the bound of the drift is the smaller of the two. The member $1/(\tau\sqrt{w_g})$ of $\lambda_a$ is attained, up to a numerical factor, when $\gamma$ contains the frequency $2M_k$ ((12.19), Step 9). For a $\gamma$ without a frequency near $2M_k$ a longer range for the acceleration is not excluded; it is not derived here.

### 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). In the limit functions (12.11), (12.12) the only part that grows is the drift $-\tau^2\gamma_0\lambda$ of the velocity; the acceleration has no growing part. This is a statement on the limit (I); a fixed start is described by it within the ranges of (12.18), and in the long run its velocity carries the growing oscillation of (12.15). 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 following are statements on the limit (I) of a class $\mathcal S_\chi$, i.e. on the limit functions (12.11), (12.14), not on a fixed start at all $\lambda$. The limit velocity carries an oscillation of frequency $2M_k$ whose amplitude grows linearly, $\tfrac12\lvert\chi\gamma_1\rvert\tau^2\lambda$ (the limit acceleration: $\lvert\chi\gamma_1\rvert\tau^2M_k\lambda$). In every level $k'$ with $2M_{k'}\neq\Omega$ the oscillating parts of the limit are bounded, by (12.11). For $\chi=0$ nothing grows in the limit (I) besides the drift, and no level is distinguished there.

*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 $\Omega>2\sqrt{M_k^2+\tau^2}$).

Before this limit the coefficient does not vanish: for a fixed smooth $g$ and $0<\Omega<2M_k$, $c_g(\Omega)=-\langle4M^2\omega'^2/(4\omega^2-\Omega^2)\rangle_g<0$, because the integrand is positive except at the four zeros of $\omega'$. By (12.15) the velocity of a fixed start therefore carries, in every level and at every field frequency below the gap, an oscillation of frequency $\Omega$ with the linearly growing amplitude $\lvert\gamma_1c_g(\Omega)\rvert\lambda$; at rest it is of order $\delta p^2$ and disappears only in the subsequent limit $\delta p\to0$.

*Between the two orders: a fixed real start at resonance.* Let $g$ be real and at rest ($j_g=0$, hence $\chi_g=\rho_g=0$; a member of a sequence of $\mathcal S_0$) and $\gamma=\gamma_1\cos(2M_k\lambda+\vartheta)$ (a $\gamma_0$ adds the terms of Step 5, by linearity). Put $j_g=0$ into (12.7) and into $\partial_\lambda\mathcal K_g$ of Step 6 and use, with $\eta(p):=\omega-M=\tau^2\sin^2p/(\omega+M)\le\tau^2\sin^2p/2M$, the exact integral $\int_0^\lambda\mathrm ds\,\cos(2Ms+\vartheta)\sin(2\omega\Delta)=\frac\lambda2\frac{\sin\eta\lambda}{\eta\lambda}\sin((\omega+M)\lambda+\vartheta)+\frac{\cos(2M\lambda+\vartheta)-\cos(2\omega\lambda-\vartheta)}{4(\omega+M)}$ and its analogue with $\cos(2\omega\Delta)$ (first term with $\cos((\omega+M)\lambda+\vartheta)$, second term $\frac{\sin(2M\lambda+\vartheta)+\sin(2\omega\lambda-\vartheta)}{4(\omega+M)}$). With $q_g:=\langle\sin^2p\rangle_g\le\delta p^2$, for all $\lambda\ge0$ (exact):

$$\begin{aligned}v^{(1)}_g(\lambda)&=-\tau^2\!\int_0^\lambda\!\mathrm ds\,\gamma(s)-\gamma_1\lambda^2\Bigl\langle\frac{M^2\omega'^2}{\omega}\,\frac{\sin\eta\lambda}{\eta\lambda}\,\sin\bigl((\omega+M)\lambda+\vartheta\bigr)\Bigr\rangle_g+r^v_g(\lambda),\qquad&\lvert r^v_g(\lambda)\rvert&\le4\tau^2u\,\lvert\gamma_1\rvert\,\lambda\,q_g,\\a^{(1)}_g(\lambda)&=-\tau^2\gamma(\lambda)-2\gamma_1\lambda^2\Bigl\langle M^2\omega'^2\,\frac{\sin\eta\lambda}{\eta\lambda}\,\cos\bigl((\omega+M)\lambda+\vartheta\bigr)\Bigr\rangle_g+r^a_g(\lambda),\qquad&\lvert r^a_g(\lambda)\rvert&\le\tau^2u\,\lvert\gamma_1\rvert\,(1+5M\lambda)\,q_g .\end{aligned}\tag{12.19}$$

The rests collect the terms without the factor $\lambda^2$ (those of $k_0+\tau^2$, of $k_1$, and the second terms of the two integrals); they are bounded with the elementary bounds of Step 6 and $\lvert\int_0^\lambda\mathrm ds\cdots\rvert\le\lambda$. The velocity and the acceleration are each obtained from their own kernel.
- *Before dephasing*, $(2u+\tau^2\lambda/M)\langle\sin^4p\rangle_g\ll q_g$ (for a narrow Gaussian: $u\,\delta p^2\ll1$ and $\lambda\ll\lambda_g/3$). Since $\lvert M^2\omega'^2-\tau^4\sin^2p\rvert\le\tau^4u\sin^4p$ and $\lvert\frac{\sin x}{x}\cos(a+x)-\cos a\rvert\le2\lvert x\rvert$ (the same with sines), the two averages are $\frac{\tau^4}{M}q_g\sin(2M\lambda+\vartheta)$ and $\tau^4q_g\cos(2M\lambda+\vartheta)$ up to the relative error $(2u+\tau^2\lambda/M)\langle\sin^4p\rangle_g/q_g$. The finite-width terms are then $-\frac{\tau^4\gamma_1q_g}{M}\lambda^2\sin(2M_k\lambda+\vartheta)$ in the velocity and $-2\tau^4\gamma_1q_g\,\lambda^2\cos(2M_k\lambda+\vartheta)$ in the acceleration. They grow like $\lambda^2$, faster than the terms linear in $\lambda$ of the bounds. Against the amplitudes $\tau^2\lvert\gamma_1\rvert/2M$ and $\tau^2\lvert\gamma_1\rvert$ of the sharp-label velocity and acceleration both have the relative size $2\tau^2q_g\lambda^2$, which is one at $\lambda=1/(\tau\sqrt{2q_g})\ll\lambda_g$. The term $8\tau^2\lambda^2w_g$ of (12.10) is thus attained up to the factor 4.
- *After dephasing*, $\lambda\to\infty$ at fixed $g$: $\lambda\frac{\sin\eta\lambda}{\eta\lambda}\sin((\omega+M)\lambda+\vartheta)=[\cos(2M\lambda+\vartheta)-\cos(2\omega\lambda+\vartheta)]/2\eta$, and the velocity of (12.19), with the term of $r^v_g$ proportional to $\lambda$, becomes $\lambda\gamma_1c_g(2M_k)\cos(2M_k\lambda+\vartheta)+O(\lambda^{1/2})$, which is (12.15).
- *Not derived:* a closed form for $\lambda\sim\lambda_g$ through $\delta p$ and $\chi_g$ alone (the averages of (12.19) depend on the whole weight $\lvert g\rvert^2$); the analogue of (12.19) for starts with $j_g\neq0$, for which only (12.6), (12.7) and the bounds (12.18), (12.10) hold.

**Result 3(b), the two orders.** Each statement is on the named quantity, in the named order and class.
- *Order (I), class $\mathcal S_\chi$, $\Omega=2M_k$.* The limit velocity has the growing part $-\sigma\chi\tau^2\gamma_1\frac\lambda2\sin(2M_k\lambda+\vartheta)$ and the limit acceleration the growing part $-\sigma\chi\tau^2\gamma_1M_k\lambda\cos(2M_k\lambda+\vartheta)$, by (12.14). A fixed start has these growing parts, with $\chi=\chi_g$, up to the relative errors $8(\Gamma_\lambda/\lvert\chi_g\gamma_1\rvert)(u+\tau^2\lambda/M)\,w_g$ (velocity, from $E_v$) and, for $M\lambda\ge1$, $5(\Gamma_\lambda/\lvert\chi_g\gamma_1\rvert)(u+2\tau^2\lambda/M)\,w_g$ (acceleration, from $E_a$): sufficient is $u\,w_g\ll\lvert\chi_g\gamma_1\rvert/\Gamma_\lambda$ and $\lambda\ll(\lvert\chi_g\gamma_1\rvert/\Gamma_\lambda)\,\lambda_g$, for the acceleration together with $M\lambda\ge1$. The parts of (12.14) that do not grow have sufficient ranges of their own, one per quantity. *Velocity:* its bounded part contains, for every $\chi$, the oscillation $-\tau^2\gamma_1\sin(2M_k\lambda+\vartheta)/2M_k$; the error is small against the amplitude of this oscillation if $E_v\ll\tau^2\lvert\gamma_1\rvert/2M_k$, i.e. $8M\lambda\,(u+\tau^2\lambda/M)\,w_g\ll\lvert\gamma_1\rvert/\Gamma_\lambda$ (the second range of (12.18) at $\Omega=2M_k$). *Acceleration:* its bounded part contains $-\tau^2\gamma(\lambda)$; the error is small against $\tau^2\Gamma_\lambda$ if $E_a\ll\tau^2\Gamma_\lambda$, i.e. $u\,w_g\ll1$ and $\lambda\ll\lambda_a$ (the third range of (12.18)). Whether the bounded parts are described beyond these ranges is not derived; for a real start the exact (12.19) holds at all $\lambda\ge0$. For $\chi_g=0$ the range of the growing parts is empty: before dephasing, a fixed real start has no oscillation growing like $\lambda$ but the one growing like $q_g\lambda^2$ of (12.19); after dephasing its velocity has the oscillation growing like $\lambda$ of (12.15).
- *Order (II), fixed smooth start, $\lambda\to\infty$.* By (12.15) the velocity has, in every level and at every $\Omega>0$, an oscillation of frequency $\Omega$ with the amplitude $\lvert\gamma_1c_g(\Omega)\rvert\lambda$; $c_g(\Omega)<0$ for $0<\Omega<2M_k$. After the subsequent limit $\delta p\to0$ at rest the coefficient is: $-\tau^2\gamma_1\lambda\cos(2M_k\lambda+\vartheta)$ at $\Omega=2M_k$, for every sequence at rest and independent of $\chi$; zero for $\Omega<2M_{k'}$ and for $\Omega>2\sqrt{M_{k'}^2+\tau^2}$, for every sequence at rest; zero for $2M_{k'}<\Omega\le2\sqrt{M_{k'}^2+\tau^2}$ for the sequences that vanish on a fixed neighbourhood of $\{p:2\omega_{k'}(p)=\Omega\}$, and not determined without this restriction (Open issues).
- *Comparison.* The two orders give different growing oscillations at $\Omega=2M_k$ (sine with the coefficient $-\sigma\chi\tau^2\gamma_1/2$ against cosine with $-\tau^2\gamma_1$), so the limits do not commute there; (12.19) shows the passage from one to the other for a real start. Both agree on the drift $-\tau^2\gamma_0\lambda$ and, for $\Omega<2M_k$ and after $\delta p\to0$, 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. In the limit (I) with $\chi\neq0$ it is the only level whose limit velocity has a growing oscillation. In order (II) it is the only level whose growing oscillation survives the sharp-label limit, for the sequences named above. It is not true that an oscillation of the velocity of a fixed start grows only in this level.

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

**Quantities compared.** All statements of this step are on the limit (I). On the level $k$ the starts form a class $\mathcal S_{\chi_k}$ on the branch $\sigma_k$; $c_k:=\sigma_k\chi_k$ and $h_k(\Omega):=2M_k\Omega/(4M_k^2-\Omega^2)$, so that $\mathcal A_{k,\sigma_k}(\Omega)=-\tau^2[1+i\,c_kh_k(\Omega)]$ by (12.13). Two quantities are kept apart: **(A)** the *response at the field frequency* $\mathcal A_{k,\sigma}(\Omega)$, the coefficient of the part of frequency $\Omega$ of (12.12), defined for $\Omega\neq2M_k$ (for every $\Omega>0$ if $\chi_k=0$, by (12.14)); **(B)** the *complete acceleration* $a^{(1)}_{k,\sigma}(\lambda)$ of (12.9), which contains in addition the parts of frequency $2M_k$, depending on the level and on the start. Relative accuracies are normalized by $\tau^2=\lvert\mathcal A_{k,\sigma}(0)\rvert$, the response to a constant gradient, which is common to all levels.

**(A1) Distance from the quasistatic value.** 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:* one class $\mathcal S_\chi$ on all levels, 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). *What (12.16) is the criterion for:* in every level and both branches the response at $\Omega$ lies within $\delta\tau^2$ of $-\tau^2$, i.e. $\lvert\chi\rvert\max_k\lvert h_k(\Omega)\rvert\le\delta$. It is not the criterion for the agreement between the levels; that is (A2). Below, $\Omega^{(k)}_\mp(\delta')$ are the numbers of (12.16) at the tolerance $\delta'$.

**(A2) Agreement between the levels.** The quantity compared is the **level spread**

$$\mathcal D(\Omega):=\max_{k,k'}\frac{\bigl\lvert\mathcal A_{k,\sigma_k}(\Omega)-\mathcal A_{k',\sigma_{k'}}(\Omega)\bigr\rvert}{\tau^2}=\max_kc_kh_k(\Omega)-\min_kc_kh_k(\Omega),\qquad h_k(\Omega)-h_{k'}(\Omega)=\frac{2\Omega\,(M_{k'}-M_k)\,(4M_kM_{k'}+\Omega^2)}{(4M_k^2-\Omega^2)(4M_{k'}^2-\Omega^2)},\tag{12.20}$$

and "the response at the field frequency is the same for all levels to the relative accuracy $\delta$" means $\mathcal D(\Omega)\le\delta$. (Normalizing by $\lvert\mathcal A_{k,\sigma}\rvert\ge\tau^2$ instead gives a smaller number, so $\mathcal D\le\delta$ is then sufficient.) *Class, order, status:* classes $\mathcal S_{\chi_k}$, order (I), $\Omega$ different from every $2M_k$ with $\chi_k\neq0$; exact.
1. *Necessary and sufficient.* $\max_kc_kh_k(\Omega)-\min_kc_kh_k(\Omega)\le\delta$. For one class $\mathcal S_\chi$ and one branch on all levels: $\lvert\chi\rvert\max_{k,k'}\lvert h_k(\Omega)-h_{k'}(\Omega)\rvert\le\delta$, with the second formula of (12.20). If both branches of all levels are compared in one class, $\mathcal D(\Omega)=2\lvert\chi\rvert\max_k\lvert h_k(\Omega)\rvert$, and the criterion is (12.16) at the tolerance $\delta/2$: $\Omega\notin\bigcup_k(\Omega^{(k)}_-(\delta/2),\Omega^{(k)}_+(\delta/2))$.
2. *Relation to (12.16), one class and one branch.* By the triangle inequality $\mathcal D\le2\lvert\chi\rvert\max_k\lvert h_k\rvert$: (12.16) at the tolerance $\delta/2$ is **sufficient** at every $\Omega$. If $\Omega<2M_0=2m$ or $\Omega>2M_{N-1}$, all $h_k$ have one sign and $\mathcal D<\lvert\chi\rvert\max_k\lvert h_k\rvert$: (12.16) at $\delta$ is sufficient, and not necessary. If $2M_0<\Omega<2M_{N-1}$, the $h_k$ have both signs and $\mathcal D\ge\lvert\chi\rvert\max_k\lvert h_k\rvert$: (12.16) at $\delta$ is **necessary**, and (12.16) at $\delta/2$ sufficient.
3. *Below and above all gaps, one class and one branch.* $h_k(\Omega)$ decreases with $M_k$ on each side of $M_k=\Omega/2$, so the extreme levels are $k=0$ and $k=N-1$ ($M_{N-1}=m+d_{N-1}$ is the largest):

$$\mathcal D(\Omega)=\lvert\chi\rvert\,\frac{2\Omega\,(M_{N-1}-M_0)\,(4M_0M_{N-1}+\Omega^2)}{(4M_0^2-\Omega^2)(4M_{N-1}^2-\Omega^2)}\quad(\Omega<2M_0\ \text{or}\ \Omega>2M_{N-1});\qquad0<\Omega<2m:\ \ \mathcal D(\Omega)\le\delta\iff\Omega\le\Omega_\delta,\quad\Omega_\delta=\frac{2\delta}{\lvert\chi\rvert}\,\frac{M_0M_{N-1}}{M_{N-1}-M_0}\Bigl(1+O\bigl(\Omega_\delta^2/m^2\bigr)\Bigr).\tag{12.21}$$

   On $(0,2m)$ the numerator increases and the denominator decreases, so $\mathcal D$ increases from $0$ to $\infty$ and $\Omega_\delta$ is its unique solution of $\mathcal D=\delta$; $\Omega_\delta\ge\Omega^{(0)}_-(\delta)$, larger by the factor $M_{N-1}/(M_{N-1}-M_0)$ at leading order. On $(2M_{N-1},\infty)$ the same expression decreases from $\infty$ to $0$, and $\mathcal D\le\delta$ above its unique solution there ($\approx2\lvert\chi\rvert(M_{N-1}-M_0)/\delta$ when this is $\gg2M_{N-1}$).
4. *Exceptional members.* (i) $\chi_k=0$ on all levels: $\mathcal D(\Omega)=0$ at every $\Omega>0$, the $2M_k$ included. (ii) Levels with nearly equal data: a pair contributes $\lvert\chi\rvert\lvert h_k-h_{k'}\rvert\propto\lvert M_k-M_{k'}\rvert$, so $\mathcal D\le\delta$ can hold where (12.16) fails; e.g. two levels $M_1=M_0(1+\varepsilon)$, $\Omega=M_0$, $\chi=1$, one branch: $\mathcal D=\frac{10}9\varepsilon+O(\varepsilon^2)$ while $\lvert\chi h_0\rvert=\frac23$. Exactly equal data of two levels do not occur in Model D ($N\ge2$), so for one class with $\chi\neq0$ and one branch $\mathcal D(\Omega)>0$ at every $\Omega$ (Step 8). (iii) $\Omega$ at or near a $2M_k$ with $\chi_k\neq0$: at $\Omega=2M_k$ the response $\mathcal A_{k,\sigma}$ does not exist (the part grows, (12.14)); $\mathcal D(\Omega)\to\infty$ for $\Omega\to2M_k$. In a window of length $\Lambda$ the part of frequency $\Omega$ is separated from the part of frequency $2M_k$ only if $\Lambda\gg1/\lvert\Omega-2M_k\rvert$; for a fixed start, (12.12) describes the acceleration in the window up to an error small against $\tau^2\Gamma_\lambda$ if $u\,w_g\ll1$ and the window lies in $\lambda\ll\lambda_a$ (the third range of (12.18), sufficient).

**(B) The complete acceleration.** By (12.9), with $y_k$ the solution of $y_k''+4M_k^2y_k=2M_k\gamma$, $y_k(0)=y_k'(0)=0$ (Step 7),

$$a^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma(\lambda)+\sigma\chi\,R_k[\gamma](\lambda),\qquad R_k[\gamma](\lambda):=-2M_k\tau^2\!\int_0^\lambda\!\mathrm ds\,\gamma(s)\cos\bigl(2M_k(\lambda-s)\bigr)=-\tau^2\,y_k'(\lambda).\tag{12.22}$$

- *Exactly the same, necessary and sufficient.* For a continuous $\gamma$ that does not vanish identically on $[0,\lambda_1]$: the complete accelerations of all levels coincide on $[0,\lambda_1]$ **iff $\chi_k=0$ on every level**. Proof for two levels $k\neq k'$: $c_ky_k'=c_{k'}y_{k'}'$ gives $z:=c_ky_k=c_{k'}y_{k'}$, and the difference of the two equations gives $4(M_k^2-M_{k'}^2)\,z=2(c_kM_k-c_{k'}M_{k'})\,\gamma$ with $M_k\neq M_{k'}$. If not both $c$ vanish, either $z\equiv0$, or $\gamma$ is a constant multiple of $z$ and $z$ solves a homogeneous linear equation of second order with $z(0)=z'(0)=0$; in both cases $z\equiv0$, hence $\gamma\equiv0$. Exceptional member: $\gamma\equiv0$ (for example $y=0$ in Step 11), where all accelerations vanish.
- *To the accuracy $\delta$, sufficient only.* For $\gamma$ of Step 7 and $\Omega\neq2M_k$, (12.12) gives $\lvert a^{(1)}_{k,\sigma}(\lambda)+\tau^2\gamma(\lambda)\rvert\le\lvert\chi\rvert\tau^2\bigl[\lvert\gamma_0\rvert+\lvert\gamma_1\rvert\frac{2M_k+\Omega}{\lvert2M_k-\Omega\rvert}\bigr]$ for all $\lambda\ge0$. Hence $2\max_k\lvert\chi_k\rvert\frac{2M_k+\Omega}{\lvert2M_k-\Omega\rvert}\le\delta$ is sufficient for the complete accelerations of all levels and both branches to agree within $\delta\,\tau^2(\lvert\gamma_0\rvert+\lvert\gamma_1\rvert)$ at every $\lambda\ge0$. A necessary and sufficient condition is not derived. The condition is one on the start rather than on the field frequency: the part of frequency $2M_k$ of (12.12) has the amplitude $\lvert\chi\rvert\tau^2\lvert\gamma(0)\rvert$ for $\Omega\to0$, so lowering $\Omega$ does not remove it.
- *Window average.* The average of (12.12) over $[\lambda_1,\lambda_1+\Lambda]$ equals $[v^{(1)}_{k,\sigma}(\lambda_1+\Lambda)-v^{(1)}_{k,\sigma}(\lambda_1)]/\Lambda$, which by (12.11) is $-\tau^2\gamma_0+O(1/\Lambda)$ for every level, branch and $\chi$ if $\Omega\neq2M_k$: the drift acceleration of the limit (I) is the same for all levels at every such field frequency. At $\Omega=2M_k$ with $\chi\neq0$ the average does not converge, by (12.14).

**General $\gamma(\lambda)$.** By (12.9), linear in $\gamma$, and (12.1): (12.22) holds for every continuous $\gamma$; 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 $\mathcal A_{k,\sigma}(-\nu)=\mathcal A_{k,\sigma}(\nu)^{*}$, and (A1), (A2) apply to each $\lvert\nu\rvert$ separately. 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 $\gamma(\cdot)$ (limit (I); a fixed start within the ranges of (12.18)). In order (II) the coefficient of $\lambda$ in the velocity, after $\delta p\to0$ along any sequence at rest, is $-\tau^2\gamma_0$ ($\gamma_0=\hat\gamma_0$) for all levels if every field frequency in $\gamma$ is below $2m$; this condition is sufficient, and a frequency $\lvert\nu\rvert=2M_k$ adds the term of (12.15) in the level $k$.

### 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, $y\neq0$.* $\varphi$ is sharp ($n_1=0$) and not stationary: (2.11) fails at $k=1$, $YC_f\varphi=y\gamma_f\lvert1\rangle\neq0$. $\gamma$ is not constant, so $C_{\rm eff}(\lambda)$ 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 1, $y=0$* (allowed by the question). $Y=0$ and the pair term is absent. (2.11) holds with $n_1=0$, $\gamma(\lambda)=0$ for all $\lambda$, and $C_{\rm eff}=C_{br}$ is a constant, admissible contract; by (2.12) the view of $\{b,c\}$ is pure and is that of the body and the medium alone, at all orders. The state of the field still changes ($V_f$ of (2.12)); the body does not see it. Items 2 and 3 are then empty: $\gamma_0=\gamma_1=0$, all first-order velocities and accelerations vanish, and (12.5) gives zero.
- *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, $y\neq0$, limit (I).* $\mathcal A_{k,\sigma}(2\lvert\gamma_f\rvert)=-\tau^2\bigl[1+i\sigma\chi\,M_k\lvert\gamma_f\rvert/(M_k^2-\gamma_f^2)\bigr]$ for $\lvert\gamma_f\rvert\neq M_k$; in one class $\mathcal S_\chi$ it is the same for all levels iff $\chi=0$, and then $a^{(1)}_{k,\sigma}=-\tau^2y\sin^2\gamma_f\lambda$ for every $\gamma_f$. It lies within $\delta\tau^2$ of $-\tau^2$ in every level iff $\lvert\chi\rvert M_k\lvert\gamma_f\rvert\le\delta\lvert M_k^2-\gamma_f^2\rvert$ for all $k$, by (12.16). It is the same for all levels to the accuracy $\delta$ (one branch) iff $\mathcal D(2\lvert\gamma_f\rvert)=\lvert\chi\rvert\max_{k,k'}\bigl\lvert\frac{M_k\lvert\gamma_f\rvert}{M_k^2-\gamma_f^2}-\frac{M_{k'}\lvert\gamma_f\rvert}{M_{k'}^2-\gamma_f^2}\bigr\rvert\le\delta$, by (12.20); for $\lvert\gamma_f\rvert<m$ iff $2\lvert\gamma_f\rvert\le\Omega_\delta$, by (12.21). The complete acceleration displayed above is the same for all levels iff $\chi=0$, by (12.22). At $\lvert\gamma_f\rvert=M_k$: in the limit (I) of $\mathcal S_\chi$ the growing part of the velocity of the level $k$ is $+\sigma\chi\tfrac14\tau^2y\,\lambda\sin2M_k\lambda$ and no other level has one, by (12.14); in order (II), after $\delta p\to0$ along any sequence at rest, it is $+\tfrac12\tau^2y\,\lambda\cos2M_k\lambda$, by (12.15), while a fixed start has the growing oscillation $-\tfrac y2\lambda\operatorname{Re}[c_g(2\lvert\gamma_f\rvert)e^{2i\lvert\gamma_f\rvert\lambda}]$ in every level.

**(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 $-\tau^2\gamma_0$. A fixed start, sufficient ranges of (12.18): drift of the velocity for $u\,w_g\ll\lvert\gamma_0\rvert/\Gamma_\lambda$ and $\lambda\ll(\lvert\gamma_0\rvert/\Gamma_\lambda)\lambda_g$; acceleration, against $\tau^2\Gamma_\lambda$, for $u\,w_g\ll1$ and $\lambda\ll\lambda_a$. (II): coefficient of $\lambda$ in the velocity $\gamma_0\langle\alpha_0\rangle_g+\gamma_1\operatorname{Re}[c_ge^{i(\Omega\lambda+\vartheta)}]$ | $Y^1$ | all. Limit (I) of $\mathcal S_\chi$, $\chi\neq0$: an oscillation of the limit velocity grows only in the level with $2M_k=\nu$; $\chi=0$: in none. (II), fixed start: an oscillation with the amplitude $\lvert\gamma_1c_g(\nu)\rvert\lambda$ in every level at every $\nu$, $c_g(\nu)<0$ for $0<\nu<2M_k$; after $\delta p\to0$ it remains at $\nu=2M_k$, and vanishes for $\nu<2M_k$ (every sequence at rest) and for $\nu>2M_k$ along sequences vanishing near $\{2\omega_k=\nu\}$ | (12.8)–(12.12), (12.14), (12.15), (12.18), (12.19) |
| Independence of the clock level | yes, $-\tau^2$ | Limit (I). Response at $\Omega$: $\chi=0$: $-\tau^2$ in every level at every frequency; $\chi\neq0$: distance from $-\tau^2$ of relative size $\lvert\chi\rvert\,2M_k\Omega/\lvert4M_k^2-\Omega^2\rvert$, spread between levels $\mathcal D(\Omega)$; not defined at $\Omega=2M_k$. Complete acceleration: the same for all levels iff $\chi=0$ ($\gamma\not\equiv0$). Window-averaged acceleration: $-\tau^2\gamma_0+O(1/\Lambda)$ for all levels, $\Omega\neq2M_k$. (II): the level with $2M_k=\Omega$ differs for every $\chi$ | $Y^1$ | Response the same for all levels to the accuracy $\delta$: iff $\mathcal D(\Omega)\le\delta$, (12.20); for $0<\Omega<2m$, one class and one branch, iff $\Omega\le\Omega_\delta$. Within $\delta\tau^2$ of $-\tau^2$ in every level (one class): iff $\Omega\notin\bigcup_k(\Omega^{(k)}_-,\Omega^{(k)}_+)$. For one class and one branch, (12.16) at the tolerance $\delta/2$ is sufficient for $\mathcal D\le\delta$ at every $\Omega$; (12.16) at $\delta$ is sufficient for $\Omega<2m$ or $\Omega>2M_{N-1}$ and necessary for $2m<\Omega<2M_{N-1}$. Complete acceleration: no condition on $\Omega$ alone; for $\gamma$ of Step 7 and $\Omega\neq2M_k$, $2\max_k\lvert\chi_k\rvert\frac{2M_k+\Omega}{\lvert2M_k-\Omega\rvert}\le\delta$ is sufficient for agreement within $\delta\tau^2(\lvert\gamma_0\rvert+\lvert\gamma_1\rvert)$ at every $\lambda\ge0$ | (12.13), (12.14), (12.16), (12.20)–(12.22) |

## 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 $Y^2$, $\lambda^4$, if $(Y-\gamma)C_f\varphi\neq0$ and $\Gamma\neq0$ ($\lambda^{2k_0+2}$ if (2.11) first fails at $k_0>1$).
- **(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).
- **(12.18)** — every smooth $g$ at rest, every continuous $\gamma$; exact bounds $E_v$, $E_a$; sufficient ranges, one per quantity: the drift of the velocity ($\gamma$ of Step 7, $\gamma_0\neq0$) for $u\,w_g\ll\lvert\gamma_0\rvert/\Gamma_\lambda$ and $\lambda\ll(\lvert\gamma_0\rvert/\Gamma_\lambda)\lambda_g$; the part of frequency $\Omega$ of the velocity, against $\tau^2\lvert\gamma_1\rvert/\Omega$, for $4\Omega\lambda(u+\tau^2\lambda/M)w_g\ll\lvert\gamma_1\rvert/\Gamma_\lambda$; the acceleration, against $\tau^2\Gamma_\lambda$, for $u\,w_g\ll1$ and $\lambda\ll\lambda_a$. $\lambda_a\le\lambda_g$; the conditions of the drift do not imply $\lambda\ll\lambda_a$ where $\lvert\gamma_0\rvert/\Gamma_\lambda>\lambda_a/\lambda_g$ (sufficient for this: $\lvert\gamma_0\rvert/\Gamma_\lambda>\tau\sqrt{w_g}/M$), so the range of the drift then does not cover the acceleration.
- **(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 $-\tau^2\gamma_0\lambda$, the only growing part of the limit functions.
- **(12.13)** — $\mathcal S_\chi$, (I), $\Omega\neq2M_k$; exact. $\mathcal A_{k,\sigma}(\Omega)=-\tau^2[1+i\sigma\chi\,2M_k\Omega/(4M_k^2-\Omega^2)]$; $\to-\tau^2$, i.e. (4.16), for $\Omega\to0$; for one class $\mathcal S_\chi$ on all levels and $\Omega>0$ different from all $2M_k$, $\mathcal A_{k,\sigma}(\Omega)$ is the same for all levels iff $\chi=0$; leading level dependence $-i\sigma\chi\tau^2\Omega/2M_k$ for $\Omega\ll2M_k$.
- **(12.14), (12.15), (12.19)** — $\Omega=2M_k$. Limit (I) of $\mathcal S_\chi$: the limit velocity of the level $k$, and of no other level, has an oscillation of frequency $2M_k$ growing linearly, coefficient $-\sigma\chi\tau^2\gamma_1/2$ (sine); none for $\chi=0$; a fixed start has it, with $\chi=\chi_g$, if $u\,w_g\ll\lvert\chi_g\gamma_1\rvert/\Gamma_\lambda$ and $\lambda\ll(\lvert\chi_g\gamma_1\rvert/\Gamma_\lambda)\lambda_g$ (sufficient, from $E_v$); the parts that do not grow have the sufficient ranges of Result 3(b). Order (II), fixed start: leading order with rest $O(\lambda^{1/2})$, a growing oscillation $\lambda\gamma_1\operatorname{Re}[c_g(\Omega)e^{i(\Omega\lambda+\vartheta)}]$ in every level at every $\Omega$; after $\delta p\to0$ at rest its coefficient is $-\tau^2\gamma_1$ (cosine) at $\Omega=2M_k$ for every sequence, zero below $2M_k$ for every sequence, zero above $2M_k$ for sequences vanishing near $\{2\omega_k=\Omega\}$. The limits do not commute. (12.19): real start at resonance, exact; before dephasing the finite-width terms are $-(\tau^4\gamma_1q_g/M)\lambda^2\sin(2M_k\lambda+\vartheta)$ (velocity) and $-2\tau^4\gamma_1q_g\lambda^2\cos(2M_k\lambda+\vartheta)$ (acceleration), of relative size one, against the amplitudes $\tau^2\lvert\gamma_1\rvert/2M$ and $\tau^2\lvert\gamma_1\rvert$ of the sharp-label velocity and acceleration, at $\lambda=1/(\tau\sqrt{2q_g})$.
- **(12.16), (12.20), (12.21)** — limit (I), response at the field frequency, normalized by $\tau^2$. (12.16), one class $\mathcal S_\chi$, exact iff: the response lies within $\delta\tau^2$ of $-\tau^2$ in every level iff $\Omega\notin\bigcup_k(\Omega^{(k)}_-,\Omega^{(k)}_+)$. (12.20): the level spread $\mathcal D(\Omega)$; the response is the same for all levels to the accuracy $\delta$ iff $\mathcal D(\Omega)\le\delta$; for one class and one branch, (12.16) at $\delta/2$ is sufficient at every $\Omega$, and (12.16) at $\delta$ is sufficient for $\Omega<2M_0$ or $\Omega>2M_{N-1}$ and necessary for $2M_0<\Omega<2M_{N-1}$; if both branches of all levels are compared in one class, (12.16) at $\delta/2$ is necessary and sufficient. (12.21): for $0<\Omega<2m$, one class and branch, iff $\Omega\le\Omega_\delta\approx\frac{2\delta}{\lvert\chi\rvert}\frac{M_0M_{N-1}}{M_{N-1}-M_0}$. Exceptional members: $\chi=0$ (spread zero), nearly equal level data, $\Omega$ at or near a $2M_k$.
- **(12.22)** — limit (I), every continuous $\gamma$: $a^{(1)}_{k,\sigma}(\lambda)=-\tau^2\gamma(\lambda)+\sigma\chi R_k[\gamma](\lambda)$. For $\gamma\not\equiv0$ the complete acceleration is the same for all levels iff $\chi_k=0$ on every level. To the accuracy $\delta$ only a sufficient condition is derived: for $\gamma$ of Step 7 and $\Omega$ different from every $2M_k$, $2\max_k\lvert\chi_k\rvert\frac{2M_k+\Omega}{\lvert2M_k-\Omega\rvert}\le\delta$ is sufficient for the complete accelerations of all levels and both branches to agree within $\delta\,\tau^2(\lvert\gamma_0\rvert+\lvert\gamma_1\rvert)$ at every $\lambda\ge0$. Its average over a window of length $\Lambda$ is $-\tau^2\gamma_0+O(1/\Lambda)$ for all levels if $\Omega\neq2M_k$.
- **(12.17)** — the two-place field: $\gamma(\lambda)=y\sin^2\gamma_f\lambda$, with the evaluations of Step 11 and the two controls; sharp and not stationary, with a generator that is not a contract, for $y\neq0$; for $y=0$ the body sees no field and $C_{\rm eff}=C_{br}$ is a contract.

## 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. Between the two orders only (12.19) is derived, for a real start at resonance; a formula uniform in $\lambda$ for starts with $j_g\neq0$, or through $\delta p$ and $\chi_g$ alone, is not derived; the exact (12.6), (12.7) cover it implicitly.
- The ranges of (12.18) and of Result 3(b) are sufficient; they rest on the bounds $E_v$, $E_a$, which are uniform in $\gamma$. Whether a fixed start is described beyond them is not derived, except by the exact (12.19) for a real start at resonance. For a $\gamma$ without a frequency near $2M_k$ the acceleration may be described on a longer range than $\lambda_a$; this is not derived.
- For the complete acceleration only the exact-equality criterion and a sufficient condition for the accuracy $\delta$ are derived, (12.22); a necessary and sufficient condition for the accuracy $\delta$ is not.
- 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), (12.18) 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); uniqueness for linear ordinary differential equations
- Stationary phase for the long-$\lambda$ behaviour; order of limits
- Triangle inequality and monotonicity for the spread of a response between levels

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