Version 4 · current

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 12-changing-field
- **Version:** v4
- **Mode:** external regeneration
- **Date:** 2026-10-11

## Changes from previous version

Changes against v3, for the two issues of `verification.r2.md` (external round 2), both in summary-level sentences.

- **Unchanged, word for word:** Setup; Steps 1–6; Step 7 except one sentence of Result 2(b); Step 8; Step 9 except the paragraph before "Between the two orders" and two items of Result 3(b); Step 10 except the end of "General $\gamma(\lambda)$"; Step 11 except the last clause of item 3 of (a); the three consistency checks; Methods used except one added keyword (the Sokhotski–Plemelj formula, used in the new paragraph of Step 9). All displays and the tags (12.1)–(12.22) are unchanged; no tag is renumbered, withdrawn or added, and no result is changed. The section "Changes from previous version" of v3 is replaced by this one.
- **I1 (the growing oscillation of a fixed start in order (II)).** (12.15) gives the coefficient $c_g(\Omega)$ of a term; the term grows where $\gamma_1c_g(\Omega)\neq0$. Changed sentences:
  - Step 9, the paragraph before "Between the two orders" ("Before this limit the coefficient does not vanish: ..."): replaced by the sign of $c_g(\Omega)$ in three ranges of $\Omega$ (below the gap and at it; inside the band, with the imaginary part of $c_g$ and the verifier's construction as the control; at the upper edge and above the band), and by a statement that names the range of $\Omega$ and the condition on the start. The clause "at rest it is of order $\delta p^2$" is now restricted to $\Omega$ below the gap, where v3 derived it.
  - Step 7, Result 2(b): the sentence "This is a statement on the limit (I); ... carries the growing oscillation of (12.15)".
  - Step 9, Result 3(b), item "Order (II)": the first sentence ("in every level and at every $\Omega>0$") is replaced by three sentences. In the sentence on the limit $\delta p\to0$ the upper edge $\Omega=2\sqrt{M_{k'}^2+\tau^2}$ moves from "zero for the restricted sequences, not determined otherwise" to "zero for every sequence at rest": a consequence of the closed form of $c_g$ at the edge, which the new paragraph of Step 9 contains.
  - Step 9, Result 3(b), item "What distinguishes the level": the last sentence.
  - Step 10, "General $\gamma(\lambda)$": the last sentence gets "with $\hat\gamma_\nu\neq0$" and "with $c_g\to-\tau^2$ for every sequence at rest"; one sentence on a fixed smooth start is added.
  - Step 11(a), item 3: the last clause ("a fixed start has the growing oscillation ... in every level").
  - Step 12, row "Drift acceleration from rest", last column, the entry "(II), fixed start"; row "Independence of the clock level", third column, the entry "(II)".
  - Result, item (12.14), (12.15), (12.19): the sentence "Order (II), fixed start: ..."; "The limits do not commute" now ends with "at $\Omega=2M_k$".
  - Open issues, fourth item: the band is the open one ("$<$" for "$\le$" at the upper edge), and one sentence is added.
- **I2 (row "Factor that the body sees" without its exceptions).** Changed sentences:
  - Step 12: a lead sentence before the table is added; the heading of the third column gets "(through $\gamma(\lambda)$)"; row "Factor that the body sees", columns three to five, now quotes the exceptions of Steps 2, 3 and 11.
  - Step 12, row "Independence of the clock level", read against (4.16) and (12.22): the second column names the quantity ("drift acceleration"), and the entry "Complete acceleration" of the third column says that it holds for a constant $\gamma\neq0$ as well. The row "Drift acceleration from rest" is unchanged except for the entry of I1.
  - Result, items (12.3), (12.4) and (12.5): "at first order in $Y$", the fields that are not stationary and have a constant $\gamma$, the class of (12.5) and its exceptional members.
- **Twin:** key results (12.4), (12.5), (12.15), the fourth open issue, `response_to_verification`.

## Response to verification

Round 2 (`verification.r2.md`, verdict "minor issues").

- **I1: Accepted and fixed in** Step 9 (the paragraph before "Between the two orders"). The verifier is right: inside the band $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$ the coefficient $c_g(\Omega)$ vanishes for particular starts, and the construction of the verifier is now the control. What is derived: $c_g(\Omega)<0$ for $0<\Omega\le2M_k$ and $c_g(\Omega)>0$ for $\Omega\ge2\sqrt{M_k^2+\tau^2}$, for every smooth start; inside the band $c_g(\Omega)\neq0$ for every smooth start with $g\neq0$ at one of the labels with $2\omega_k(p)=\Omega$. Every sentence that quotes (12.15) now says "where $\gamma_1c_g(\Omega)\neq0$" with this range and this condition: Step 7 (Result 2(b)), Step 9 (Result 3(b), items "Order (II)" and "What distinguishes the level"), Step 10 ("General $\gamma(\lambda)$"), Step 11(a) item 3, Step 12 (rows two and three), Result (item (12.14), (12.15), (12.19)), Open issues (fourth item).
- **I2: Accepted and fixed in** Step 12 (lead sentence, heading of the third column, row "Factor that the body sees") and in Result, items (12.3), (12.4), (12.5). The row now states: the generator is not a contract iff $\gamma$ is not constant, and a field that is not stationary can have a constant $\gamma$; under (2.11) the view is pure exactly, by (2.12); the first difference at $Y^2$ and $\lambda^4$ holds under the two conditions of (12.5), with the exceptional members of Step 3.

## 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 oscillation of frequency $\Omega$ of (12.15), whose amplitude $\lvert\gamma_1c_g(\Omega)\rvert\lambda$ grows where $\gamma_1c_g(\Omega)\neq0$: for $\gamma_1\neq0$, for every smooth start if $\Omega<2M_k$ or $\Omega\ge2\sqrt{M_k^2+\tau^2}$, and for the smooth starts with $g\neq0$ at one of the labels with $2\omega_k(p)=\Omega$ if $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$ (Step 9). 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 of a fixed smooth $g$ is governed by the sign of the denominator $4\omega^2-\Omega^2$ of (12.15), in three ranges of $\Omega$:
- *Below the gap and at it, $0<\Omega\le2M_k$.* $c_g(\Omega)=-\langle4M^2\omega'^2/(4\omega^2-\Omega^2)\rangle_g<0$ for every smooth $g$, because the integrand is positive except at zeros of $\omega'$ (at $\Omega=2M_k$ it is $M^2\tau^2\cos^2p/\omega^2$, as above).
- *Inside the band, $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$.* The denominator has simple zeros at the four labels $p_r$ with $2\omega_k(p_r)=\Omega$, where $\omega'\neq0$, and the limit $\epsilon\to0^+$ gives $\operatorname{Im}c_g(\Omega)=\frac{M^2}{2\Omega}\sum_{p_r}\lvert\omega'(p_r)\rvert\,\lvert g(p_r)\rvert^2$ (Sokhotski–Plemelj formula). Hence $c_g(\Omega)\neq0$ for every smooth $g$ with $g(p_r)\neq0$ at one of the four labels. If $g$ vanishes at all four, $c_g(\Omega)$ is real, and it can vanish. *Control:* let $g_1$, $g_2$ be smooth and normalized with $\lvert g_i(-p)\rvert=\lvert g_i(p)\rvert$, $g_1$ supported where $2\omega_k<\Omega$ and $g_2$ where $2\omega_k>\Omega$, both away from the $p_r$. Then $c_{g_1}(\Omega)>0>c_{g_2}(\Omega)$, and $g=\sqrt{1-t}\,g_1+\sqrt t\,g_2$ is a smooth start at rest with $c_g(\Omega)=(1-t)\,c_{g_1}(\Omega)+t\,c_{g_2}(\Omega)=0$ at $t=c_{g_1}/(c_{g_1}-c_{g_2})\in(0,1)$.
- *At the upper edge and above the band, $\Omega\ge2\sqrt{M_k^2+\tau^2}$.* $c_g(\Omega)=\langle4M^2\omega'^2/(\Omega^2-4\omega^2)\rangle_g>0$ for every smooth $g$, because the integrand is positive except at zeros of $\omega'$ (at the edge it is $M^2\tau^2\sin^2p/\omega^2$, so that there $c_g\le\tau^2\delta p^2\to0$ for every sequence at rest).

By (12.15) the velocity of a fixed smooth start therefore carries an oscillation of frequency $\Omega$ with the linearly growing amplitude $\lvert\gamma_1c_g(\Omega)\rvert\lambda$ where $\gamma_1c_g(\Omega)\neq0$: for $\gamma_1\neq0$, in a level with $\Omega\le2M_k$ or $\Omega\ge2\sqrt{M_k^2+\tau^2}$ for every smooth start, and in a level with $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$ for every smooth start with $g\neq0$ at one of the labels $p_r$. Where $\gamma_1c_g(\Omega)=0$, (12.15) gives $o(\lambda)$ for the part of frequency $\Omega$ and no more. Below the gap the amplitude is, at rest, 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, $\lambda\to\infty$.* (12.15) gives for the velocity the term $\lambda\gamma_1\operatorname{Re}[c_g(\Omega)e^{i(\Omega\lambda+\vartheta)}]$: an oscillation of frequency $\Omega$ whose amplitude $\lvert\gamma_1c_g(\Omega)\rvert\lambda$ grows in the levels and for the starts with $\gamma_1c_g(\Omega)\neq0$; where $\gamma_1c_g(\Omega)=0$, (12.15) gives $o(\lambda)$ for this part. For every smooth start, $c_g(\Omega)<0$ for $0<\Omega\le2M_k$ and $c_g(\Omega)>0$ for $\Omega\ge2\sqrt{M_k^2+\tau^2}$. For $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$, $c_g(\Omega)\neq0$ for every smooth start with $g\neq0$ at one of the labels with $2\omega_k(p)=\Omega$, and $c_g(\Omega)=0$ for particular starts that vanish at all of them (the control above). 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 $\Omega\ge2\sqrt{M_{k'}^2+\tau^2}$, for every sequence at rest; zero for $2M_{k'}<\Omega<2\sqrt{M_{k'}^2+\tau^2}$ 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: for $\gamma_1\neq0$ it grows like $\lambda$ in every level $k'$ with $\Omega\le2M_{k'}$ or $\Omega\ge2\sqrt{M_{k'}^2+\tau^2}$, for every smooth start, and in a level $k'$ with $2M_{k'}<\Omega<2\sqrt{M_{k'}^2+\tau^2}$ for every smooth start with $g\neq0$ at one of the labels with $2\omega_{k'}(p)=\Omega$.

### 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 $\lvert\nu\rvert=2M_k$ with $\hat\gamma_\nu\neq0$ adds the term of (12.15) in the level $k$, with $c_g\to-\tau^2$ for every sequence at rest. Before $\delta p\to0$, for a fixed smooth start, the coefficient of $\lambda$ is $\gamma_0\langle\alpha_0\rangle_g+\sum_{\nu>0}2\operatorname{Re}[c_g(\nu)\hat\gamma_\nu e^{i\nu\lambda}]$, by (12.15) and linearity in $\gamma$; the term of a frequency $\nu>0$ with $\hat\gamma_\nu\neq0$ is present in the level $k$ where $c_g(\nu)\neq0$: for every smooth start if $\nu\le2M_k$ or $\nu\ge2\sqrt{M_k^2+\tau^2}$, and for the smooth starts with $g\neq0$ at one of the labels with $2\omega_k(p)=\nu$ if $2M_k<\nu<2\sqrt{M_k^2+\tau^2}$ (Step 9).

### 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 smooth start has in a level $k'$ the term $-\tfrac y2\lambda\operatorname{Re}[c_g(2\lvert\gamma_f\rvert)e^{2i\lvert\gamma_f\rvert\lambda}]$ ($c_g$ with the data of the level $k'$), a growing oscillation where $c_g(2\lvert\gamma_f\rvert)\neq0$: for every smooth start in the levels with $M_{k'}\ge M_k$ or $M_{k'}^2+\tau^2\le M_k^2$, and for the smooth starts with $g\neq0$ at one of the labels with $\omega_{k'}(p)=M_k$ in the levels with $M_{k'}<M_k<\sqrt{M_{k'}^2+\tau^2}$.

**(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)

The third column states the change through the gradient seen $\gamma(\lambda)$ of (12.1), in the second and third rows with $\gamma_0$, $\gamma_1$, $\Omega$ of Step 7 ($\nu$ is a frequency of a general $\gamma$). A field that is not stationary can have a constant $\gamma$ (under (2.11); for $[Y,C_f]=0$; for $y=0$ in Step 11). At first order in $Y$ it then changes nothing: the entries of the third column reduce to those of the second, with $\gamma_1=0$.

| Quantity | Stationary sharp field | What a field that is not stationary changes (through $\gamma(\lambda)$) | Order in $Y$ | Field frequencies | Tags |
|---|---|---|---|---|---|
| Factor that the body sees | the number $1+\gamma x$; contract $C_{\rm red}$, exact | at order $Y^1$ the function $1+\gamma(\lambda)x$, with the generator $C_{\rm eff}(\lambda)$, which is not a contract iff $\gamma$ is not constant. A field that is not stationary can have a constant $\gamma$ (under (2.11); for $[Y,C_f]=0$; for $y=0$ in Step 11); $C_{\rm eff}$ is then a contract. Under (2.11) the view is pure and is that of $C_{\rm red}$ exactly, by (2.12). Beyond first order the view can be mixed: by (12.5), for a sharp start with (2.11) violated, if $(Y-\gamma)C_f\varphi\neq0$ and $\Gamma\neq0$ | $Y^1$: only $\gamma(\cdot)$. First difference from the view of the generator, for a sharp start with (2.11) violated: $Y^2$ and $\lambda^4$ if $(Y-\gamma)C_f\varphi\neq0$ and $\Gamma\neq0$; $Y^2$ and $\lambda^{2k_0+2}$ if (2.11) first fails at $k_0>1$; the coefficient of $\lambda^4$ vanishes if $\Gamma=0$. No difference at any order under (2.11). A start that is not sharp: $Y^2$ and $\lambda^2$ in the control $[Y,C_f]=0$ of Step 11(b), for $\Gamma\neq0$; not derived otherwise | at order $Y^1$ 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 smooth start: an oscillation with the amplitude $\lvert\gamma_1c_g(\nu)\rvert\lambda$, which grows where $\gamma_1c_g(\nu)\neq0$; for every smooth start $c_g(\nu)<0$ for $0<\nu\le2M_k$ and $c_g(\nu)>0$ for $\nu\ge2\sqrt{M_k^2+\tau^2}$; for $2M_k<\nu<2\sqrt{M_k^2+\tau^2}$, $c_g(\nu)\neq0$ if $g\neq0$ at one of the labels with $2\omega_k(p)=\nu$, and $c_g(\nu)=0$ for particular starts that vanish at all of them; 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 | drift acceleration: yes, $-\tau^2\gamma$ in every level and both branches, (4.16) | 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 $\chi=0$ ($\gamma\not\equiv0$; a constant $\gamma\neq0$ included). Window-averaged acceleration: $-\tau^2\gamma_0+O(1/\Lambda)$ for all levels, $\Omega\neq2M_k$. (II), $\gamma_1\neq0$, after $\delta p\to0$: the growing oscillation remains in the level with $2M_k=\Omega$, for every $\chi$ and every sequence at rest; in a level $k'$ it vanishes for $\Omega<2M_{k'}$ or $\Omega\ge2\sqrt{M_{k'}^2+\tau^2}$ (every sequence at rest) and, for $2M_{k'}<\Omega<2\sqrt{M_{k'}^2+\tau^2}$, along sequences vanishing near $\{2\omega_{k'}=\Omega\}$ | $Y^1$ | 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. At first order in $Y$ the view of $\{b,c\}$ and every schedule depend on the field only through $\gamma(\cdot)$ of (12.1), and are reproduced up to $O(\lVert Y\rVert^2)$ by the generator $C_{\rm eff}(\lambda)=C_{br}+\gamma(\lambda)B$, which is an admissible contract iff $\gamma$ is constant. A field that is not stationary can have a constant $\gamma$ (under (2.11); for $[Y,C_f]=0$; for $Y=y\,\mathbb 1_f$). Exact at all orders for a sharp stationary field (2.3) and under (2.11), by (2.12).
- **(12.5)** — sharp start with (2.11) violated (the case "sharp and not stationary" of the question); expansion in $\lambda$ at fixed operators; exact leading term. First difference from the view of the generator at order $Y^2$ and $\lambda^4$, if $(Y-\gamma)C_f\varphi\neq0$ and $\Gamma\neq0$; at $Y^2$ and $\lambda^{2k_0+2}$ if (2.11) first fails at $k_0>1$; the coefficient of $\lambda^4$ vanishes if $\Gamma=0$. No difference at any order under (2.11), by (2.12).
- **(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 smooth start, every $\Omega>0$: leading order with rest $O(\lambda^{1/2})$, the term $\lambda\gamma_1\operatorname{Re}[c_g(\Omega)e^{i(\Omega\lambda+\vartheta)}]$, a growing oscillation where $\gamma_1c_g(\Omega)\neq0$; for every smooth start $c_g(\Omega)<0$ for $0<\Omega\le2M_k$ and $c_g(\Omega)>0$ for $\Omega\ge2\sqrt{M_k^2+\tau^2}$; for $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$, $c_g(\Omega)\neq0$ if $g\neq0$ at one of the labels with $2\omega_k(p)=\Omega$, and $c_g(\Omega)=0$ for particular starts that vanish at all of them; 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 at $\Omega=2M_k$. (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, $2M_k<\Omega<2\sqrt{M_k^2+\tau^2}$: $c_g(\Omega)$ 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. For a fixed start that vanishes at all these labels $c_g(\Omega)$ is real and can vanish (the control of Step 9); which of these starts have $c_g(\Omega)=0$ is not characterized, and where $\gamma_1c_g(\Omega)=0$ only the estimate $o(\lambda)$ of (12.15) is derived for the part of frequency $\Omega$ of the velocity.
- 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
- Sokhotski–Plemelj formula for a resonant denominator (sign and zeros of the long-run coefficient)
- 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