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

# Fall of a bound pair of two bodies in a sharp field

- **Subproject:** 05-ilang-free-fall
- **Package:** 13-pair-fall
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## Setup and assumptions

- **Setting.** Objects, contract $C$, class $\mathcal W$, records, field, levels, bound level, packets, orders and the regime $\mathsf R$ are those of the question, used literally. Assumptions of the package (M8, A11): two Model D bodies, one medium with one operator and one strength, a binding of the class $\mathcal W$ without a factor of the field, the field coupled to the rest and clock terms only, the field start $\varphi$ sharp and stationary, and a packet of a bound level as the start (M9). Equal parameters of the two types are never assumed.
- **Inputs, as quoted in the question:** (11.1), (11.2), (11.5)–(11.9), (11.12)–(11.14), (11.16), (11.19)–(11.22) of 11-bound-pair; (8.10), (8.11) of 08-two-types; (2.1), (2.3) of 02-field-object; (4.7), (4.10), (4.11), (4.16), (4.24) of 04-branch-fall; (5.1), (5.5), (5.6), (5.11) of 04-ilang-time/05-moving-clock; (17.15) of 03-ilang-space/17-motion; (16.11) of 03-ilang-space/16-common-space.
- **Notation added here.**
  - The packet weight is written $g(P)$ and enters only through $g$, $g'$ and $\langle F\rangle_g:=\int\mathrm dP\,\lvert g(P)\rvert^2F(P)$. The common gradient ($g$ in the question) is written $g_{\rm c}:=g_1=g_2$.
  - $\mathfrak h:=\ell^2(\mathbb Z)\otimes\mathbb C^2\otimes\mathbb C^2$ is the fibre space; $\lvert P;\psi\rangle(x_1,x_2)=(2\pi)^{-1/2}e^{iP(\eta x_1+(1-\eta)x_2)}\psi(x_1-x_2)$, so that $\langle P;\psi\vert P';\psi'\rangle=\delta(P-P')\langle\psi\vert\psi'\rangle$ and $\lVert\Phi_g\rVert=1$.
  - $\Gamma(P)>0$ is the distance of $E_b(P)$ from the rest of the spectrum of $H(P)$; $R_P:=Q_P(H(P)-E_b(P))^{-1}Q_P$ with $\lVert R_P\rVert=1/\Gamma(P)$; $\mathcal U_P(\lambda):=\mathbb 1-e^{-i(H(P)-E_b(P))\lambda}$; $I_g$ as in (11.8); $\Gamma_g:=\min_{I_g}\Gamma>0$.
  - $\mathsf b:=\lvert g_1\rvert M_1+\lvert g_2\rvert M_2+2\kappa\lvert\langle X\rangle\rvert$, $\mathsf d_\eta:=\lvert1-\eta\rvert\lvert g_1\rvert M_1+\lvert\eta\rvert\lvert g_2\rvert M_2+\lvert1-2\eta\rvert\kappa\lvert\langle X\rangle\rvert$, $\tau_{\min}:=\min(\tau_1,\tau_2)$.
- **Standard facts used.**
  - F1 (first order in a term of the contract): $e^{-i(H_0+\Delta)\lambda}=e^{-iH_0\lambda}-i\int_0^\lambda\mathrm ds\,e^{-iH_0(\lambda-s)}\Delta\,e^{-iH_0s}+O(\Delta^2)$ at fixed $\lambda$.
  - F2 (isolated simple eigenvalue of an analytic family): $E_b$ and $\psi_P$ are analytic in $P$ and in the data, $Q_P\psi_P'=-R_PV_\eta\psi_P$, and $\langle\psi_P\vert\psi_P'\rangle\in i\mathbb R$.
  - F3: an eigenvector of $H(P)$ for an eigenvalue outside the bands (11.2) decays exponentially in $\lvert r\rvert$ (finite range of the hopping and of $W$), locally uniformly in $P$; hence $\hat r\psi_P\in\mathfrak h$.
  - F4 (long chain): at fixed $\lambda$ the first-order coefficients on the chain $\lvert x_i\rvert\le L$ converge for $L\to\infty$ to those of the infinite chain, because the hopping has range one and $\Phi_g$, $\hat x_i\Phi_g$ are square summable.
- **Approximations and their range.** First order in $(g_1,g_2,\kappa\langle X\rangle)$ at fixed $\lambda$ (the size of the second order is not estimated). Records at leading order in $\kappa$ with the convention of the question: the bodies evolve by $C_{\rm bod}$. Items 3 and 4 at leading order in $\mathsf R$ at fixed $u$; their remainders are not uniform in $u$.

## Derivation

### Step 1. Reduced contract (item 1a)

1. Every term of $C$ is the identity on $\mathcal H_f$, or $\theta_i\,\hat x_i\otimes G_i\otimes\Phi$, or $C_f$. With $\Phi\varphi=\gamma\varphi$ and $C_f\varphi=e_f\varphi$, for every $\Psi\in\mathcal H_{b_1}\otimes\mathcal H_{b_2}\otimes\mathcal H_c$ (the computation of (2.3) and (8.10); $U$ is carried along unchanged):

$$
C\,(\Psi\otimes\varphi)=\bigl((C_{\rm red}+e_f)\Psi\bigr)\otimes\varphi,\qquad C_{\rm red}=\sum_{i=1,2}\Bigl[A_i+\sum_xP^{(i)}_x\otimes(1+g_ix)\,G_i+\kappa\,\hat x_i\otimes X\Bigr]+U .
\tag{13.1}
$$

2. Hence $e^{-iC\lambda}(\Psi\otimes\varphi)=e^{-ie_f\lambda}\,(e^{-iC_{\rm red}\lambda}\Psi)\otimes\varphi$ for all $\lambda$: the view of $\{b_1,b_2,c\}$ is pure and evolves by $C_{\rm red}$ (exact, for this start of the field only).
3. $C_{\rm red}$ is the contract of an admissible description of $b_1,b_2,c$. It is self-adjoint and independent of $\lambda$; $U$ is a pair term of $b_1,b_2$; $\kappa\,\hat x_i\otimes X$ are pair terms of $b_i,c$; $A_i+\sum_xP^{(i)}_x\otimes(1+g_ix)G_i$ acts on one object and is included in a pair term (A5). Its position-dependent rest and clock term is then a property of the type of $b_i$ (A10).
4. A binding with a factor of the field needs a term on $\mathcal H_{b_1}\otimes\mathcal H_{b_2}\otimes\mathcal H_f$, for example $\sum_{x_1,x_2}\lvert x_1\rangle\langle x_1\rvert\otimes\lvert x_2\rangle\langle x_2\rvert\otimes W(x_1-x_2)\otimes\Phi_{x_1x_2}$, that is not a sum of pair terms. Decompose the operators of each object into $\mathbb C\mathbb 1$ and the traceless ones: a sum of pair terms has no component that is traceless on all three objects, while $U'\otimes\Phi'$ has one whenever $U'\neq u_1\otimes\mathbb 1+\mathbb 1\otimes u_2$ and $\Phi'\notin\mathbb C\mathbb 1$. Such a three-object term is not of the form (1.10).

### Step 2. The operator on the two bodies (item 1b)

1. The partial expectation of $C-e_f$ in $\chi\otimes\varphi$ is, by (13.1),

$$
C_{\rm bod}=C^{\rm D}_1+C^{\rm D}_2+U+\sum_{i=1,2}g_i\,\hat x_i\otimes G_i+\kappa\langle X\rangle\,(\hat x_1+\hat x_2).
\tag{13.2}
$$

2. $[C_{\rm bod},D_i]=0$, so the levels are conserved, and $G_i=M_i\sigma^{(i)}_3$ on the levels $k_1,k_2$. With $\hat x_1=\hat X_\eta+(1-\eta)\hat r$ and $\hat x_2=\hat X_\eta-\eta\hat r$ the gradient terms $\Delta:=C_{\rm bod}-C_{\rm pair}$ are

$$
\begin{aligned}
\Delta&=\hat X_\eta\,\mathcal K+\hat r\,\mathcal D_\eta,\qquad \mathcal K:=g_1M_1\sigma^{(1)}_3+g_2M_2\sigma^{(2)}_3+2\kappa\langle X\rangle,\\
\mathcal D_\eta&:=(1-\eta)\,g_1M_1\sigma^{(1)}_3-\eta\,g_2M_2\sigma^{(2)}_3+(1-2\eta)\,\kappa\langle X\rangle .
\end{aligned}
\tag{13.3}
$$

3. Well-definedness (M6). A replacement of the listed terms that leaves $C$ unchanged leaves $\langle\chi\otimes\varphi\vert C\vert\chi\otimes\varphi\rangle$ unchanged. It can change the listed $C_f$ by a multiple of $\mathbb 1$, hence $e_f$: $C_{\rm bod}$ is fixed by $C$, $\chi$, $\varphi$ up to an additive real constant, which drops out of everything below. Let $b(x_1,x_2)$ be the block of $C_{\rm bod}$ that is diagonal in both positions, an operator on the internal labels. For $\lvert x_1-x_2\rvert>R$ it is $(1+g_1x_1)G_1+(1+g_2x_2)G_2+\kappa\langle X\rangle(x_1+x_2)$ up to the constant. $G_1$ and $G_2$ are traceless and act on the labels of one body each, so the three parts are linearly independent: $\kappa\langle X\rangle$ is the normalized trace of $b(x_1+1,x_2)-b(x_1,x_2)$ (both arguments in that region), $g_1G_1$ is its traceless part, and $G_1$ is the traceless part of $b(0,x_2)$ on the labels of $b_1$; likewise for $b_2$.
   - $g_1$, $g_2$, $\kappa\langle X\rangle$: fixed by $C_{\rm bod}$, well defined, independent of $\eta$. They are the coefficients read from $C_{\rm bod}$. The listed numbers $\gamma\theta_i$ and $\kappa\langle\chi\vert X\vert\chi\rangle$ can change under a relisting such as $X\to X-c$, $C^{\rm D}_i\to C^{\rm D}_i+\kappa c\,\hat x_i$. Used: the label $0$ of the two types (A2) and the finite range of $W$.
   - The operator that multiplies the centre label, $\mathcal K$ (on all levels $g_1G_1+g_2G_2+2\kappa\langle X\rangle$): fixed by $C_{\rm bod}$, independent of $\eta$.
   - The operator that multiplies $\hat r$: fixed by $C_{\rm bod}$ once $\eta$ is given. It depends on the convention, $\mathcal D_\eta=\mathcal D_0-\eta\,\mathcal K$, and is independent of $\eta$ iff $\mathcal K=0$, i.e. (as $M_i>0$) iff all three gradients vanish.

### Step 3. First-order velocities from a bound level (item 2a)

Class: every $W\in\mathcal W$, all data, every bound level, every $g\in\mathcal G(\mathcal P_0)$, all gradients. On the levels $k_1,k_2$ write $H_0:=C_{\rm pair}$ and $\Phi_g(\lambda):=e^{-iH_0\lambda}\Phi_g$.

1. $\Delta$ is diagonal in the positions, so $i[C_{\rm bod},\hat x_i]=V_i$ and $\mathrm d\bar m_i/\mathrm d\lambda=\langle V_i\rangle$ in the evolved state, exactly. The first-order velocity $v^{(1)}_i$ is the part of this expectation that is linear in the gradients. By F1,

$$
v^{(1)}_i(\lambda)=2\,\mathrm{Re}\,\bigl\langle\Phi_g(\lambda)\big\vert V_i\big\vert\Phi^{(1)}(\lambda)\bigr\rangle,\qquad
\Phi^{(1)}(\lambda):=-i\int_0^\lambda\mathrm ds\;e^{-iH_0(\lambda-s)}\,\Delta\,\Phi_g(s),
\tag{13.4}
$$

and the same formula with $V_i\to\hat r$ gives the first-order change of $\bar m_1-\bar m_2$. On a finite chain all operators are bounded; the long-chain limit is F4, with (13.5).
2. In the fibres $\Phi_g(s)$ is $g\,e^{-iE_bs}\psi_P$, and $\hat X_\eta$ acts as $i\partial_P$ ($g$ has compact support in $\mathcal P_0$), so

$$
\Delta\,\Phi_g(s)\ \longleftrightarrow\ e^{-iE_b(P)s}\bigl[s\,E_b'\,g\,\mathcal K\psi_P+\xi_P\bigr],\qquad
\xi_P:=i\,g'\,\mathcal K\psi_P+i\,g\,\mathcal K\psi_P'+g\,\hat r\,\mathcal D_\eta\psi_P ,
\tag{13.5}
$$

a vector of $\mathfrak h$ with $\int\mathrm dP\,\lVert\xi_P\rVert^2<\infty$ (F2, F3, $g\in C^1$ with compact support).
3. Split (13.5) along $\psi_P$ and in $Q_P\mathfrak h$. On $\psi_P$ the propagator is $e^{-iE_b(\lambda-s)}$. On $Q_P\mathfrak h$, $\int_0^\lambda\mathrm ds\,e^{-i(H-E_b)(\lambda-s)}=-iR_P\,\mathcal U_P(\lambda)$ and $\int_0^\lambda\mathrm ds\,s\,e^{-i(H-E_b)(\lambda-s)}=-i\lambda R_P+R_P^2\,\mathcal U_P(\lambda)$.
4. Part along $\psi_P$. $\langle V_i\rangle_P=E_b'$ by (11.7), and $\langle\mathcal K\rangle_P$, $\langle\hat r\mathcal D_\eta\rangle_P$ are real. The terms with $\lambda^2$ and with $\mathcal D_\eta$ are imaginary and drop out. With $2\,\mathrm{Re}\langle\psi_P\vert\mathcal K\vert\psi_P'\rangle=\partial_P\langle\mathcal K\rangle_P$ the rest is $\lambda\int\mathrm dP\,E_b'\,\partial_P\bigl(\lvert g\rvert^2\langle\mathcal K\rangle_P\bigr)=-\lambda\,\langle E_b''\langle\mathcal K\rangle\rangle_g$ (partial integration).
5. Part in $Q_P\mathfrak h$. The term $-i\lambda R_P$ gives $-\lambda\int\mathrm dP\,\lvert g\rvert^2E_b'\,2\,\mathrm{Re}\langle V_i\psi_P\vert R_P\vert\mathcal K\psi_P\rangle$. Since $V_1-V_2=i[H(P),\hat r]$ and $(H-E_b)\psi_P=0$,

$$
\begin{aligned}
R_PV_1\psi_P&=R_PV_\eta\psi_P+i(1-\eta)\,Q_P\hat r\psi_P,\qquad R_PV_2\psi_P=R_PV_\eta\psi_P-i\eta\,Q_P\hat r\psi_P,\\
2\,\mathrm{Re}\,\langle V_i\psi_P\vert R_P\vert\mathcal K\psi_P\rangle&=-\partial_P\langle\mathcal K\rangle_P\qquad(i=1,2).
\end{aligned}
\tag{13.6}
$$

The second line holds because $\partial_P\langle\mathcal K\rangle_P=-2\,\mathrm{Re}\langle\mathcal K\psi_P\vert R_P\vert V_\eta\psi_P\rangle$ (F2) and $\langle Q_P\hat r\psi_P\vert\mathcal K\psi_P\rangle=\langle\hat r\mathcal K\rangle_P-\langle\hat r\rangle_P\langle\mathcal K\rangle_P$ is real ($[\hat r,\mathcal K]=0$).
6. Result, for $i=1,2$ and every $\lambda$:

$$
\boxed{\;v^{(1)}_i(\lambda)=\lambda\,\langle\alpha\rangle_g+\rho_i(\lambda)\;},\qquad \alpha(P):=-E_b''(P)\,\langle\mathcal K\rangle_P+E_b'(P)\,\partial_P\langle\mathcal K\rangle_P ,
\tag{13.7}
$$

$$
\alpha=g_1\alpha_1+g_2\alpha_2+\kappa\langle X\rangle\,\alpha_X,\qquad
\alpha_i(P)=-M_i\bigl[E_b''\,\langle\sigma^{(i)}_3\rangle_P-E_b'\,\partial_P\langle\sigma^{(i)}_3\rangle_P\bigr],\qquad \alpha_X(P)=-2E_b''(P),
\tag{13.8}
$$

$$
\begin{aligned}
\rho_i(\lambda)&=-2\,\mathrm{Re}\int\mathrm dP\,\Bigl[i\lvert g\rvert^2E_b'\,\langle V_i\psi_P\vert R_P^2\,\mathcal U_P(\lambda)\vert\mathcal K\psi_P\rangle+g^{*}\langle V_i\psi_P\vert R_P\,\mathcal U_P(\lambda)\vert\xi_P\rangle\Bigr],\\
\lvert\rho_i(\lambda)\rvert&\le\tau_i\,\mathsf B_g,\qquad \mathsf B_g:=\frac4{\Gamma_g}\Bigl[\mathsf b\Bigl(\frac{\tau_{\min}}{\Gamma_g}+\lVert g'\rVert_2+\max_{I_g}\lVert\psi_P'\rVert\Bigr)+\mathsf d_\eta\max_{I_g}\lVert\hat r\psi_P\rVert\Bigr].
\end{aligned}
\tag{13.9}
$$

The bound uses $\lVert V_i\rVert\le\tau_i$, $\lvert E_b'\rvert\le\tau_{\min}$, $\lVert\mathcal U_P\rVert\le2$, $\lVert\mathcal K\rVert\le\mathsf b$, $\lVert\mathcal D_\eta\rVert\le\mathsf d_\eta$ and $\int\lvert g\rvert\lvert g'\rvert\le\lVert g'\rVert_2$.
7. Status. Both parts have the same coefficient $\langle\alpha\rangle_g$ of the growing part. The rests differ in general, and $\rho_i(0)=0$. The rest is bounded for all $\lambda$, in particular for all $\lambda\ge0$, under the conditions of the class alone: the level is isolated on the compact $I_g\subset\mathcal P_0$ ($\Gamma_g>0$) and $g\in\mathcal G(\mathcal P_0)$ ($\lVert g'\rVert_2<\infty$). $\rho_i$ does not depend on $\eta$ or on the phase of $\psi_P$; the single terms of the bound do. By (11.5) and F2, $\langle\sigma^{(i)}_3\rangle_P=\partial E_b/\partial M_i$ and $\partial_P\langle\sigma^{(i)}_3\rangle_P=\partial E_b'/\partial M_i$. Numerical check: section "Code".

### Step 4. Mean label separation (item 2b)

1. The computation of Step 3 with $V_i\to\hat r$ ($\langle\hat r\rangle_P$ is real; (13.6) is not used) gives, for every member of the class of item 2,

$$
\begin{aligned}
(\bar m_1-\bar m_2)^{(1)}(\lambda)&=\lambda\,\langle\beta\rangle_g+\sigma(\lambda),\qquad \lvert\sigma(\lambda)\rvert\le\max_{I_g}\lVert\hat r\psi_P\rVert\;\mathsf B_g,\\
\beta(P)&:=-\langle\mathcal K\rangle_P\,\partial_P\langle\hat r\rangle_P-2E_b'(P)\,\mathrm{Re}\,\langle\hat r\psi_P\vert R_P\vert\mathcal K\psi_P\rangle ,
\end{aligned}
\tag{13.10}
$$

where $\sigma$ is the first line of (13.9) with $V_i\psi_P\to\hat r\psi_P$. There is no term $\propto\lambda^2$: the first-order relative velocity $\rho_1-\rho_2=\langle\beta\rangle_g+\dot\sigma$ has no growing part. This holds alike for the common gradient and for $g_1\neq g_2$.
2. Bounded case (sufficient condition). Let $W(-r)^{*}=W(r)$ for all $r$ (complex conjugate of the matrix in the branch basis); every diagonal on-site binding has this property. The antiunitary $(\Theta\psi)(r):=\psi(-r)^{*}$ then commutes with $H(P)$ for every $P$ and $\eta$ ($\Theta S\Theta^{-1}=S^\dagger$ in (11.1)) and with $\mathcal K$, and $\Theta\hat r\Theta^{-1}=-\hat r$. For a simple level $\Theta\psi_P=e^{i\phi}\psi_P$, hence $\langle\hat r\rangle_P=0$ for all $P$ and $\langle\hat r\psi_P\vert R_P\vert\mathcal K\psi_P\rangle\in i\mathbb R$. So $\beta\equiv0$, and the first-order change of the separation is $\sigma(\lambda)$: bounded for all $\lambda\ge0$, for all values of the gradients.
3. Vanishing case (sufficient condition). Let $\tau_1=\tau_2$, $M_1=M_2$, $W(-r)=\mathsf S\,W(r)\,\mathsf S$ with $\mathsf S$ the exchange of the two branch labels, and $g_1=g_2$ (any $\kappa\langle X\rangle$). The exchange $\Sigma$ of the labels of the two bodies then commutes with $C_{\rm bod}$ and with the joint shift, $\Sigma\Phi_g=\pm\Phi_g$ (simple level) and $\Sigma\hat r\Sigma=-\hat r$. Hence $\bar m_1-\bar m_2=0$ for all $\lambda$, to all orders in the gradients.
4. Otherwise. For a binding without the property of 2, $\beta\not\equiv0$ in general, and the first-order separation grows linearly in $\lambda$: the internal structure of the level follows the first-order change $-\lambda\langle\mathcal K\rangle_P$ of the total label. This rests on a numerical control (`code/v1/pair_fall_check.py`, set S5), not on an analytic example. With the property of 2 and without the conditions of 3 the change is bounded and does not vanish in general (same file, sets S1–S4: it relaxes to a non-zero value).

### Step 5. Drift acceleration (items 2c, 2d)

1. The first-order acceleration is $a^{(1)}_i=\mathrm dv^{(1)}_i/\mathrm d\lambda=\langle\alpha\rangle_g+\dot\rho_i(\lambda)$; $\dot\rho_i$ is bounded, but neither small nor decaying in general. Its window average is a difference quotient of the velocity, exactly: $\overline{a^{(1)}_i}=\langle\alpha\rangle_g+[\rho_i(\lambda_1+\Lambda)-\rho_i(\lambda_1)]/\Lambda$, so $\lvert\overline{a^{(1)}_i}-\langle\alpha\rangle_g\rvert\le2\tau_i\mathsf B_g/\Lambda$ for all $\lambda_1\ge0$, $\Lambda>0$.
2. Sharp total label. Class of sequences $\mathcal S(P_0,I)$: packets of $\mathcal G(\mathcal P_0)$ with supports in one compact interval $I\subset\mathcal P_0$, mean $P_0$ and $\delta P\to0$. Taylor's formula with $\int(P-P_0)\lvert g\rvert^2=0$ gives $\lvert\langle\alpha\rangle_g-\alpha(P_0)\rvert\le\frac12\max_I\lvert\alpha''\rvert\,\delta P^2$. Hence, with the window first and the sharp label second, for both parts:

$$
\boxed{\;a^{\rm drift}(P_0):=\lim_{\delta P\to0}\,\lim_{\Lambda\to\infty}\overline{a^{(1)}_i}=g_1\,\alpha_1(P_0)+g_2\,\alpha_2(P_0)+\kappa\langle X\rangle\,\alpha_X(P_0)\;},\qquad \alpha_i=-M_i\Bigl[E_b''\,\frac{\partial E_b}{\partial M_i}-E_b'\,\frac{\partial E_b'}{\partial M_i}\Bigr],\quad\alpha_X=-2E_b'' .
\tag{13.11}
$$

Here $E_b'=\langle V_i\rangle_P$ (11.7), $\partial E_b/\partial M_i=\langle\sigma^{(i)}_3\rangle_P$ (11.5), and $E_b''=-\eta^2\langle A_1\rangle_P-(1-\eta)^2\langle A_2\rangle_P-2\mathcal S_\eta(P)$ (11.9).
3. The other order. At a fixed window the limit $\delta P\to0$ does not exist on $\mathcal S(P_0,I)$ in general. The joint shift by $n$ labels, $g\to g\,e^{-inP}$, maps the class to itself with the same $\lvert g\rvert^2$. In (13.9) only the term with $g'$ changes: $\rho_i\to\rho_i-2n\,\mathrm{Re}\langle F_{i,\lambda}\rangle_g$ with $F_{i,\lambda}(P):=\langle V_i\psi_P\vert R_P\,\mathcal U_P(\lambda)\vert\mathcal K\psi_P\rangle$. The window averages of a sequence and of its shifted copy therefore differ by $-\frac{2n}{\Lambda}\mathrm{Re}\langle F_{i,\lambda_1+\Lambda}-F_{i,\lambda_1}\rangle_g$, which tends to $-2n\,\mathsf f_i/\Lambda$ with $\mathsf f_i:=\mathrm{Re}\bigl[F_{i,\lambda_1+\Lambda}(P_0)-F_{i,\lambda_1}(P_0)\bigr]$. For $\mathsf f_i\neq0$ two sequences that differ by a fixed shift have different limits, and a sequence shifted by $n\to\infty$ has none. On a subclass on which the limit exists it defines the window average of the first-order acceleration of a start with a sharp total label. That quantity depends on the window, on the part and on the place of the packet, so it is not a property of the level; whether it tends to $\alpha(P_0)$ for $\Lambda\to\infty$ is not derived.
4. What multiplies the gradients. $E_b''(P_0)$ multiplies all three, but it is the whole coefficient only for $\kappa\langle X\rangle$: $\alpha_X=-2E_b''$, exactly, on the whole class. For $g_i$ it is weighted by $M_i\,\partial E_b/\partial M_i=M_i\langle\sigma^{(i)}_3\rangle_{P_0}$, and for $E_b'(P_0)\neq0$ the term $M_iE_b'\,\partial E_b'/\partial M_i$ enters in addition. Exceptional members:
   - $E_b''(P_0)=0$: $\alpha_X=0$, $\alpha_i=M_iE_b'\,\partial E_b'/\partial M_i$; at $P_0=0$ the inertia $\mathfrak m_b$ is not defined.
   - $E_b'(P_0)=0$: $\alpha_i=-M_iE_b''\langle\sigma^{(i)}_3\rangle_{P_0}$; at $P_0=0$ then $a^{\rm drift}(0)=-\langle\mathcal K\rangle_0/\mathfrak m_b$. The class does not imply $E_b'(0)=0$; a sufficient condition is $W(-r)=\sigma^{(1)}_3\sigma^{(2)}_3W(r)\sigma^{(1)}_3\sigma^{(2)}_3$ together with a level that the map $P\to-P$ takes to itself.
   - $\langle\sigma^{(i)}_3\rangle_{P_0}=0$: the first term of $\alpha_i$ vanishes.
5. Common gradient (item 2d). With (11.6),

$$
\alpha_1+\alpha_2=-E_b''\,\mathcal M_P+E_b'\,\mathcal M_P',\qquad
\mathcal M_P:=M_1\langle\sigma^{(1)}_3\rangle_P+M_2\langle\sigma^{(2)}_3\rangle_P=E_b(P)-\langle A_1\rangle_P-\langle A_2\rangle_P-\langle U\rangle_P .
\tag{13.12}
$$

The coefficient of $g_{\rm c}$ is built from the two rest terms of the scaling relation only. The motion terms $\langle A_i\rangle_P$ and the binding term $\langle U\rangle_P$, which the field does not multiply, do not enter; $\mathcal M_P=E_b(P)$ iff $\langle A_1\rangle_P+\langle A_2\rangle_P+\langle U\rangle_P=0$.

### Step 6. Diagonal on-site binding in the regime $\mathsf R$ (item 3)

Class: every diagonal on-site binding with $\hat w_{11}\neq0$, the level (11.12), fixed hatted data, $P_0=0$.

1. At rest. $(\mathcal R\psi)(r):=\sigma^{(1)}_3\sigma^{(2)}_3\psi(-r)$ gives $\mathcal RH(P)\mathcal R^{-1}=H(-P)$ for a diagonal on-site binding, and by (11.12) the level near $\mathcal B_{++}$ is the only one for small $\epsilon$. So $E_b(-P)=E_b(P)$ and $E_b'(0)=0$, exactly. By (11.8), without gradients $\mathrm d\bar m_i/\mathrm d\lambda=\langle E_b'\rangle_g=O(\delta P^2)$ for all $\lambda$ (zero for even $\lvert g\rvert^2$): the pair is at rest in the sharp-total-label limit.
2. Closed form. (13.11) at $P_0=0$ is $a_{\rm pair}:=a^{\rm drift}(0)=-E_b''(0)\bigl[g_1M_1\langle\sigma^{(1)}_3\rangle_0+g_2M_2\langle\sigma^{(2)}_3\rangle_0+2\kappa\langle X\rangle\bigr]$, exactly. With (11.14) and (11.16):

$$
\boxed{\;a_{\rm pair}=\frac{\varsigma}{\sqrt{1+u^2}}\;\frac{g_1M_1(1+r_1)+g_2M_2(1+r_2)+2\kappa\langle X\rangle(1+r_X)}{\mathfrak m_1+\mathfrak m_2}\;},\qquad r_1,r_2,r_X=O(\epsilon^2)\ \text{at fixed }u .
\tag{13.13}
$$

Equivalently $a_{\rm pair}=-(g_1M_1+g_2M_2+2\kappa\langle X\rangle)/\mathfrak m_b$. Since $\mathfrak m_ia^{(i)}=-(g_iM_i+\kappa\langle X\rangle)$, at leading order $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\,\bar a$ with the inertia-weighted mean $\bar a:=(\mathfrak m_1a^{(1)}+\mathfrak m_2a^{(2)})/(\mathfrak m_1+\mathfrak m_2)$ of the free falls. Each coefficient is of order $\epsilon^2$; the remainders are not uniform in $u$.
3. Ordered limits of the leading coefficient (first $\epsilon\to0$ at fixed $u$, then $u$):

$$
u\to0:\ \ a_{\rm pair}\to-\varsigma\,\bar a\,\bigl[1-\tfrac12u^2+O(u^4)\bigr];\qquad
u\to\infty:\ \ a_{\rm pair}\to-\varsigma\,\frac{\bar a}{u}\,\bigl[1+O(u^{-2})\bigr]\to0 .
\tag{13.14}
$$

For $w_{11}<0$ the wide pair falls with $\bar a$: the sum of the weights $g_iM_i+\kappa\langle X\rangle$ over the sum of the inertias. For $w_{11}>0$ the level lies above the band, $\mathfrak m_b<0$, and the pair accelerates opposite to $\bar a$ at every $u$; by (4.7) its wide form $-\bar a$ is, at leading order, the same mean of the drift accelerations of the free parts at $p=\pi/2$ instead of at rest. A tight pair does not fall at this order: its inertia grows like $u$, its weight does not.
4. Ratios for $\langle X\rangle=0$ (item 3b), where $a^{(i)}=-\tau_i^2g_i$ and $g_i\neq0$:

$$
\frac{a_{\rm pair}}{a^{(i)}}=-\frac{\varsigma}{\sqrt{1+u^2}}\cdot\frac{g_1M_1+g_2M_2}{g_i\,\tau_i^2\,(\mathfrak m_1+\mathfrak m_2)}\,\bigl(1+O(\epsilon^2)\bigr),\qquad
\tau_{\rm p}^2:=\frac{M_1+M_2}{M_1/\tau_1^2+M_2/\tau_2^2}.
\tag{13.15}
$$

   - Common gradient, $\tau_1=\tau_2$ (the free parts fall alike): both ratios are $-\varsigma/\sqrt{1+u^2}$. For $w_{11}<0$ this is $<1$ at every fixed $u$. The pair falls like its parts neither exactly nor to any order in $\epsilon$ (the difference is of order $\epsilon^0$), but only in the ordered wide-pair limit, with the leading difference $-\tfrac12u^2$. For $w_{11}>0$: not at all (ratio $\to-1$).
   - Common gradient, $\tau_1\neq\tau_2$ (the free parts differ, $a^{(1)}/a^{(2)}=\tau_1^2/\tau_2^2$): the ratios are $-\varsigma(1+u^2)^{-1/2}\tau_{\rm p}^2/\tau_i^2$, with $\tau_{\rm p}^2$ strictly between $\tau_1^2$ and $\tau_2^2$. The pair falls like its parts at no $u$ and in no limit. In the wide limit ($w_{11}<0$) the ratio differs from $1$ by $M_j(\tau_j^2-\tau_i^2)/(M_i\tau_j^2+M_j\tau_i^2)$, $j\neq i$. Exceptional: for the part with the smaller $\tau_i$ and $w_{11}<0$ the ratio is $1$ at the single value $u_*=(\tau_{\rm p}^4/\tau_i^4-1)^{1/2}$.
   - $g_1\neq g_2$: $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\bar a$. If $\tau_1^2g_1=\tau_2^2g_2$ the free parts fall alike and the first case applies. Otherwise not at all: in the wide limit ($w_{11}<0$) $a_{\rm pair}-a^{(1)}\to\mathfrak m_2(a^{(2)}-a^{(1)})/(\mathfrak m_1+\mathfrak m_2)$. Exceptional: $g_1M_1+g_2M_2=0$ (no fall at this order).
5. Clock levels (item 3c). With $u=4\lvert w\rvert\mathfrak m_1\mathfrak m_2/(\mathfrak m_1+\mathfrak m_2)$, (13.13) reads

$$
a_{\rm pair}=\varsigma\,\frac{g_1M_1+g_2M_2+2\kappa\langle X\rangle}{\sqrt{(\mathfrak m_1+\mathfrak m_2)^2+16\,w^2\,\mathfrak m_1^2\mathfrak m_2^2}}\,\bigl(1+O(\epsilon^2)\bigr),\qquad \mathfrak m_i=\frac{M^{(i)}_{k_i}}{\tau_i^2}.
\tag{13.16}
$$

   - Claim: the leading coefficient is the same for all $N_1N_2$ pairs of levels iff $g_1=g_2=0$ and $\kappa\langle X\rangle=0$. Proof: call the numerator $\mathsf N$ and the root $h$. If $\mathsf N=\varsigma c\,h$ on four pairs $\{M_1,M_1'\}\times\{M_2,M_2'\}$ ($N_i\ge2$), take the mixed second difference. It is $0$ for $\mathsf N$ and strictly positive for $h$, because $4h^3\,\partial_{\mathfrak m_1}\partial_{\mathfrak m_2}h=4q\bigl[(\mathfrak m_1+\mathfrak m_2)^2+(1+q)\mathfrak m_1\mathfrak m_2\bigr]>0$ with $q:=16w^2\mathfrak m_1\mathfrak m_2$. So $c=0$, $\mathsf N=0$ on the four pairs, and all three gradients vanish.
   - Hence: for $\langle X\rangle=0$ iff $g_1=g_2=0$ (no fall); for $\langle X\rangle\neq0$ never.
   - Leading dependence: $\partial\ln a_{\rm pair}/\partial\ln M_i=g_iM_i/\mathsf N-\mathfrak m_i/(\mathfrak m_1+\mathfrak m_2)-\frac{u^2}{1+u^2}\frac{a_i}{a}$. For the common gradient with $\tau_1=\tau_2=\tau$ and $\langle X\rangle=0$ only the last term remains: $a_{\rm pair}=\varsigma g_{\rm c}\tau^2/\sqrt{1+u^2}$ with $u=\frac{4\lvert w\rvert}{\tau^2}\frac{M_1M_2}{M_1+M_2}$.
   - In the ordered wide-pair limit the leading coefficient $\varsigma\mathsf N/(\mathfrak m_1+\mathfrak m_2)$ is the same for all levels iff $\langle X\rangle=0$ and $\tau_1^2g_1=\tau_2^2g_2$ (two affine functions of $M_1,M_2$ that agree on the four pairs).
   - Comparison: a free part falls alike on all levels iff $\langle X\rangle=0$ (4.16). The pair does not, at any fixed $u$, because the binding $w$ does not scale with the levels while $a$ does.
6. $\langle X\rangle\neq0$ (item 3d). It adds $-2\kappa\langle X\rangle E_b''(P_0)$ to the drift acceleration, exactly and on the whole class of item 2 (13.8). At $P_0=0$ this is $-2\kappa\langle X\rangle/\mathfrak m_b$, in $\mathsf R$ equal to $\frac{\varsigma}{\sqrt{1+u^2}}\frac{2\kappa\langle X\rangle}{\mathfrak m_1+\mathfrak m_2}$. For a free part it adds $-\kappa\langle X\rangle/\mathfrak m_i$ (4.16). In both cases the plain gradient enters as the sum of the plain gradients over the curvature inertia, without a factor $\langle\sigma_3\rangle$, and it makes the fall depend on the levels. For the pair it is in addition reduced by $(1+u^2)^{-1/2}$ against the inertia-weighted mean, and reversed for $w_{11}>0$.

### Step 7. Fall and the clocks of the parts (item 4)

Class as in Step 6, with $\langle X\rangle=0$.

1. Rate (item 4a). At the label $x_0$ put $n_i:=1+g_ix_0>0$. The operator $(1+g_ix_0)G_i$ is the rest and clock operator of Model D with $m_i\to n_im_i$ and $\lambda_{0,i}\to\lambda_{0,i}/n_i$, hence $M^{(i)}_k\to n_iM^{(i)}_k$ and $\bar M_i\to n_i\bar M_i$. The zeroth order in the gradients is the pair without gradients and with these data. It is invariant under the joint shift, so (11.19) applies to a packet placed at $x_0$: the reading of the clock label of the part $i$ is $P_{N_i}(a;\vartheta_i)+\mathcal R_i$ with the advance $\vartheta_i=\Gamma_i(x_0)\,n_i\lambda/\lambda_{0,i}$. The rate is $\mathrm d\vartheta_i/\mathrm d\lambda$. With (11.20):

$$
\begin{aligned}
r_i(x_0)&=\frac{1+g_ix_0}{\lambda_{0,i}}\;\Gamma_i(x_0),\qquad \Gamma_i(x_0)=1-\frac{\tau_i^2}{4\,n_i^2\bar M_i^2}\,\Lambda(x_0)+O(\epsilon^4),\\
\Lambda(x_0)&=1+\frac{\varsigma}{\sqrt{1+u(x_0)^2}},\qquad u(x_0)=\frac{\lvert w\rvert}{a_1/n_1+a_2/n_2}
\end{aligned}
\tag{13.17}
$$

($a_i$ with $\bar M_i$ for the part that is read and $M_j$ for the other part).
   - Conditions, from (11.19): the clock of the part $i$ starts in $\lvert0\rangle$; the other part is on its level; the positions and branches are in a packet of the bound level of the data at $x_0$ with mean total label $0$ and spread $\delta P$; leading order in $\kappa$.
   - Range of $\lambda$: $\mathcal R_i$ is small for $\lambda\ll\lambda_{0,i}/\delta_i$ (in $\mathsf R$ with $\delta_i\to D_i/M_i$) and $\delta P^2(1+\lambda/\lambda_{0,i})\ll1$.
   - As a statement on the pair in the field, (13.17) is the zeroth order in $(\lvert g_1\rvert+\lvert g_2\rvert)\,s$, where $s\gtrsim1/(2\delta P)$ is the width of the packet in labels. The first-order corrections to the reading are not derived here.
2. Relative change per label, to first order in the gradients ($\varrho_i=\partial_{x_0}\ln r_i$ at $x_0=0$):

$$
\varrho_i=g_i+\frac{\tau_i^2}{4\bar M_i^2}\Bigl[2\Lambda\,g_i+\frac{\varsigma\,u^2}{(1+u^2)^{3/2}}\,\frac{g_1a_1+g_2a_2}{a}\Bigr]+O(\epsilon^4)\,(\lvert g_1\rvert+\lvert g_2\rvert) .
\tag{13.18}
$$

At leading order $\varrho_i=g_i$, as for the free part. The term of order $\epsilon^2$ follows from (13.17) by differentiation in the data; it assumes that the remainder of (11.20) stays $O(\epsilon^4)$ under one derivative in $M_1,M_2$, and it contains the gradient of the other part.
3. The number $\mathsf w_i^2$ (item 4b), for $g_i\neq0$ (then $\varrho_i\neq0$ for small $\epsilon$):

$$
\mathsf w_i^2:=-\frac{a_{\rm pair}}{\varrho_i}=-\frac{\varsigma}{\sqrt{1+u^2}}\cdot\frac{g_1M_1+g_2M_2}{g_i\,(\mathfrak m_1+\mathfrak m_2)}\,\bigl(1+O(\epsilon^2)\bigr)\qquad\Bigl[\,=-\frac{\varsigma\,\tau_{\rm p}^2}{\sqrt{1+u^2}}\,\bigl(1+O(\epsilon^2)\bigr)\ \text{for } g_1=g_2\Bigr].
\tag{13.19}
$$

   - Equal to $\tau_i^2$: no. For the common gradient $\mathsf w_i^2/\tau_i^2=-\varsigma(1+u^2)^{-1/2}\tau_{\rm p}^2/\tau_i^2$, which is $\neq1$ except at $u_*$ of Step 6 for one part; for $w_{11}>0$ it is negative. For $g_1\neq g_2$ equality holds only for one special value of $g_1/g_2$ at given $u$.
   - The same for $i=1,2$: $\mathsf w_1^2/\mathsf w_2^2=\varrho_2/\varrho_1$ (exact), which is $g_2/g_1$ at leading order. So: the same at leading order iff $g_1=g_2$; and then, by (13.18), not at relative order $\epsilon^2$ unless $\tau_1/\bar M_1=\tau_2/\bar M_2$.
   - Data of the pair: yes. It contains the binding ($\varsigma$, $u$) and the clock levels (through $u$, and through $\tau_{\rm p}$ if $\tau_1\neq\tau_2$).
   - Exceptional: for $g_i=0\neq g_j$, $\varrho_i=O(\epsilon^2)g_j$; (13.19) does not apply, and $\mathsf w_i^2$ follows from (13.13) and (13.18).
4. Free parts alike with one speed (item 4c): $g_1=g_2=g_{\rm c}$ and $\tau_1=\tau_2=\tau$, so that $a^{(i)}=-\tau^2g_{\rm c}$ is (4.24) with $v_{\max}=\tau d_0$. Both parts of the pair have the drift acceleration $a_{\rm pair}$, and $\bar x=d_0\,\bar m\,\hat e$ for both (17.15). Hence

$$
\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\Big\vert_{\rm drift}=-\,\frac{\mathsf w^2}{\tau^2}\;\frac{v_{\max}^2}{d_0}\;\varrho_i\;\hat e,\qquad \frac{\mathsf w^2}{\tau^2}=-\frac{\varsigma}{\sqrt{1+u^2}}\,\bigl(1+O(\epsilon^2)\bigr).
\tag{13.20}
$$

The pair obeys a relation of the form (4.24) in which $v_{\max}^2$ is replaced by $-\varsigma\,v_{\max}^2/\sqrt{1+u^2}$. With the speed $\tau d_0$ itself it holds neither exactly nor to any order in $\epsilon$ at fixed $u$. For $w_{11}<0$ it holds only in the ordered wide-pair limit, with the leading deviation $\mathsf w^2/\tau^2-1=-\tfrac12u^2+O(u^4)$; for $w_{11}>0$ not at all ($\mathsf w^2/\tau^2\to-1$).

## Result

Every statement is at first order in $(g_1,g_2,\kappa\langle X\rangle)$ at fixed $\lambda$, followed by the long chain, with the records at leading order in $\kappa$. "Derived" means derived here from the quoted inputs and standard facts.

- **Item 1** (class: the contract $C$ of the question and the field start $\varphi$; exact; derived).
  - The two bodies and the medium evolve by $C_{\rm red}$ (13.1). It is the contract of an admissible description of $b_1,b_2,c$: yes. A binding with a factor of the field would need a term on three objects, which (1.10) does not contain.
  - $C_{\rm bod}$ is (13.2), fixed by $C,\chi,\varphi$ up to an additive constant; its gradient terms are (13.3).
  - Fixed by $C_{\rm bod}$ and well defined: $g_1$ yes; $g_2$ yes; $\kappa\langle X\rangle$ yes; the operator $\mathcal K$ at the centre label yes; the operator $\mathcal D_\eta$ at $\hat r$ yes, for given $\eta$. Only $\mathcal D_\eta$ (with the centre label itself) depends on $\eta$, unless all gradients vanish.
- **Item 2** (class: every $W\in\mathcal W$, all data, every bound level, every $g\in\mathcal G(\mathcal P_0)$, all gradients; exact relations at first order; derived).
  - (a) $v^{(1)}_i=\lambda\langle\alpha\rangle_g+\rho_i(\lambda)$ (13.7), (13.8), with the same coefficient for both parts, and $\lvert\rho_i\rvert\le\tau_i\mathsf B_g$ for all $\lambda$ (13.9).
  - (b) $(\bar m_1-\bar m_2)^{(1)}=\lambda\langle\beta\rangle_g+\sigma(\lambda)$ with bounded $\sigma$ (13.10). There is never a term $\propto\lambda^2$. It stays bounded if $W(-r)^{*}=W(r)$ (all diagonal on-site bindings); it is zero under the sufficient condition of Step 4.3; otherwise it grows linearly in general (numerical control only). The answer is the same for the common gradient and for $g_1\neq g_2$, except that the vanishing case needs $g_1=g_2$.
  - (c) $a^{\rm drift}(P_0)=g_1\alpha_1+g_2\alpha_2+\kappa\langle X\rangle\alpha_X$ at $P_0$ (13.11): window average first, then $\delta P\to0$ on $\mathcal S(P_0,I)$. In the other order the limit does not exist on the class in general. $E_b''(P_0)$ is not the only property of the level: $\partial E_b/\partial M_i$, $E_b'$ and $\partial E_b'/\partial M_i$ enter the coefficients of $g_i$.
  - (d) The coefficient of the common gradient contains only the rest terms $\mathcal M_P$ of (11.6), not its motion and binding terms (13.12).
- **Item 3** (class: diagonal on-site binding, $\hat w_{11}\neq0$, level (11.12), $P_0=0$; leading order in $\mathsf R$ at fixed $u$, then limits in $u$; derived from (11.14), (11.16)). Closed form (13.13), limits (13.14), ratios (13.15), levels (13.16). The pair falls like its parts only for $w_{11}<0$, for free parts that fall alike, and in the ordered wide-pair limit. Its fall is the same for all pairs of levels only if all gradients vanish.
- **Item 4** (same class, $\langle X\rangle=0$; derived from (11.19), (11.20)). Rate (13.17), relative change (13.18), number $\mathsf w_i^2$ (13.19), relation (13.20). The term of order $\epsilon^2$ in (13.18) rests on an assumption on the remainder of (11.20); (13.19) and (13.20) do not use it.
- **Item 5.**

| | Free part $i$ (upper branch, at rest) | Bound pair (level (11.12), $P_0=0$, leading order in $\mathsf R$) | Pair agrees with its parts |
|---|---|---|---|
| Drift acceleration from rest | $a^{(i)}=-(g_iM_i+\kappa\langle X\rangle)/\mathfrak m_i$; (4.16), (8.11) | $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\,\bar a$; (13.11), (13.13) | at no fixed $u$ with a non-zero fall; in the ordered limit $u\to0$ iff $a^{(1)}=a^{(2)}$ and $w_{11}<0$; (13.14), (13.15) |
| Independent of the clock levels | iff $\langle X\rangle=0$; (4.16) | iff $g_1=g_2=0$ and $\langle X\rangle=0$; (13.16) | at no fixed $u$ with a non-zero fall; in the ordered limit $u\to0$ iff $\langle X\rangle=0$ and $\tau_1^2g_1=\tau_2^2g_2$; (13.16) |
| Number that ties it to the relative change of a clock rate ($\langle X\rangle=0$) | $\tau_i^2$, from $a^{(i)}=-\tau_i^2g_i$; (4.24) | $\mathsf w_i^2=-\varsigma(1+u^2)^{-1/2}(g_1M_1+g_2M_2)/(g_i(\mathfrak m_1+\mathfrak m_2))$; (13.18), (13.19) | for $g_1=g_2$, $\tau_1=\tau_2$: $\mathsf w^2/\tau^2=-\varsigma(1+u^2)^{-1/2}$, equal to $1$ only for $w_{11}<0$ in the ordered limit $u\to0$; (13.20) |

## Consistency checks

1. **One body, and no binding.** (13.8) applied to one free body on a level ($E=\sigma\omega$, $\langle\sigma_3\rangle=\partial E/\partial M=\sigma M/\omega$, $\mathcal K=G\sigma_3-\mathcal F$) gives $\alpha=-GM(\omega\omega')'/\omega^2+\sigma\mathcal F\omega''=-GM\tau^2\cos2p/\omega^2+\sigma\mathcal F\tau^2(M^2\cos2p-\tau^2\sin^4p)/\omega^3$. This is (4.7) with $g_\zeta=0$, for all $p$ and both branches. In the pair, the ordered limit of no binding ($u\to0$, $w_{11}<0$) of (13.13) is $\bar a$, equal to $a^{(1)}=a^{(2)}$ when the free parts fall alike. Passed.
2. **Common gradient against (11.6).** With (11.14) at $P=0$: $\mathcal M_0=M_1+M_2-a\Lambda$ from $\langle\sigma^{(i)}_3\rangle_0$, and $E_b-\langle A_1\rangle_0-\langle A_2\rangle_0-\langle U\rangle_0=M_1+M_2+a+\varsigma\sqrt{w^2+a^2}-2a\Lambda-\varsigma w^2/\sqrt{w^2+a^2}=M_1+M_2-a\Lambda$. Passed, in agreement with (13.12).
3. **Dimensions.** Labels and $P$ are pure numbers and costs are $1/\lambda$. $g_i$ is per label, $\kappa\langle X\rangle$ a cost per label, $E_b''$ and $1/\mathfrak m$ are costs. (13.8), (13.13): cost$^2$, i.e. labels$/\lambda^2$. (13.10): $\beta$ is a cost, i.e. labels$/\lambda$. (13.17): $1/\lambda$. (13.19): $\mathsf w_i^2$ is a cost$^2$, like $\tau_i^2$. Passed.

## Open issues

- The second order in the gradients, and the range of $\lambda$ over which the first order describes the motion, are not estimated (out of scope); (13.9) and (13.10) bound first-order terms only.
- The limit $\delta P\to0$ at a fixed window is only shown not to exist on the class in general; the quantity it defines on a subclass is not characterized.
- A binding of $\mathcal W$ with $\beta\not\equiv0$ is exhibited numerically (set S5), not analytically.
- The relative remainders $O(\epsilon^2)$ of (13.13)–(13.20) are not determined (numerically up to 4 % at $\tau_i/M_i=0.2$) and are not uniform in $u$. The $\epsilon^2$ term of (13.18) assumes a differentiable remainder of (11.20).
- The first-order corrections to the clock reading of a pair in the field (the gradient across the packet) are not derived.
- The results hold for a binding and for motion terms without a factor of the field, the class of the question. Whether another admissible coupling makes the pair fall like its parts is outside this package.

## Code

- **File:** `code/v1/pair_fall_check.py`, role: check. Outputs: `code/v1/output/summary.txt` (all numbers below), `code/v1/output/results.json`, `code/v1/output/first_order_motion.png`.
- **What it does.**
  - It builds the fibre contract (11.1) on $\lvert r\rvert\le48$, finds the bound level attached to $\mathcal B_{++}$ on a grid of 161 values of $P$, and computes $\alpha$ (13.8) and $\beta$ (13.10) from it ($E_b''$ by (11.9); the reduced resolvent by full diagonalization).
  - It builds $\Phi_g$ with $g\propto\cos^4\bigl(\pi(P-P_0)/(2P_c)\bigr)$ on the chain $\lvert x_i\rvert\le60$ (121 sites, dimension 58564), evolves it by $C_{\rm bod}$ (13.2) with small gradients of both signs (Chebyshev expansion), and takes the symmetric difference.
  - The slope of a first-order quantity is its mean rate over $[0,\lambda_{\max}]$ with the weight $\propto\sin^4(\pi\lambda/\lambda_{\max})$. The sampling step is chosen so that no multiple of $2\pi/\text{step}$ lies near a frequency $\lvert E_n(P)-E_b(P)\rvert$ of the rest.
- **Sets.** Each with the common gradient, with $g_1$ only ($g_2=0$), and with a plain gradient in the place of $\kappa\langle X\rangle$.
  - S1: $\tau_i=0.2$, $M_i=1$, $w_{11}=-0.02$ ($u=1.00$). S2: $\tau=(0.20,0.18)$, $M=(1,1.25)$, $w_{11}=-0.02$ ($u=1.21$). S3: $\tau=(0.20,0.17)$, $M=(1,0.9)$, $w_{11}=+0.022$ ($u=1.22$). S4: $\tau=(1,0.8)$, $M=(1,1.2)$, $w_{11}=-0.9$. In all four the other $w_{s_1s_2}$ are non-zero, $P_0=0$ and $P_c=0.5$ ($\delta P=0.106$).
  - S5 (control): the data of S4 with a binding of range 1 that is neither diagonal nor symmetric, $P_0=0.2$, $P_c=0.4$ ($\delta P=0.085$).
  - Range: $\lambda\le3000$ (S1–S3), $200$ (S4), $400$ (S5). Gradients: $5\cdot10^{-7}$ (S1–S3), $2\cdot10^{-6}$ (S4, S5).
- **Item 2a.** The slopes of $v^{(1)}_1$ and $v^{(1)}_2$ agree with $\langle\alpha\rangle_g$ of (13.7), in all three configurations, to a relative $2\cdot10^{-5}$ (S1), $1.3\cdot10^{-5}$ (S2), $6\cdot10^{-6}$ (S3), $4\cdot10^{-7}$ (S4), $3\cdot10^{-5}$ (S5). The rest stays bounded, at most 3 % of the linear part at $\lambda_{\max}$.
- **Item 2b.** For S1–S4 the first-order separation stays bounded and relaxes to a constant (slope times $\lambda_{\max}$ below 0.5 % of its maximum). For S1 with the common and the plain gradient it is zero within the noise of the difference ($3\cdot10^{-8}$ against $270$ for $g_1$ only), as in Step 4.3. For S5 it grows linearly: slopes $-0.08662$, $-0.02819$, $-0.07457$ against $\langle\beta\rangle_g=-0.08651$, $-0.02813$, $-0.07444$; with $\lambda\le600$ the second slope is $-0.028139$.
- **Item 3.** The coefficients of (13.13) deviate from those of the numerical level at $P_0=0$ by $-5\cdot10^{-4}$ to $+2.5\cdot10^{-3}$ (S1), up to $2.9\cdot10^{-3}$ (S2), and $-2.2\cdot10^{-2}$ to $-3.8\cdot10^{-2}$ (S3), against $(\tau_i/M_i)^2\le0.04$. With $\tau\to s\tau$, $w\to s^2w$ and $s=1,\tfrac12,\tfrac14$ the deviations fall by factors $3.8$ to $4.5$ per step. The slopes of the evolution agree, to the same deviations, with (13.13) averaged over the packet through (11.12).
- **Convergence** (configuration $g_1$ only; relative change of the slopes).
  - Chain: $45$ against $60$: $3\cdot10^{-7}$ (S1), $5\cdot10^{-8}$ (S4); $75$ against $60$: $1.5\cdot10^{-6}$ (S5).
  - Gradient doubled: $2\cdot10^{-6}$. Other sampling step: below $10^{-6}$. Fibre range $48$ against $64$: $10^{-13}$ in $\alpha$.
  - Narrower packet ($P_c=0.35$): the same agreement with its own $\langle\alpha\rangle_g$.
  - Range of $\lambda$: on half of the range the slopes deviate by up to $1.2\cdot10^{-2}$; the full range reduces this to the values of item 2a.
- **Run.** Once, in the foreground: 305 s. No package was installed.

## Methods used

- Reduction of a contract on an eigenstate of the field operators; partial expectation
- Fibre decomposition under the joint shift of two bodies
- First-order perturbation theory in a term of the contract (Duhamel formula)
- Perturbation theory of an isolated simple eigenvalue; reduced resolvent
- Commutator identities for velocity operators; Hellmann–Feynman relations
- Antiunitary and exchange symmetries of the fibre contract
- Window averaging of bounded oscillating terms; packets with a sharp total label
- Leading-order expansion in a scaling regime with ordered limits
- Mixed second differences (strict supermodularity) for a uniqueness argument
- Numerical evolution by a Chebyshev expansion with a symmetric difference in the gradients

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