Version 7 · current

# Inertia of a pair bound through a field of oscillator cells

- **Subproject:** 08-ilang-propagation
- **Package:** 07-bond-inertia
- **Version:** v7
- **Mode:** external regeneration
- **Date:** 2026-10-25

## Changes from previous version

- Step 6 (vi), Step 7 (paragraph after (7.9)), Step 9 ((7.13)), Result (7.9), (7.13): the inversion $I_{\rm pair}=I+D+D^2/I+\dots$ needs $|D/I|\ll1$; its next term has the relative order $|D/I|=O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|)+2\langle k^2\rangle$, the lattice part giving $2\langle k^2\rangle=O(a_b^{-2})$, not $O(|\Delta E|/\bar M)$ (verification r2, I1).
- Step 9 ((7.13) and *Variation of the field over the pair*), Result (7.13): the left side of (7.13) is now defined with every weight referred to the members' inertia-weighted field gradient, so that the members' finite-extent term cancels exactly; the unsuppressed members' term that a reference at the centre of inertia would give is stated (I2).
- Step 4 ((7.6) and the paragraph after it), Result (7.5), (7.6): the $V^2$ expansion above the pinning length needs $V^2|r|/(c_0^2\ell)\ll1$; the unexpanded continuum one-quantum kernel is given for larger separations (I3).
- Step 9 ((7.14)), Result (7.14), Open issues: the linear grip at $r=\sqrt2$ is the excluded cancellation of the leading terms, not a member of the fall criterion; (7.13) is evaluated there at the first nonzero order of its computed terms (I4).
- Code: both scripts unchanged from v6, rerun into `code/v7/output/`; every cited number is from that run.

## Response to verification

- I1: Accepted and fixed in Step 7 (paragraph after (7.9)), Step 6 (vi), Step 9 and Result (7.9), (7.13). The inversion condition is now $|D/I|\ll1$, and $D/I$ is split into the part $O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|)$ of the first three parts of (7.9) and the lattice part $4m_{\rm red}\langle T\rangle=2\langle k^2\rangle$ ($=\frac{2\alpha}3a_b^{-2}$ deep, $2a_b^{-2}$ shallow); levels with small $|\sigma_1\bar M_1+\sigma_2\bar M_2|$ remain excluded by (vi). (7.13), whose exact form has $I+D$ in the denominator, carries the same relative correction.
- I2: Accepted and fixed in Step 9 and Result (7.13). $w_{\rm pair}$ is defined as the force on the pair over minus the members' inertia-weighted gradient $\overline{\nabla\phi}=\sum_iI_i\langle\nabla\phi(x_i)\rangle/I$; since $w_i=\kappa I_i$, the members' force is then exactly $\kappa I\,\overline{\nabla\phi}$, and only the bond carries the finite-extent factor, $1+O(\langle\xi^2\rangle/\ell^2)$. Referred to the field at the centre of inertia, the members add $\frac{m_{\rm red}\langle\xi^2\rangle}{2I\ell^2}+O(a_b^4/\ell^4)$ ($\langle\xi^2\rangle/(8\ell^2)$ for equal members, the verifier's example), without the factor $|\Delta E|/\bar M$; this term is not part of the contribution of the mutual term.
- I3: Accepted and fixed in Step 4 and Result (7.5), (7.6): the second line of (7.6), and (7.5) at $|r|\gg\ell$, need $V^2|r|/(c_0^2\ell)\ll1$; for larger separations the unexpanded one-quantum kernel (range $\ell\sqrt{1-V^2/c_0^2}$) is stated. The coefficient of $V^2$ used in Step 7 is unaffected.
- I4: Accepted and fixed in Step 9 and Result (7.14): $s=0$, $r=\sqrt2$ is excluded from (7.14), since there both the leading $w_b$ and the leading $D$ vanish. At the first nonzero order the members', form-factor and retardation parts of (7.13) cancel there, and the lattice part $-2\langle k^2\rangle$ remains beside corrections that are not computed; so in the deep regime the linear grip does not fall like its members at $r\neq\sqrt2$ (leading term $(r^2-2)\langle U\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)\neq0$), and at $r=\sqrt2$ the computed order does not decide.

## Setup and assumptions

**Description (M8, M9, A15).** Ring of $L$ oscillator cells, contract $H=\sum_x\tfrac12p_x^2+\tfrac{\kappa_h}2\sum_x(q_{x+1}-q_x)^2+\tfrac{m_h^2}2\sum_xq_x^2$ plus the terms of the bodies, $\Phi_x=q_x$; $c_0=\sqrt{\kappa_h}$, $\ell=c_0/m_h$, $\ell\gg1$. Two Model D bodies of different types; the own contract of body $i$ is its motion term $A_i$ (strength $\tau$ for both, a condition, M7), its grip is $\sum_x\lvert x\rangle\langle x\rvert\otimes S_i\otimes F(q_x)$, $S_i=\sigma_3\otimes(m_i\mathbb 1+D_i)$, with the common grip $F(q)=F_0+F_1q+F_2q^2$, $F_0>0$, $F_1\neq0$ (a condition, M7). No pair term between the bodies. A body step is a field step ($k=1$ in (4.17)), so by (5.11) and (4.17) the maximal speed of each body is $\tau$ cells per unit $\lambda$ and equals the speed of the field iff $r:=\tau/c_0=1$. $s:=4F_0F_2/F_1^2$ labels the grip: $s=1$ is the square grip of (2.18), $s=0$ the linear grip.

**Quoted inputs.** (2.11) of 08-ilang-propagation/02-grip-weight: $U(\mathbf r)=-\mu_1\mu_2\bigl[F_1^{2}G(\mathbf r)+4F_2^{2}J_2(\mathbf r)\bigr]+O(\mu^3)$ in (G2), with $G=(K^{-1})_{x+r,x}$, $J_2=\int_0^\infty\Gamma_r(t)^2dt$, $\Gamma_r(t)=\langle q_{x+r}e^{-(H-E_f)t}q_x\rangle$; $U$ has the sign of $-\mu_1\mu_2$. (2.18) there: the static bond of held bodies is weighed like the rest cost, $\rho_{{\rm bond},i}=m_{\mathbf r}$, for every level and both branches iff the grip is square, at leading order in the grip. (4.17) of 04-formation, (5.11) of 04-ilang-time/05-moving-clock, and (6.7) of 04-ilang-time/06-inertia-internal-energy: $M_k=m+d_k$, gap $2M_k$, $\omega_k''(0)=\tau^2/M_k$.

**Notation.** Field modes $q\in\frac{2\pi}L\mathbb Z_L$, $\omega_q^2=m_h^2+4\kappa_h\sin^2\frac q2$ (the eigenvalues of $K$), $w_0=\langle q_x^2\rangle=\frac1L\sum_q\frac1{2\omega_q}$, $\bar a=\langle F(q_x)\rangle=F_0+F_2w_0$. Clock level $k_i$ and branch $\sigma_i$ of body $i$; $\mu_i=\sigma_iM_{k_i}$, $\bar M_i=\bar aM_{k_i}$, $\lambda_{C,i}=\tau/\bar M_i$ (the length, in cells, below which the band of body $i$ is not quadratic).

**Assumptions and regime.**
- (A) $M\to\infty$ first: the eigenvalues of $C_M$ plus the body terms below any fixed bound converge to those of $H$ plus the body terms (strong resolvent convergence of the compressions, standard), and every statement is about this limit. Then $L\to\infty$ for the separated reference (with $L\gg\ell,a_b$ for the bound level), then the regime limits below.
- (B) Orders: second order in $\delta F=F_1q+F_2q^2$ at fixed $F_0$ for the mutual term (Step 2), and leading order in $|\Delta E|/\bar M_i$ for $D$ (Steps 7–8). The remainder of the elimination is $O(\delta F^3)$: three vertices give terms $\propto F_1^2F_2$ and $F_2^3$ (Gaussian parity removes only odd total powers of $q$); for held bodies, for example, the exact shift gives $-\frac12j^{\rm T}(K+2\Lambda)^{-1}j$ with $j\propto F_1$ and $\Lambda$ diagonal, $\propto F_2$, whose cubic part is $j^{\rm T}K^{-1}\Lambda K^{-1}j$. These terms are smaller than the second-order ones by the factor $u=2\max_iM_{k_i}|F_2|G(0)$, assumed $\ll1$. For $F_2=0$ the odd orders vanish, and the fourth-order terms vanish for held bodies (a linear term on a quadratic contract is removed exactly by a shift of $q$), so the remainder is $O(\delta F^4)$ and enters only through the motion of the bodies, smaller by $r^2|U(0)|/\bar M_i$ (recoil of the virtual quanta), assumed $\ll1$. The single-body self-energies $\Sigma_i(p)$ at second order are one-body terms, the same in the pair as for the body alone; they cancel from $\Delta E$, and enter $D$ only at relative order $\delta F^2$ (Step 7); under (E) none of their denominators vanishes.
- (C) Slow bodies (continuum regime): the bound level has size $a_b$ with $a_b\gg1$ and $a_b\gg\lambda_{C,i}$; corrections are of relative order $a_b^{-2}$ and $(\lambda_{C,i}/a_b)^2$.
- (D) No Cherenkov denominators: $r\min(1,\lambda_{C,i}/2)\le1$ for $i=1,2$ (Step 2; equality is allowed since $\omega_q>2c_0|\sin\frac q2|$ strictly), so that the elimination holds on the whole band.
- (E) No decay at the order computed (note of the question): every member in the branch $+$ satisfies the no-decay condition of (7.15) (Step 6), the range (N), with margins large against $|\Delta E|$, $\langle T\rangle$ and $\tau^2/(\bar M_ia_b^2)$. By Step 6, a bound level is an isolated eigenvalue of the full contract iff it lies in the range (I) of (7.15). (I) is part of (N$_l$) and needs $r<1/(2\ell)$; there the coefficients of items 2 and 3 are the ordinary second-order coefficients of the level. In the rest of (N) (all of (N$_h$), and (N$_l$) outside (I)) the levels are not isolated, and every coefficient of items 2 and 3 is a *formal coefficient*: the second-order coefficient of a level whose width is of fourth or higher order in $\delta F$.

## Derivation

### Step 1. Bodies at first order

The first-order part of the grip is $\langle0\vert\delta F(q_x)\vert0\rangle=F_2w_0$ in the lowest eigenvector $\vert0\rangle$ of the field; it is put into $H_0$, together with the motion terms and $H$. The clock label is conserved by every term, so $S_i\to M_{k_i}\sigma_3$. In plane waves $\vert p\rangle=L^{-1/2}\sum_xe^{ipx}\vert x\rangle$ the body contract is $h_i(p)=\bar M_i\sigma_3-\tau\sin p\,\sigma_1$, with bands $\varepsilon_i^{\sigma}(p)=\sigma E_i(p)$, $E_i=\sqrt{\bar M_i^2+\tau^2\sin^2p}$, eigenvectors $u_+=(\cos\frac\theta2,-\sin\frac\theta2)$, $u_-=(\sin\frac\theta2,\cos\frac\theta2)$, $\tan\theta_i(p)=\tau\sin p/\bar M_i$. Hence

$$
\varepsilon_i(p)=\sigma_i\bar M_i+\frac{p^2}{2I_i}-\Bigl(\frac1{8I_i^3\tau^2}+\frac1{6I_i}\Bigr)p^4+O(p^6),\qquad I_i=\frac{\sigma_i\bar M_i}{\tau^2},\qquad \langle u_{\sigma}(p')\vert\sigma_3\vert u_\sigma(p)\rangle=\sigma\,c_i(p',p),\ \ c_i=\cos\tfrac{\theta_i(p)+\theta_i(p')}2 .
\tag{7.1}
$$

$I_i$ is the signed inertia; for $\sigma_i=+$ it is (6.7) with $M_k\to\bar aM_k$ at this order. The band has the same form near $p=\pi$ (the factor $\sin p$): every body has two minima of $E_i$, at $p=0$ and $p=\pi$. In (7.1) the $p^4$ term $-1/(8I_i^3\tau^2)$ is the relativistic part, $-1/(6I_i)$ the lattice part ($\sin^2p=p^2-p^4/3$). Near $p=0$ the velocity is $v_i=\varepsilon_i'(p)=p/I_i$ and $c_i=1-\bar v_i^2/(2\tau^2)+O(v^4)$, $\bar v_i=(v_i+v_i')/2$.

### Step 2. Elimination of the field: the mutual term

The modes $q_x=\sum_q(2L\omega_q)^{-1/2}(b_qe^{iqx}+b_q^\dagger e^{-iqx})$ diagonalize $H$ (standard). $V=\sum_i\sum_x\lvert x\rangle\langle x\rvert\otimes M_{k_i}\sigma_3\otimes(\delta F(q_x)-F_2w_0)$ has no component in the band space $\mathcal P$ (bodies in bands $\sigma_1,\sigma_2$, field in $\vert0\rangle$). The symmetric second-order effective contract on $\mathcal P$ (des Cloizeaux/Schrieffer–Wolff) is $\langle f\vert H_{\rm eff}\vert i\rangle=\langle f\vert H_0\vert i\rangle+\frac12\sum_{n\notin\mathcal P}\langle f\vert V\vert n\rangle\langle n\vert V\vert i\rangle\bigl[(E_i-E_n)^{-1}+(E_f-E_n)^{-1}\bigr]$. From $\vert0\rangle$, $F_1q_x$ creates one quantum and $F_2(q_x^2-w_0)$ two; the mixed products vanish (odd in $q$). Terms with both vertices on one body are the single-body self-energies; terms with one vertex on each body form the mutual term. For one vertex per body, body $i$ ends in its own band, so only same-band intermediate states enter. With the vertex amplitudes $\mu_iF_1c_i/\sqrt{2L\omega_q}$ (one quantum) and $\mu_iF_2c_i/(L\sqrt{\omega_a\omega_b})$ (two quanta, $a\neq b$; $\sqrt2\mu_iF_2c_i/(2L\omega_a)$ for $a=b$), and the two time orders, the mutual term is, for $p_1'=p_1-q$, $p_2'=p_2+q$, $\Delta_1=\varepsilon_1(p_1)-\varepsilon_1(p_1')$, $\Delta_2=\varepsilon_2(p_2')-\varepsilon_2(p_2)$:

$$
\langle p_1'p_2'\vert U\vert p_1p_2\rangle=-\frac{\mu_1\mu_2c_1c_2}{2L}\sum_{j=1,2}\Bigl[\frac{F_1^2}{\omega_q^2-\Delta_j^2}+\frac{F_2^2}{L}\sum_{a+b=q}\frac{\Omega_{ab}}{\omega_a\omega_b(\Omega_{ab}^2-\Delta_j^2)}\Bigr],\qquad \Omega_{ab}=\omega_a+\omega_b ,
\tag{7.2}
$$

with $c_1=c_1(p_1',p_1)$, $c_2=c_2(p_2',p_2)$. The effective joint contract in the sector $P=p_1+p_2$ is

$$
H_{\rm eff}(P)=\varepsilon_1(p_1)+\varepsilon_2(p_2)+U+O(\delta F^2\ \text{self-energies})+O(\delta F^3)\quad(O(\delta F^4)\ \text{for}\ F_2=0).
\tag{7.3}
$$

Code check (i) (`code/v7/check_mutual_term.py`, output `code/v7/output/mutual_term_check.json`): on a ring of 16 cells, (7.2) agrees with second-order perturbation theory in the total contract restricted to at most two quanta, built in the position basis with numerically obtained spinors and both branches as intermediate states, to a relative deviation of at most $3.9\times10^{-14}$ over 32 matrix elements (four descriptions: a generic (G2) grip, the square grip and mixed branches at $r=0.8$, and the generic grip at $r=1.3$).

*No Cherenkov denominators.* $\lvert E(p)-E(p')\rvert\le\tau\lvert\sin p-\sin p'\rvert\le2\tau\lvert\sin\frac q2\rvert$ and $\lvert E(p)-E(p')\rvert\le\frac{\tau^2}{2\bar M}\lvert\sin(p+p')\sin(p-p')\rvert\le\frac{\tau^2}{\bar M}\lvert\sin\frac q2\rvert$, while $\omega_q>2c_0\lvert\sin\frac q2\rvert$ and $\Omega_{ab}>2c_0\lvert\sin\frac{a+b}2\rvert$. Hence $\lvert\Delta_j\rvert<\omega_q,\Omega_{ab}$ on the whole band under (D); the elimination is then valid for every label.

### Step 3. Held limit and time representation

With $\frac1{\Omega-\Delta}+\frac1{\Omega+\Delta}=2\int_0^\infty e^{-\Omega t}\cosh(\Delta t)\,dt$ and $\frac1L\sum_qe^{iqr}e^{-\omega_qt}/(2\omega_q)=\Gamma_r(t)$, (7.2) becomes, with $r=x_1-x_2$ and the transfer $q$ acting as $-i\partial_r$,

$$
U=-c_1c_2\int_0^\infty dt\ \tfrac12\textstyle\sum_j\cosh(\Delta_jt)\,W(r,t),\qquad W(r,t)=2\mu_1\mu_2\bigl[F_1^2\Gamma_r(t)+2F_2^2\Gamma_r(t)^2\bigr].
\tag{7.4}
$$

For held bodies ($\tau=0$: $c_i=1$, $\Delta_j=0$) it reduces to $U(r)=-\int_0^\infty W\,dt=-\mu_1\mu_2[F_1^2G(r)+4F_2^2J_2(r)]$, since $\int_0^\infty\Gamma_r\,dt=G(r)/2$: this is (2.11) in (G2), here without an $O(\mu^3)$ remainder at second order in $\delta F$. $U(r)$ below always denotes this static bond; it has the sign of $-\mu_1\mu_2$ at every $r$, as stated with (2.11) ($\Gamma_r(t)>0$).

### Step 4. The bond in motion (item 1)

In the sector $P$ write $p_1=\frac{I_1}IP+k$, $p_2=\frac{I_2}IP-k$, $I=I_1+I_2$, $V=P/I$, $m_{\rm red}=I_1I_2/I$; then $\sum_ip_i^2/(2I_i)=P^2/(2I)+k^2/(2m_{\rm red})$ exactly, and $v_i=V+u_i$ with $u_1=k/I_1$, $u_2=-k/I_2$. Near the band minimum (at $0$, and identically at $\pi$), Step 1 gives $c_1c_2=1-(\bar v_1^2+\bar v_2^2)/(2\tau^2)$, and $\Delta_j=q\bar v_j$ up to relative $O(q^2,\lambda_C^2k^2)$ (midpoint rule), so $\cosh(\Delta_jt)=1+q^2\bar v_j^2t^2/2$. Keeping in (7.4) the terms of second order in $V$, with $q^2\to-\partial_r^2$:
$U_P=U(r)+V^2[-U(r)/\tau^2+R(r)]+(\text{odd in }k)+(\text{terms in }k\text{ only})$, $R(r)=\frac12\partial_r^2\int_0^\infty t^2W(r,t)\,dt$.

*Identity for $R$.* For $\ell\gg1$ and $L\to\infty$ the relevant transfers are $q\ll1$, and $\Gamma_r(t)=(2\pi)^{-2}\int dq\,d\nu\,e^{iqr+i\nu t}/(\nu^2+c_0^2q^2+m_h^2)$ depends on $(r,t)$ only through $t^2+r^2/c_0^2$ (rotation of the integration variables $(\nu,c_0q)$; corrections $O(q^2)$ from $\sin^2\frac q2=\frac{q^2}4-\frac{q^4}{48}$). So $W=w(t^2+\rho^2)$, $\rho=r/c_0$. With $\Psi(\rho)=\int_0^\infty t^2w\,dt$: $\Psi'(\rho)=\rho\int_0^\infty t\,\partial_tw\,dt=-\rho\int_0^\infty w\,dt=\rho U$ (integration by parts; $tw\to0$ at both ends). Hence $R=\Psi''/(2c_0^2)=(rU)'/(2c_0^2)$, for both the one- and the two-quantum part, and

$$
U_P(r)=U(r)+\frac{P^2}{I^2}\Bigl[-\frac{U(r)}{\tau^2}+\frac{(rU(r))'}{2c_0^2}\Bigr]+O\bigl(V^4,\ V\cdot u,\ u^2\bigr),
\tag{7.5}
$$

with relative corrections $O(\ell^{-2},a_b^{-2},\lambda_C^2/a_b^2)$ to the $V^2$ term. The coefficient of $V^2$ holds at every $r$; the truncation at $V^2$ needs in addition $V^2|r|/(c_0^2\ell)\ll1$ at $|r|\gg\ell$, where the remainder $O(V^4)$ contains $(V^2|r|/(c_0^2\ell))^2U$ (below). The first bracket term is the reduction of the grip of a moving body (form factor), the second the motion of the field between the bodies (retardation). For the one-quantum part, dropping $u$, the kernel is $1/(m_h^2+(c_0^2-V^2)q^2)$, i.e. the static $G$ with $c_0^2\to c_0^2-V^2$: with $U_G(r)=-\mu_1\mu_2F_1^2\ell e^{-|r|/\ell}/(2c_0^2)$ and $\ell_V:=\ell\sqrt{1-V^2/c_0^2}$, the unexpanded continuum kernel is $U_{G,P}(r)\simeq-c_1c_2\,\mu_1\mu_2F_1^2\,\ell_Ve^{-|r|/\ell_V}/\bigl(2(c_0^2-V^2)\bigr)$, $c_1c_2=1-V^2/\tau^2+O(V^4)$, for $V<c_0$ and every $|r|$. Its expansion to second order in $V$, below and above the pinning length:

$$
U_{G,P}(r)\simeq\begin{cases}U_G(0)\Bigl[1-V^2\bigl(\tfrac1{\tau^2}-\tfrac1{2c_0^2}\bigr)\Bigr]+\dfrac{\mu_1\mu_2F_1^2}{2c_0^2}\,|r|\Bigl[1-V^2\bigl(\tfrac1{\tau^2}-\tfrac1{c_0^2}\bigr)\Bigr], & |r|\ll\ell,\\[2mm] U_G(r)\Bigl[1-\dfrac{V^2}{\tau^2}+\dfrac{V^2}{2c_0^2}\bigl(1-\tfrac{|r|}\ell\bigr)\Bigr], & \ell\ll|r|\ll\dfrac{c_0^2\ell}{V^2} .\end{cases}
\tag{7.6}
$$

Below $\ell$ the constant and the slope of the bond change with $P$ by different factors; above $\ell$ the bond's range shrinks to $\ell_V$. The second line expands $e^{-|r|/\ell_V}=e^{-|r|/\ell}e^{-V^2|r|/(2c_0^2\ell)+O(V^4|r|/\ell)}$ and holds only for $V^2|r|/(c_0^2\ell)\ll1$, where the moving bond is weaker by the relative amount $V^2|r|/(2c_0^2\ell)$ (besides the $|r|$-independent factors); at larger $|r|$ the unexpanded kernel holds, and the moving bond is smaller than the static one by the factor $c_1c_2(1-V^2/c_0^2)^{-1/2}e^{-|r|(1/\ell_V-1/\ell)}$, which tends to zero.

### Step 5. Branches with a bound level

Near the minimum of the bands, $H_{\rm eff}(0)=\sigma_1\bar M_1+\sigma_2\bar M_2+k^2/(2m_{\rm red})+U(r)$ with $U$ of the sign of $-\sigma_1\sigma_2$ at every $r$ (Step 3). In one dimension a level splits off from the edge of the relative band iff the potential pulls toward that edge (standard), i.e. iff $\sigma_1\sigma_2\,m_{\rm red}>0$:

$$
\text{bound level near labels }0\iff\begin{cases}\sigma_1=\sigma_2=+: & \text{always; lowest level of the sector}\\ \sigma_1=-\sigma_2: & \text{iff the member in the branch }+\text{ has the larger }\bar M\ (I>0,\ m_{\rm red}<0);\ \text{highest level}\\ \sigma_1=\sigma_2=-: & \text{never}\end{cases}
\tag{7.7}
$$

For $(-,-)$ bound levels exist only near the labels $\pm\pi/2$, where the bands of the branch $-$ have their minima; there the continuum regime (C) does not hold, and they are out of scope (note of the question after the review of v2: members near their rest levels). The extreme level of (7.7) comes in two copies, the pair near labels $(0,0)$ and near $(\pi,\pi)$; the mutual term couples them only through transfers $q\approx\pi$, with $t\simeq\tilde U(\pi)|\phi(0)|^2$, $\tilde U(\pi)\simeq-\mu_1\mu_2F_1^2/(m_h^2+4c_0^2)$, $\phi$ the relative wave function. The extreme level is the even combination, unique for $t\neq0$; $|t|/|\Delta E|=O(1/(a_b\ell))$ in the deep and $O(\ell^{-2})$ in the shallow regime, so $\Delta E$ and $D$ below are those of one copy up to these orders.

### Step 6. Binding regime and isolation

With $L\to\infty$ the separated pair has $\sigma_1\bar M_1+\sigma_2\bar M_2$, so $\Delta E$ is the eigenvalue of the relative contract. Two regimes, for $\sigma_1=\sigma_2=+$ (for the mixed case replace $m_{\rm red}\to-|m_{\rm red}|$; $\Delta E>0$):

$$
\begin{aligned}
&\text{deep }(1,\lambda_C\ll a_b\ll\ell,\ \epsilon_J\ll1): &&U\simeq U(0)+\lambda|r|,\ \ \lambda=\frac{\mu_1\mu_2F_1^2}{2c_0^2},\ \ a_b=(2m_{\rm red}\lambda)^{-1/3},\\
& &&\Delta E=U(0)+\alpha\Bigl(\frac{\lambda^2}{2m_{\rm red}}\Bigr)^{1/3},\ \ \langle T\rangle=\frac\alpha3\Bigl(\frac{\lambda^2}{2m_{\rm red}}\Bigr)^{1/3},\ \ U(0)=-\mu_1\mu_2\bigl[F_1^2G(0)+4F_2^2J_2(0)\bigr],\ G(0)\simeq\frac{\ell}{2c_0^2};\\
&\text{shallow }(a_b\gg\ell\gg1): &&U\to-g\,\delta_{r0},\ \ g=\mu_1\mu_2\Bigl[\frac{F_1^2}{m_h^2}+\frac{F_2^2}L\sum_q\frac1{2\omega_q^3}\Bigr],\ \ a_b=\frac1{m_{\rm red}g},\ \ \Delta E=-\frac{m_{\rm red}g^2}2=-\langle T\rangle .
\end{aligned}
\tag{7.8}
$$

Here $\alpha=1.0188$ is the first zero of $-{\rm Ai}'(-x)$ (even ground state of $k^2/(2m)+\lambda|r|$), $\langle T\rangle=\langle k^2/(2m_{\rm red})\rangle$, and $g=-\sum_rU(r)$ is exact on the ring ($\sum_rG=1/m_h^2$, $\sum_r4J_2=\frac1L\sum_q\frac1{2\omega_q^3}$). Corrections: deep, relative $O(a_b/\ell)$ to the Airy part (the $r^2/\ell$ term of $G$) and $O(\epsilon_J)$ from the two-quantum part $U_J$ of the bond, whose slope ($\propto|r|\ln(\ell/|r|)$) is smaller than $\lambda$ by
$$\epsilon_J:=\frac{2F_2^2}{\pi F_1^2c_0}\ln\frac\ell{a_b},$$
and whose constant is $U_J(0)/U_G(0)=F_2^2/(2F_1^2c_0)<\epsilon_J$ (continuum values, $J_2(0)\simeq\ell/(16c_0^3)$ from $\int_0^\infty K_0^2=\pi^2/4$; $\ln(\ell/a_b)>\pi/4$); shallow, relative $O(\ell/a_b)$. The deep regime requires $\epsilon_J\ll1$: for grips of (G2) with $\epsilon_J$ not small ($F_2^2\ln(\ell/a_b)\gtrsim F_1^2c_0$) the two-quantum bond, not linear near $r=0$, sets the binding, and they are excluded from the deep regime (note (c) of the question). With $s$ fixed, $\epsilon_J=\frac{s^2F_1^2}{8\pi F_0^2c_0}\ln\frac\ell{a_b}$ is small for weak grips; $\epsilon_J=0$ for the linear grip. In the deep regime $\langle T\rangle/|\Delta E|=O(a_b/\ell)$, since $|U(0)|\simeq\lambda\ell$.

*Isolation and decay.* Let $\Omega_{\max}=\max_q\omega_q=\sqrt{m_h^2+4\kappa_h}$ ($\simeq2c_0$ for $\ell\gg1$; $\Omega_{\max}\ge2m_h$).
- (i) From the relative continuum: by $|\Delta E|$ (shallow) or by $|U(0)|$ (deep); from the next relative level by $\sim(\lambda^2/2m_{\rm red})^{1/3}$ (deep) and from the other valley copy by $2|t|$.
- (ii) From the field sectors with both members in their own branches: for $(+,+)$ they lie at least $m_h$ above the bound level; for mixed branches (body 2 in $-$) the states reached by one vertex lie above the level iff $\Delta E<\gamma:=\min_q[\omega_q-(E_2(q)-\bar M_2)]$, and $\gamma>0$ under (D) by the bounds of Step 2.
- (iii) From the sectors in which a member of the branch $+$ is in its branch $-$, at nonzero labels of both members. In the sector $P=0$ a state with both members in the branch $-$ and $n\ge1$ quanta has the energy $\sum_j\omega_{q_j}-E_1(p_1)-E_2(p_2)$, with $p_1+p_2+\sum_jq_j=0$. Three facts bound it from below. First, $\omega_q\ge m_h$, with equality only at $q=0$. Second, $E_i(p)\le\sqrt{\bar M_i^2+\tau^2}$, with equality at $p=\pm\pi/2$. Third, $\sum_j\omega_{q_j}\ge\omega_{\sum_jq_j}$, since $\omega$ is subadditive ($|\sin\frac{a+b}2|\le|\sin\frac a2|+|\sin\frac b2|$ and the triangle inequality). The first two bounds are reached at the same point, the members at $p_1=-p_2=\pi/2$ and one quantum at $q=0$, so the lowest such energy is exactly
$$A=m_h-\sqrt{\bar M_1^2+\tau^2}-\sqrt{\bar M_2^2+\tau^2}.$$
With one quantum the energies fill $[A,-\bar M_1-\bar M_2+\Omega_{\max}]$ (a continuous function on a connected set; the maximum has the members at $0$ and $\pi$ and the quantum at $\pi$). With $n$ quanta they fill an interval from at most $-\bar M_1-\bar M_2+nm_h$ to at least $-\bar M_1-\bar M_2+n\Omega_{\max}$, and consecutive intervals overlap since $\Omega_{\max}\ge2m_h$. So the continuous spectrum of the full contract in the sector $0$ contains $[A,\infty)$. A sector with one member flipped and quanta, or with both members in their own branches and quanta, lies at or above $A$ ($+E_i\ge-E_i$), so $A$ is the bottom of every sector with quanta. The sectors without quanta in other branches have the energies $\sigma_1'E_1(p)+\sigma_2'E_2(p)$ ($p_2=-p_1$). Since $E_1-E_2$ is monotone in $\sin^2p$, they lie at a distance at least $\min(2\min_i\bar M_i,|\sigma_1\bar M_1+\sigma_2\bar M_2|)-|\Delta E|$ from $E_b(0)$, which is positive under (B) and (vi). Hence, given (i), $E_b(0)$ is an isolated eigenvalue iff $E_b(0)<A$. The margin must be large against the second-order shifts of $A$: the self-energies, and the binding of a pair of members in the branch $-$ near $\pm\pi/2$, which lowers the threshold. For mixed branches $E_b(0)<A$ implies (ii).
- (iv) Decay at the order computed. A state with member $i$ in its branch $-$ is reached from the level by one vertex on member $i$ (rate of second order in $\delta F$), the other member being a spectator. From member $i$ at label $0$ (energy $\bar M_i$) one vertex reaches the energies $f_i(q)=\omega_q-E_i(q)$ (one quantum, from $F_1$) and $g_i(a,b)=\omega_a+\omega_b-E_i(a+b)$ (two quanta, from $F_2$; for $F_2=0$ two quanta need two vertices, rate $O(\delta F^4)$). Since $\omega_q$ increases on $[0,\pi]$ and $E_i(\pi)=\bar M_i$ is the minimum of $E_i$, $f_i$ fills $[\min f_i,\Omega_{\max}-\bar M_i]$ and $g_i$ fills $[\min g_i,2\Omega_{\max}-\bar M_i]$ (continuous functions on a connected set); $\min g_i\ge\min f_i$ because $\omega$ is subadditive ($|\sin\frac{a+b}2|\le|\sin\frac a2|+|\sin\frac b2|$ and the triangle inequality), and $f_i(0)=m_h-\bar M_i$. No resonance means $\bar M_i$ outside the range of $f_i$ and, for $F_2\neq0$, of $g_i$. This is (7.15): $\bar M_i<\min f_i$ implies $\bar M_i<f_i(0)$, i.e. $\bar M_i<m_h/2$, and $\bar M_i<\min g_i$; and $\Omega_{\max}/2<\bar M_i\le\Omega_{\max}$ lies in the range of $g_i$, since $g_i(0,0)=2m_h-\bar M_i<\bar M_i$.

$$
\begin{aligned}
&\text{no decay of member }i\ (\sigma_i=+)\text{ at second order}\iff\bar M_i>\tfrac12\Omega_{\max}\ (F_2=0),\ \ \bar M_i>\Omega_{\max}\ (F_2\neq0)\ \ (\text{N}_h),\ \ \text{or}\ \ \bar M_i<\min_q\bigl[\omega_q-E_i(q)\bigr]\ \ (\text{N}_l);\\
&E_b(0)\ \text{is an isolated eigenvalue}\iff\sigma_1\bar M_1+\sigma_2\bar M_2+\Delta E<m_h-\sqrt{\bar M_1^2+\tau^2}-\sqrt{\bar M_2^2+\tau^2}\ \ (\text{I}),
\end{aligned}
\tag{7.15}
$$

  The first line is for a member at label $0$, with margins large against $|\Delta E|$, $\langle T\rangle$ and $\tau^2/(\bar M_ia_b^2)$ (the shift of the member's energy in the level and the spread $O(1/a_b)$ of its label). The second holds with the margin of (iii).
- (v) Ranges.
  - (N$_h$): $\lambda_{C,i}<2\tau/\Omega_{\max}<r$ for $F_2=0$, and $<r/2$ for $F_2\neq0$; at $r\le1$ the members are heavy on the lattice scale.
  - (N$_l$): $\min f_i\le f_i(0)$ gives $\bar M_i<m_h/2$, i.e. $\lambda_{C,i}>2r\ell$, so (C) needs $a_b\gg2r\ell$: only the shallow regime unless $r\ll1$. On the lattice $\omega_q^2-E_i(q)^2=m_h^2-\bar M_i^2+4\sin^2\frac q2\,[c_0^2-\tau^2\cos^2\frac q2]$, with minimum $m_h^2-\bar M_i^2-(\tau^2-c_0^2)^2/\tau^2<0$ for $r>1$ with $(r^2-1)/r\ge1/\ell$: (N$_l$) is empty there.
  - (I) lies in (N$_l$): with $p_j=0$ in (iii), $A\le\min_q[\omega_q-E_i(q)]-\bar M_j$, so (I) gives $\bar M_i<\min_q[\omega_q-E_i(q)]-(1+\sigma_j)\bar M_j-\Delta E\le\min_q[\omega_q-E_i(q)]$. Here $\Delta E<0$ only for $(+,+)$, where $|\Delta E|<2\bar M_j$ by (B); for $F_2\neq0$, $\min g_i\ge\min f_i$. Since $\sigma_1\bar M_1+\sigma_2\bar M_2+\Delta E>0$ (by (B), and by $\bar M_1>\bar M_2$, $\Delta E>0$ for mixed branches), (I) needs $2\tau<m_h$, i.e. $r<1/(2\ell)$, and $\bar M_i<m_h/2$ for the members in the branch $+$; $\lambda_{C,i}$ is not restricted. (I) is not empty: for example $(+,+)$ with $\bar M_1=0.15\,m_h$, $\bar M_2=0.1\,m_h$, $\tau=0.2\,m_h$ and any $\ell\gg1$ gives $0.72\,m_h<m_h$.
  - No bound level of (N$_h$) is isolated, since there $E_b(0)-A\ge2\sum_{\sigma_i=+}\bar M_i-|\Delta E|-m_h>\Omega_{\max}-m_h-|\Delta E|>0$. Nor is any level of (N$_l$) with $r\ge1/(2\ell)$.
  - Outside (I) the level lies in $[A,\infty)$. The states there that one vertex reaches (member $i$ in its branch $-$ with one or two quanta, the other member a spectator) are excluded by (N). The other states are reached only through two or more vertices, so the width is of fourth or higher order in $\delta F$.

  Items 2 and 3 are stated in the range (N): (7.15) for every member in the branch $+$. Their coefficients are ordinary second-order coefficients of an isolated eigenvalue in (I), and formal coefficients in the rest of (N) (assumption (E)).
- (vi) The inversion of $1/E_b''$ below needs $|D/I|\ll1$ (Step 7), which holds if $|\Delta E|\ll|\sigma_1\bar M_1+\sigma_2\bar M_2|$ and $\langle k^2\rangle\ll1$. The first is automatic for $(+,+)$ by (B); for mixed branches it excludes the levels with $\bar M_1\approx\bar M_2$ (small $I$, large $|m_{\rm red}|$). The second follows from (C).

### Step 7. The bound level near $P=0$

Take the expectation of the $P^2$ terms of $H_{\rm eff}(P)$ in the bound eigenvector $\phi$ of the relative contract $k^2/(2m_{\rm red})+U$ (first-order perturbation in $P^2$):
- *motion of the members:* the $p^4$ terms of (7.1) with $p_i=I_iV+k_i$ contain $6I_i^2V^2k^2$; with $\langle k^2\rangle=2m_{\rm red}\langle T\rangle$ they give $-V^2[\frac3{2\tau^2}\langle T\rangle+2m_{\rm red}I\langle T\rangle]$, the first from the relativistic, the second from the lattice part;
- *labels in the mutual term:* by (7.5), $V^2\langle-U/\tau^2+(rU)'/(2c_0^2)\rangle$;
- *terms odd in $k$* (from $p_i^4$, $c_1c_2$ and $\Delta_j^2$): zero at first order, since $\phi$ is real (the relative contract at $V=0$ is real in the position basis); at second order they are of relative order $|\Delta E|/|\bar M_i|$;
- *self-energies:* the terms of Step 2 with both vertices on one body form the self-energy $\Sigma_i(p_i)$ of body $i$ alone, a one-body term (the other body is a spectator, whose energy cancels from the denominators). Body $i$ dressed alone at second order has $1/I_i=\varepsilon_i''(0)+\Sigma_i''(0)$, i.e. $I_i=\sigma_i\bar M_i/\tau^2+I_i^{(2)}$ with $I_i^{(2)}=-(\sigma_i\bar M_i/\tau^2)^2\Sigma_i''(0)$ of relative order $\delta F^2$ (the dressed single body is the subject of 06-mark-inertia). With the dressed $I_i$ in the decomposition $p_i=\frac{I_i}IP\pm k$, the $p^2$ parts of $\Sigma_i$ enter $P^2/(2I)$ exactly as for the bodies alone, so $I_i^{(2)}$ cancels from $D=I_{\rm pair}-I_1-I_2$. The members' own second-order terms enter $D$ only through the dressed $I_i$ and $m_{\rm red}$ in the parts below and through the $p^4$ parts of $\Sigma_i$ ($O(\delta F^2\langle k^2\rangle)$): at relative order $\delta F^2$, i.e. at order $\delta F^4$. At the order that $D$ needs, $I_i=\sigma_i\bar M_i/\tau^2$, (7.1).

With $1/I_{\rm pair}=E_b''(0)=1/I+2I^{-2}[\dots]$ ($I_1,I_2$ of the bodies dressed alone), at first order in $D/I$ (inversion below):

$$
E_b(P)=E_b(0)+\frac{P^2}{2I_{\rm pair}}+O(P^4),\qquad I_{\rm pair}=I_1+I_2+D,\qquad D=\underbrace{\frac{3\langle T\rangle}{\tau^2}}_{\text{members}}+\underbrace{\frac{2\langle U\rangle}{\tau^2}-\frac{\langle(rU)'\rangle}{c_0^2}}_{\text{labels in }U}+\underbrace{4I_1I_2\langle T\rangle}_{\text{lattice}} .
\tag{7.9}
$$

The term $2\langle U\rangle/\tau^2$ is the form factor, $-\langle(rU)'\rangle/c_0^2$ the retardation. The first three parts are of order $|\Delta E|/\tau^2$ with relative corrections $O(|\Delta E|/\bar M,(\lambda_C/a_b)^2,a_b^{-2},\ell^{-2},\delta F^2,u)$; the lattice part over the members' part is $4\sigma_1\sigma_2\bar M_1\bar M_2/(3\tau^2)=4\sigma_1\sigma_2/(3\lambda_{C,1}\lambda_{C,2})$, small for $\lambda_{C,i}\gg1$ and large in (N$_h$) at $r\le1$.

*Inversion.* With $D$ of (7.9), $1/I_{\rm pair}=(1-D/I)/I$ at this order, so $I_{\rm pair}-I_1-I_2=D+D^2/I+\dots$; this needs $|D/I|\ll1$, and the second term has the relative order $|D/I|$. By (7.9), $D/I=\bigl[3\langle T\rangle+2\langle U\rangle-\tau^2\langle(rU)'\rangle/c_0^2\bigr]/(\sigma_1\bar M_1+\sigma_2\bar M_2)+4m_{\rm red}\langle T\rangle$. The first part is $O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|)$. The lattice part is $4m_{\rm red}\langle T\rangle=2\langle k^2\rangle$, set by the size of the level, not by $|\Delta E|/\bar M$: $\langle k^2\rangle=\frac\alpha3a_b^{-2}$ (deep) and $a_b^{-2}$ (shallow) by (7.8); it dominates $D/I$ where the lattice part dominates $D$ (code check (ii), Step 8: $|D/I|\le0.006$, `code/v7/output/bound_pair_check.json`). So the inversion adds the relative corrections $O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|,\ a_b^{-2})$, small by (vi) and (C); near $\sigma_1\bar M_1+\sigma_2\bar M_2=0$ (mixed branches, $\bar M_1\approx\bar M_2$) it fails, and these levels are excluded (Step 6 (vi)).

### Step 8. Virial form and the factor of $\Delta E$ (item 2)

For an eigenvector of $k^2/(2m_{\rm red})+U(r)$, $2\langle T\rangle=\langle rU'\rangle$ for either sign of $m_{\rm red}$ (virial theorem, from $\langle[H,rk]\rangle=0$), so $\langle(rU)'\rangle=\langle U\rangle+2\langle T\rangle$, and $\Delta E=\langle T\rangle+\langle U\rangle$ at leading order. Then (7.9) becomes

$$
D=\frac{(2-r^2)\,\Delta E+(1-r^2)\langle T\rangle}{\tau^2}+4I_1I_2\langle T\rangle .
\tag{7.10}
$$

In the two regimes of (7.8):

$$
\text{deep: }D=\frac{(2-r^2)\Delta E}{\tau^2}+O\Bigl(\bigl(1+\tfrac{1}{\lambda_{C,1}\lambda_{C,2}}\bigr)\tfrac{a_b}\ell\tfrac{|\Delta E|}{\tau^2}\Bigr);\qquad \text{shallow: }D=\Delta E\Bigl(\frac1{\tau^2}-4I_1I_2\Bigr)\Bigl[1+O\bigl(\tfrac\ell{a_b}\bigr)\Bigr]=\frac{\Delta E}{\tau^2}\Bigl(1-\frac{4\sigma_1\sigma_2}{\lambda_{C,1}\lambda_{C,2}}\Bigr)\Bigl[1+O\bigl(\tfrac\ell{a_b}\bigr)\Bigr].
\tag{7.11}
$$

In the shallow regime the retardation vanishes ($rU$ has zero derivative on average for a contact bond) and the form factor and the members' motion give $(-3+4)\Delta E/\tau^2$, for every $r$. In the deep regime $\Delta E\simeq U(0)$, the made field of each body at the other, and only the form factor ($2U(0)/\tau^2$) and the retardation ($-U(0)/c_0^2$) remain.

*Size of the lattice part in (N).* Write $\rho:=\langle T\rangle/\Delta E$; it runs from $0^-$ (deep) to $-1$ (shallow) and depends on $F_1$, $\ell$ and, through $m_{\rm red}$, on the levels. Since $4\tau^2I_1I_2=4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2})$, the lattice part is $4\sigma_1\sigma_2\rho/(\lambda_{C,1}\lambda_{C,2})$ times $\Delta E/\tau^2$. For $(+,+)$ in (N$_h$), $\lambda_{C,1}\lambda_{C,2}<4\tau^2/\Omega_{\max}^2<r^2$ ($<r^2/4$ for $F_2\ne0$): in the deep regime the lattice part is $O(a_b/(\ell\lambda_{C,1}\lambda_{C,2}))$, larger than $O(a_b/(\ell r^2))$, so the deep term of (7.11) leads only for $a_b\ll\ell\lambda_{C,1}\lambda_{C,2}$; in the shallow regime $4/(\lambda_{C,1}\lambda_{C,2})>4/r^2$, so for $r<2$ the factor $\tau^{-2}-4I_1I_2$ of (7.11) is negative and $D$ has the sign of $-\Delta E$: the lattice part dominates. For mixed branches (N$_h$) restricts only $\lambda_{C,1}<r$. In (N$_l$), $\lambda_{C,1}\lambda_{C,2}>4r^2\ell^2$, and for $r\ell\gg1$ the lattice part is $O((r\ell)^{-2})$ in either regime. (7.11) then gives $D=\frac{\Delta E}{\tau^2}[1+O(\ell/a_b,(r\ell)^{-2})]$ in the shallow regime, the only one there unless $r\ll1$. In (N$_l$) with $r\ell$ not large ($r\ell=O(1)$, so $r=O(1/\ell)$; this part contains (I), where $r\ell<\frac12$), (N$_l$) bounds $\lambda_{C,i}$ only from below, by $\lambda_{C,i}>2r\ell$ with $2r\ell=O(1)$ (for the member in the branch $-$ of a mixed pair through $\bar M_2<\bar M_1$), so the lattice part is at most $|\rho|/(r\ell)^2$ times $|\Delta E|/\tau^2$ and can be of order one; the $r^2$ terms of (7.10) there are $O(\ell^{-2})$ (below $1/(4\ell^2)$ in (I)), the order of the corrections dropped in (7.5).

*One speed.* By (7.10), $\tau^2D/\Delta E=2-r^2+\rho\,[1-r^2+4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2})]$. Hence, at fixed levels, with (7.10) taken at every value of $\rho$ (every binding strength),

$$
D=\frac{\Delta E}{v^2}\ \text{with one }v\text{ for every }\rho\iff r^2=1+\frac{4\sigma_1\sigma_2}{\lambda_{C,1}\lambda_{C,2}},\ \ v=\frac{\tau}{\sqrt{2-r^2}};\qquad (+,+):\ \text{no such }r\text{ under (D)};\qquad \text{at }r=1:\ D=\frac{\Delta E}{\tau^2}+4I_1I_2\langle T\rangle .
\tag{7.12}
$$

For $(+,+)$ the condition needs $r^2>1$. (D) gives either $\lambda_{C,1}\lambda_{C,2}\le4/r^2$ (both $\lambda_{C,i}<2$), so that $1+4/(\lambda_{C,1}\lambda_{C,2})\ge1+r^2>r^2$, or $r\le1$: no solution. For mixed branches it is $\lambda_{C,1}\lambda_{C,2}=4/(1-r^2)$ with $r<1$, and then $v=\tau(1+4/(\lambda_{C,1}\lambda_{C,2}))^{-1/2}=\tau/\sqrt{2-r^2}$. Such levels lie outside (N$_l$) whenever $1-r^2\ge(r\ell)^{-2}$, since there $\lambda_{C,1}\lambda_{C,2}>4r^2\ell^2\ge4/(1-r^2)$; they lie in (I) for suitable members, for example $\tau=0.2\,m_h$, $\bar M_1=0.125\,m_h$, $\bar M_2=0.08\,m_h(1-r^2)$, branches $(+,-)$, any $\ell\gg1$ (so $r=0.2/\ell<1/(2\ell)$): member 1 is in (N$_l$), since $\min_q[\omega_q-E_1(q)]\ge m_h-\sqrt{\bar M_1^2+\tau^2}=0.76\,m_h>\bar M_1$, and the isolation inequality of (7.15) reads $0.045\,m_h+\Delta E<0.55\,m_h$, with (vi) for $\Delta E\ll0.045\,m_h$. They also occur in (N$_l$) outside (I), with formal coefficients: for $\frac12<r\ell<1$ (so $r\ll1$ and (I) is empty, since $r\ge1/(2\ell)$), $4/(1-r^2)>4>4r^2\ell^2$, and the branches $(+,-)$ with $2r\ell<\lambda_{C,1}<2/\sqrt{1-r^2}$ (member 1 in (N$_l$), as $\bar M_1<m_h/2$ up to $O(r^2)$) and $\lambda_{C,2}=4/((1-r^2)\lambda_{C,1})>\lambda_{C,1}$ (so $\bar M_2<\bar M_1$, (7.7)) meet the condition. For $\lambda_{C,i}\to\infty$ it is $r=1$, $v=\tau=c_0$. At $r=1$ only the lattice part breaks the equality, by the relative amount $4|\rho|/(\lambda_{C,1}\lambda_{C,2})$. At leading order in the regime parameters, in every part of (N) where the regime exists, (7.11) gives $v=\tau/\sqrt{2-r^2}$ in the deep regime with $a_b\ll\ell\min(1,\lambda_{C,1}\lambda_{C,2})$ (up to $O(a_b/\ell,\ a_b/(\ell\lambda_{C,1}\lambda_{C,2}))$; no real $v$ for $r\ge\sqrt2$) and $v=\tau(1-4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2}))^{-1/2}$ in the shallow regime where real (up to $O(\ell/a_b)$); the two coincide iff the condition of (7.12) holds. In the parts of (N):
- (N$_h$) (formal coefficients). Deep: $v=\tau=c_0$ at $r=1$. Shallow: for $(+,+)$ the lattice part dominates ($\lambda_{C,1}\lambda_{C,2}<r^2$), and at $r<2$ there is no real speed.
- (N$_l$) with $r\ell\gg1$ (formal coefficients). For $r$ of order one only shallow levels exist, and $v=\tau$ up to $O((1-r^2)\ell/a_b,(r\ell)^{-2})$ (since $1+\rho=O(\ell/a_b)$ and $\lambda_{C,1}\lambda_{C,2}>4r^2\ell^2$) for every allowed binding strength. At $r=1$ this is $v=\tau=c_0$ up to $O(\ell^{-2})$ for every binding strength. For $r\ll1$ deep levels exist as well, with $v=\tau/\sqrt{2-r^2}$, so there the speed depends on the binding (the mixed solution above needs $1-r^2<(r\ell)^{-2}$, which fails for $r\ll1$).
- (N$_l$) with $r\ell$ not large ($r\ell=O(1)$). This part contains (I) (isolated levels, $r<1/(2\ell)$, ordinary coefficients) and, with formal coefficients, the levels with $r<1/(2\ell)$ that violate the isolation inequality and those with $r\ell$ of order one; $\lambda_{C,i}$ is bounded only from below ($\lambda_{C,i}>2r\ell$), and $r^2=O(\ell^{-2})$. Deep: $v=\tau/\sqrt2$ up to $O(\ell^{-2},\,a_b/\ell,\,a_b/(\ell\lambda_{C,1}\lambda_{C,2}))$. Shallow: $v=\tau(1-4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2}))^{-1/2}$ where real. The speed depends on the binding, except for mixed branches at $\lambda_{C,1}\lambda_{C,2}=4/(1-r^2)$, where (7.10) gives $D=\Delta E/v^2$ with $v=\tau/\sqrt{2-r^2}$ at every binding strength (such levels occur in (I), as the example above, and in (N$_l$) outside (I), with formal coefficients); for $(+,+)$ there is no exception, since $\tau/\sqrt2$ would need $1-4/(\lambda_{C,1}\lambda_{C,2})=2$. $r=1$ lies outside this part.

The grip enters (7.10) only through $U$, so the linear grip ($F_2=0$) obeys (7.10)–(7.12) unchanged, in the wider heavy range $\bar M_i>\Omega_{\max}/2$; it is no control for item 2. Controls in the class: every $r\neq1$ (for example deep, $r=\frac12$, $a_b\ll\ell\lambda_{C,1}\lambda_{C,2}$: $D=\frac74\Delta E/\tau^2$), and at $r=1$ the shallow regime of (N$_h$) ($D\simeq-4I_1I_2\Delta E$). Cancellations, excluded: $r=\sqrt2$ in the deep regime (the leading term vanishes; $D=[-\tau^{-2}+4I_1I_2]\langle T\rangle$); $\lambda_{C,1}\lambda_{C,2}=4$ for $(+,+)$ in the shallow regime ($D=0$ at leading order), which lies outside (N) under (D) for $r\ell>1$ ((N$_h$) with (D) gives $\lambda_{C,i}<\min(r,2/r)$, so $\lambda_{C,1}\lambda_{C,2}<2$); $\bar M_1\approx\bar M_2$ for mixed branches (Step 6 (vi)). For a common grip $U$ has no zero, so no separation is tuned away.

Code check (ii) (`code/v7/check_bound_pair.py`, output `code/v7/output/bound_pair_check.json`): exact diagonalization of the effective contract (7.2)–(7.3) in sectors of the periodic ring, $P=2\pi m/L$ with the allowed labels $p_1=2\pi n/L$, $p_2=P-p_1$, on rings of $L=1024$ cells for $(+,+)$ and $3072$ for $(+,-)$, with members in the heavy range of (7.15): $\tau=1$, $M_{k_1}=5.0$, $M_{k_2}=4.5$ ($\lambda_{C,1}\lambda_{C,2}\approx0.044$), square and linear grip, $r=0.5,1,1.4$. The no-decay margin, $\bar M_i-\Omega_{\max}$ (square) or $\bar M_i-\Omega_{\max}/2$ (linear), is at least $0.49$, and $2\sum_{\sigma_i=+}\bar M_i+\Delta E-m_h\ge9.9>0$. So a necessary condition of (I) fails ($\sqrt{\bar M_i^2+\tau^2}\ge\bar M_i$): the levels checked are not isolated, and their curvatures are formal coefficients. Bound levels: deep ($\ell=30$, $a_b\approx10$, $\rho\approx-0.125$) and two shallower ($\ell=30$, $a_b\approx93$, $\rho\approx-0.59$, for $(+,-)$ $a_b\approx92$, $\rho\approx-0.60$; $\ell=8$, $a_b\approx70$, $\rho\approx-0.81$), branches $(+,+)$ with both grips, and $(+,-)$ with the linear grip (the first two). The joint labels have $(I_1/I)m$ an integer ($m=19,38,76$ for $(+,+)$, where $I_1/I=10/19$; $m=1,2,3$ for $(+,-)$, where $I_1/I=10$; pair velocity at most $0.049\,\tau$), so that the relative label $p_1-\frac{I_1}IP$ runs over the same grid as at $P=0$. Otherwise the relative motion carries the twist $e^{2\pi i(I_1/I)m}$, whose finite-size shift ($\propto e^{-L/a_b}$) depends on $P$: for the shallow $(+,+)$ level with the linear grip at $r=1$, twisted labels give $|D_{\rm num}/D_{(7.10)}-1|=0.036$ at $L=1024$ and $0.0007$ at $L=2048$, untwisted ones $0.0003$ and $0.0005$. The curvature of the extreme level agrees with (7.10), with $\Delta E$ and $\langle T\rangle$ from the same eigenvector: $|D_{\rm num}-D_{(7.10)}|\le0.0045\,|D|$, and $\le0.051\,|\Delta E|/\tau^2$, in all 24 cases; the largest $|\Delta_j|/\omega_q$, $|\Delta_j|/\Omega_{ab}$ is $0.15$. The lattice part is $11.2$, $53.2$ ($53.8$ for $(+,-)$ at $L=3072$), $72.8$ times $|\Delta E|/\tau^2$ in the three regimes and dominates $D$, as stated for (N$_h$) above: for $(+,+)$ at $r=1$, $D/(\Delta E/\tau^2)=-10.22,\,-52.24,\,-71.81$ (formula $-10.22,\,-52.22,\,-71.81$), so $D>0$ while $\Delta E<0$, also in the deep case, where $a_b\approx10$ exceeds $\ell\lambda_{C,1}\lambda_{C,2}\approx1.3$; in the deep case at $r=0.5$ and $1.4$, $D/(\Delta E/\tau^2)=-9.57$ and $-11.06$ (formula $-9.56$, $-11.07$), the dependence on $r$ of (7.10).

### Step 9. Fall of the pair (item 3)

A distant held source $g_sq_{x_s}$ translates the lowest eigenvector of the field by $\langle q_x\rangle=\phi(x)=-g_sG(x-x_s)$ (first order in $g_s$; quadratic contract), with unchanged fluctuations $\tilde q$. On the pinned ring ($\ell\gg1$, $L\gg\ell$, $|x-x_s|\gg1$), $G(x)\simeq\ell e^{-|x|/\ell}/(2c_0^2)$, so $\phi(X+y)=\phi(X)e^{\mp y/\ell}$ for placements on one side of the source: the made field varies on the length $\ell$, at every distance $d$ of the pair from the source. It is locally uniform over a pair of extent $a_b$ iff $a_b\ll\ell$; of the regimes of (7.8) only the deep one has it, and item 3 is answered there (note (a) of the question). Shallow pairs ($a_b\gg\ell$) span a change of the field by factors $e^{\pm a_b/\ell}$ and are outside item 3.

Locally the bodies hold $F(\phi+\tilde q)$: $\bar a\to\bar a+F_1\phi$, $F_1\to F_1+2F_2\phi$, $F_2$ unchanged. The weight of a cost $E$ is $w=\partial E/\partial\phi$ (the force is $-w\nabla\phi$). For the members at first order in the grip (note (b) of the question), $w_i=\mu_iF_1$ and $w_i/I_i=\kappa:=F_1\tau^2/\bar a$ for every level and both branches. The bond adds $w_b=\partial\Delta E/\partial\phi$; by Hellmann–Feynman on $k^2/(2m_{\rm red})+U$ ($m_{\rm red}\propto\bar a$, $U_G\propto F_1^2$, $U_J\propto F_2^2$): $w_b=\frac{F_1}{\bar a}(s\langle U\rangle-\langle T\rangle)$, at leading order in the grip ($F_2w_0/F_0$ dropped, and $U_J$ against $U_G$, which is $O(\epsilon_J)$ in the deep regime), the order of (2.18). For $s=1$ the static part is (2.18): the held bond is weighed like the rest cost. The contribution of the mutual term to the fall is the comparison of $w_b$ with its share $D$ of the inertia. *Reference.* The members sit at $x_1=X+\frac{I_2}I\xi$, $x_2=X-\frac{I_1}I\xi$ ($X$ the centre of inertia, $\xi$ the separation) and are weighed at their own places, the reference of (2.18). Every weight of the pair is referred to the members' inertia-weighted field gradient $\overline{\nabla\phi}:=\sum_iI_i\langle\nabla\phi(x_i)\rangle/I$: $w_{\rm pair}:=-F_{\rm pair}/\overline{\nabla\phi}$, with $F_{\rm pair}$ the force on the pair. The left side of (7.13) is then the acceleration $F_{\rm pair}/I_{\rm pair}$ of the centre of inertia over $-\kappa\overline{\nabla\phi}$, the inertia-weighted acceleration of the members alone at their own places, minus one. Since $w_i=\kappa I_i$, the members' force is exactly $-\kappa I\,\overline{\nabla\phi}$, so $w_{\rm pair}=w_1+w_2+w_b$ up to the finite-extent factor of the bond (below). With $I_{\rm pair}=I_1+I_2+D$ the left side is exactly $(w_b/\kappa-D)/(I+D)$; at first order in $D/I$ (relative corrections $O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|,a_b^{-2})$, Step 7), and with (7.10),

$$
\frac{w_{\rm pair}/I_{\rm pair}}{w_i/I_i}-1=\frac{w_b/\kappa-D}{I_1+I_2}=\frac{s\langle U\rangle-\langle T\rangle-\tau^2D}{\sigma_1\bar M_1+\sigma_2\bar M_2}=\frac{(s-2+r^2)\langle U\rangle-(4-2r^2+4\tau^2I_1I_2)\langle T\rangle}{\sigma_1\bar M_1+\sigma_2\bar M_2}.
\tag{7.13}
$$

*Variation of the field over the pair.* Since $\nabla\phi=\mp\phi/\ell$, the force on the pair is $\pm\ell^{-1}$ times its potential $\sum_iw_i\langle\phi(x_i)\rangle+w_b\phi_b$, with $\phi_b$ the field at the bodies weighed by the derivatives of $\Delta E$ in $\phi(x_1)$ and $\phi(x_2)$, and $\overline{\nabla\phi}=\mp\ell^{-1}\sum_iI_i\langle\phi(x_i)\rangle/I$. The members' part, $\sum_iw_i\langle\phi(x_i)\rangle=\kappa\sum_iI_i\langle\phi(x_i)\rangle$, is exactly $\mp\ell\kappa I\,\overline{\nabla\phi}$ and drops out of the left side of (7.13) at every extent. The term linear in $\xi$ of $\sum_iw_i\phi(x_i)$ is $F_1\phi'\xi(\mu_1I_2-\mu_2I_1)/I=0$ ($I_i=\mu_i\bar a/\tau^2$), so the field does not polarize the pair at this order, and the odd terms of $\phi_b$ average to zero in the even relative eigenvector. Hence $\phi_b$ differs from the members' inertia-weighted field $\sum_iI_i\langle\phi(x_i)\rangle/I$ by the relative amount $O(\langle\xi^2\rangle/\ell^2)$; the bond's weight enters (7.13) multiplied by $1+O(\langle\xi^2\rangle/\ell^2)$, which changes (7.13) by $O((a_b/\ell)^2|\Delta E|/\bar M)$ at every distance $d$: in the deep regime, below the $O(a_b/\ell)$ remainder of (7.14). With another reference this does not hold: referred to the field gradient at the centre of inertia, $\overline{\nabla\phi}/\nabla\phi(X)=\sum_i\frac{I_i}I\langle e^{\mp y_i/\ell}\rangle=1+\frac{m_{\rm red}\langle\xi^2\rangle}{2I\ell^2}+O(a_b^4/\ell^4)$ ($y_i=x_i-X$, $\sum_iI_iy_i=0$, $\sum_iI_iy_i^2=m_{\rm red}\xi^2$, odd moments zero), and the members alone add this term to the left side of (7.13), without the factor $|\Delta E|/\bar M$ ($\langle\xi^2\rangle/(8\ell^2)$ for equal members). It is a property of the members' places, present without the bond, and not part of the contribution of the mutual term.

In the deep regime $\langle T\rangle/\langle U\rangle=O(a_b/\ell)$, so

$$
\text{deep, }s\neq0:\ w_b=\kappa D,\ \text{i.e. the pair falls like its members at first order in the grip}\iff s=2-r^2\quad(\text{square grip: }r=1),
\tag{7.14}
$$

in the range (N), at leading order in $a_b/\ell$, $a_b/(\ell\lambda_{C,1}\lambda_{C,2})$ (the $\langle T\rangle$ term of (7.13) carries $4\tau^2I_1I_2$) and $\epsilon_J$. On the line $s=2-r^2$ the leading terms are $w_b/\kappa=D=s\Delta E/\tau^2$, nonzero iff $s\neq0$. *Excluded cancellation: the linear grip at $r=\sqrt2$* ($s=0$ on the line). There the leading $w_b$ and the leading $D$ both vanish, and their equality is no fall criterion. At the first nonzero order of the computed terms, the continuum terms of (7.13) cancel there (coefficients $s-2+r^2=0$ of $\langle U\rangle$ and $4-2r^2=0$ of the continuum part of $\langle T\rangle$; for $F_2=0$, $w_b/\kappa=-\langle T\rangle/\tau^2$ at leading order in the grip), and the lattice part remains: $-4\tau^2I_1I_2\langle T\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)=-4m_{\rm red}\langle T\rangle=-2\langle k^2\rangle=-\frac{2\alpha}3a_b^{-2}$. Beside it stand the corrections of (7.9), of relative orders $(\lambda_C/a_b)^2$, $a_b^{-2}$, $\ell^{-2}$, $\delta F^2$, $|\Delta E|/\bar M$ against $|\Delta E|/\tau^2$, i.e. of these orders times $|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|$ in (7.13); they are not computed and are not shown to be smaller than $2\langle k^2\rangle$. So at $r=\sqrt2$ the computed order does not decide whether the pair with the linear grip falls like its members; at every other $r$ of the deep regime it does not, at leading order (leading term $(r^2-2)\langle U\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)\neq0$). In (I) (isolated levels) the condition is $s=2-r^2$ with $r^2<1/(4\ell^2)$, met by neither the linear nor the square grip. The answer to item 3 is therefore no in general: the mutual term leaves the fall unchanged only on the line $s=2-r^2$, $s\neq0$. With the square grip, which weighs the held mark and bond (2.18), a pair falls like its members at first order in the grip only at $r=1$, in (N$_h$) with $a_b\ll\ell\lambda_{C,1}\lambda_{C,2}$, with formal coefficients; isolated levels miss it by $(r^2-1)\langle U\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)$. The remainder at $r=1$ is $-(2+4\tau^2I_1I_2)\langle T\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)=-2\langle T\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)-2\langle k^2\rangle=O\bigl((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\bar M}\bigr)$.

*Dressed members.* (7.13)–(7.14) give the contribution of the mutual term, against the members at first order in the grip (note (b) of the question). The members dressed alone have the weights $w_i+\partial_\phi\Sigma_i(0)$ and the inertias $I_i+I_i^{(2)}$ (Step 7). These one-body terms enter the pair and the members alike, but they change $w_i/I_i$ at relative order $u$ and $M_{k_i}F_1^2G(0)$, the order of (7.13) or larger, and in general differently for different levels and branches: in the held limit $\Sigma_i\supset-\mu_i^2F_1^2G(0)/2$, so $\partial_\phi\Sigma_i/w_i=-2\mu_iF_2G(0)$, which depends on $\mu_i$ unless $F_2=0$. A comparison with the dressed members would further need these self-energies with their weights and curvatures (06-mark-inertia), and a common dressed ratio $\tilde w_1/\tilde I_1=\tilde w_2/\tilde I_2$ at this order; given it, (7.13) and (7.14) hold with $\kappa\to\tilde w_i/\tilde I_i$, up to relative corrections of the order of the dressing.

## Result

Class for all items: the description of "Setup" (common grip of (G2) with $F_0>0$, $F_1\neq0$, every pair of levels, ring with $m_h>0$, $\ell\gg1$, $L\gg\ell,a_b$), $M\to\infty$ first, then $L\to\infty$, then the regime limits; second order in $\delta F$ for the mutual term, with the remainder $O(\delta F^3)$ (relative order $u$; $O(\delta F^4)$ for $F_2=0$) and the self-energies as in (B); continuum regime (C) and no Cherenkov denominators (D); the deep regime includes $\epsilon_J\ll1$ (7.8). For (7.7)–(7.14) in addition the range (N) of (7.15) for every member in the branch $+$ (assumption (E)). In its part (I) the bound levels are isolated eigenvalues, and (7.8)–(7.14) give their second-order coefficients. In the rest of (N) (all of N$_h$, and N$_l$ outside (I)) the levels are not isolated, and every coefficient of (7.8)–(7.14) is a formal coefficient: the second-order coefficient of a level whose width is of fourth or higher order in $\delta F$.

- **(7.1)** (definition) Bands $\varepsilon_i=\sigma_iE_i(p)$ of the bodies at first order, signed inertias $I_i=\sigma_i\bar aM_{k_i}/\tau^2$, form factor $\sigma_ic_i$ of the grip between band states; minima of $E_i$ at $p=0$ and $p=\pi$.
- **(7.2)** (formula) The mutual term between two-body states of labels $(p_1,p_2)\to(p_1-q,p_2+q)$, one- and two-quantum exchange with the form factors $c_1c_2$ and the energy transfers $\Delta_1,\Delta_2$, for every label under (D).
- **(7.3)** (formula) The effective joint contract in the sector $P$: the bands of the two bodies plus the mutual term, up to the single-body self-energies and $O(\delta F^3)$, of relative order $u$ ($O(\delta F^4)$ for $F_2=0$).
- **(7.4)** (formula) Time representation of (7.2); for held bodies it is the static bond (2.11) in (G2), with the sign of $-\mu_1\mu_2$ at every separation.
- **(7.5)** (formula) The bond of a pair moving with $V=P/I$: $U+V^2[-U/\tau^2+(rU)'/(2c_0^2)]$, at second order in the velocities, for $\ell\gg1$, with relative corrections $O(\ell^{-2},a_b^{-2},\lambda_C^2/a_b^2)$ to the coefficient of $V^2$, which holds at every $r$; the truncation at $V^2$ needs in addition $V^2|r|/(c_0^2\ell)\ll1$ at $|r|\gg\ell$.
- **(7.6)** (formula) The one-quantum part of the moving bond, at second order in $V$: below the pinning length its constant and its slope change with $P$ by the factors $1-V^2(\tau^{-2}-\frac12c_0^{-2})$ and $1-V^2(\tau^{-2}-c_0^{-2})$; above it, for $\ell\ll|r|\ll c_0^2\ell/V^2$, it is $U_G(r)[1-V^2/\tau^2+\frac{V^2}{2c_0^2}(1-|r|/\ell)]$. For every $|r|$ (unexpanded continuum kernel, $V<c_0$) the moving bond has the range $\ell\sqrt{1-V^2/c_0^2}$.
- **(7.7)** (criterion) A bound level near labels $0$ exists for $(+,+)$ (lowest level of the sector), for mixed branches iff the member in the branch $+$ has the larger $\bar M$ (highest level), never for $(-,-)$; it comes in two valley copies, and the extreme level is their even combination, unique for $t\neq0$.
- **(7.8)** (formula) $\Delta E$, $\langle T\rangle$, $a_b$ in the deep regime $1,\lambda_C\ll a_b\ll\ell$ with $\epsilon_J=\frac{2F_2^2}{\pi F_1^2c_0}\ln\frac\ell{a_b}\ll1$ (Airy, relative corrections $O(a_b/\ell,\epsilon_J)$; grips with $\epsilon_J$ not small are excluded from it) and the shallow regime $a_b\gg\ell$ (contact bond, $O(\ell/a_b)$), with the conditions (i), (ii) and (vi) of Step 6.
- **(7.9)** (formula) $E_b(P)=E_b(0)+P^2/(2I_{\rm pair})+O(P^4)$, $I_{\rm pair}=I_1+I_2+D$, with $D$ the sum of the members' part $3\langle T\rangle/\tau^2$, the labels' part $2\langle U\rangle/\tau^2-\langle(rU)'\rangle/c_0^2$ and the lattice part $4I_1I_2\langle T\rangle$, the first three parts with relative corrections $O(|\Delta E|/\bar M,(\lambda_C/a_b)^2,a_b^{-2},\ell^{-2},\delta F^2,u)$. The inversion of $1/I_{\rm pair}$ is at first order in $D/I$ and needs $|D/I|\ll1$; its next term $D^2/I$ has the relative order $|D/I|=O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|)+2\langle k^2\rangle$, the lattice part contributing $4m_{\rm red}\langle T\rangle=2\langle k^2\rangle=O(a_b^{-2})$; levels with $\sigma_1\bar M_1+\sigma_2\bar M_2\approx0$ are excluded. Here $I_i=\sigma_i\bar M_i/\tau^2+I_i^{(2)}$ is the inertia of body $i$ dressed alone; its second-order part $I_i^{(2)}$ cancels from $D$, and the members' own second-order terms enter $D$ at relative order $\delta F^2$, so $D$ needs $I_i$ only at first order.
- **(7.10)** (formula) $D=[(2-r^2)\Delta E+(1-r^2)\langle T\rangle]/\tau^2+4I_1I_2\langle T\rangle$, same class and order.
- **(7.11)** (formula) Deep (regime of (7.8), with $\epsilon_J\ll1$): $D=(2-r^2)\Delta E/\tau^2+O((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\tau^2})$; shallow: $D=\Delta E(\tau^{-2}-4I_1I_2)[1+O(\ell/a_b)]$, for every $r>0$ allowed by (D) and (N). The deep term leads only for $a_b\ll\ell\min(1,\lambda_{C,1}\lambda_{C,2})$; in (N$_h$), $\lambda_{C,1}\lambda_{C,2}<4\tau^2/\Omega_{\max}^2<r^2$ for $(+,+)$. In the shallow regime of (N$_h$), for $(+,+)$, the lattice part dominates and, for $r<2$, $D$ has the sign of $-\Delta E$. In (N$_l$) with $r\ell\gg1$ the lattice part is $O((r\ell)^{-2})$, and only the shallow regime exists unless $r\ll1$. In (N$_l$) with $r\ell$ not large ((I) included) the $r^2$ terms are $O(\ell^{-2})$, $\lambda_{C,i}$ is bounded only from below ($\lambda_{C,i}>2r\ell$), and the lattice part is at most $|\rho|/(r\ell)^2$ times $|\Delta E|/\tau^2$ and can be of order one.
- **(7.12)** (criterion) At fixed levels, with (7.10) taken at every binding strength, $D=\Delta E/v^2$ with one speed for every binding strength iff $r^2=1+4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2})$, and then $v=\tau/\sqrt{2-r^2}$; for $(+,+)$ no $r$ satisfies this under (D); for mixed branches it is $\lambda_{C,1}\lambda_{C,2}=4/(1-r^2)$, $r<1$, which lies outside (N$_l$) whenever $1-r^2\ge(r\ell)^{-2}$ and is reached in (I) (example in Step 8). At $r=1$, $D=\Delta E/\tau^2+4I_1I_2\langle T\rangle$. At leading order, in every part of (N) where the regime exists: deep regime with $a_b\ll\ell\min(1,\lambda_{C,1}\lambda_{C,2})$, $v=\tau/\sqrt{2-r^2}$ (no real speed for $r\ge\sqrt2$); shallow regime, $v=\tau(1-4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2}))^{-1/2}$ where real; the two coincide iff the condition above holds. In the parts of (N):
  - (N$_h$), formal coefficients. Deep: $v=\tau=c_0$ at $r=1$. Shallow: for $(+,+)$ the lattice part dominates, and at $r<2$ there is no real speed.
  - (N$_l$) with $r\ell\gg1$, formal coefficients. For $r$ of order one only shallow levels exist, and $v=\tau$ up to $O((1-r^2)\ell/a_b,(r\ell)^{-2})$ for every allowed binding strength; at $r=1$ this is $v=\tau=c_0$ up to $O(\ell^{-2})$. For $r\ll1$ deep levels have $v=\tau/\sqrt{2-r^2}$, and the speed depends on the binding.
  - (N$_l$) with $r\ell$ not large ($r\ell=O(1)$): (I), the isolated levels ($r<1/(2\ell)$, ordinary coefficients), and, with formal coefficients, the levels with $r<1/(2\ell)$ outside (I) and those with $r\ell$ of order one; $\lambda_{C,i}$ is bounded only from below ($\lambda_{C,i}>2r\ell$). Deep: $v=\tau/\sqrt2$ up to $O(\ell^{-2},a_b/\ell,a_b/(\ell\lambda_{C,1}\lambda_{C,2}))$. Shallow: $v=\tau(1-4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2}))^{-1/2}$ where real. The speed depends on the binding, except for mixed branches at $\lambda_{C,1}\lambda_{C,2}=4/(1-r^2)$, where $v=\tau/\sqrt{2-r^2}$ for every binding strength (such levels occur in (I), as the example of Step 8, and in (N$_l$) outside (I), with formal coefficients); for $(+,+)$ there is no exception. $r=1$ lies outside this part.

  Excluded cancellations: $r=\sqrt2$ (deep); $\lambda_{C,1}\lambda_{C,2}=4$ for $(+,+)$ (shallow; inside (N) under (D) only for $r\ell<1$); $\bar M_1\approx\bar M_2$ (mixed branches).
- **(7.13)** (formula) The contribution of the mutual term to the fall: the relative difference of weight over inertia between the pair and its members at first order in the grip, $(w_b/\kappa-D)/(I_1+I_2)$ with $\kappa=w_i/I_i=F_1\tau^2/\bar a$, every weight of the pair referred to the members' inertia-weighted field gradient $\overline{\nabla\phi}=\sum_iI_i\langle\nabla\phi(x_i)\rangle/I$ (the acceleration of the centre of inertia against the inertia-weighted acceleration of the members alone at their own places), at leading order in the grip and first order in $D/I$ (relative corrections $O(|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|,a_b^{-2})$), in the range (N), for a pair whose extent is small against the length $\ell$ on which the made field of a distant held source varies on the pinned ring ($a_b\ll\ell$: the deep regime). In this reference the members' finite-extent term cancels exactly, and the variation of the field over the pair enters only through the bond, at $O((a_b/\ell)^2|\Delta E|/\bar M)$, at every distance of the source; referred instead to the gradient at the centre of inertia, the members add $\frac{m_{\rm red}\langle\xi^2\rangle}{2I\ell^2}+O(a_b^4/\ell^4)$, without the factor $|\Delta E|/\bar M$, a term of the members' places that is not part of the contribution of the mutual term.
- **(7.14)** (criterion) In the range (N), in the deep regime (the only regime of (7.8) with $a_b\ll\ell$), the bond's weight equals its share $D$ of the inertia times $w_i/I_i$, i.e. the pair falls like its members at first order in the grip, iff $s=4F_0F_2/F_1^2=2-r^2$, for $s\neq0$, at leading order in $a_b/\ell$, $a_b/(\ell\lambda_{C,1}\lambda_{C,2})$ and $\epsilon_J$ (square grip: iff $r=1$). Excluded cancellation: $s=0$ on this line, the linear grip at $r=\sqrt2$, where the leading $w_b$ and $D$ both vanish; there (7.13) keeps, of its computed terms, only the lattice part $-2\langle k^2\rangle=-\frac{2\alpha}3a_b^{-2}$, beside uncomputed corrections not shown to be smaller, so the computed order does not decide; at every other $r$ of the deep regime the linear grip does not fall like its members at leading order. In (I) (isolated levels) neither the linear nor the square grip meets the condition. With the square grip a pair falls like its members at first order in the grip only at $r=1$, in (N$_h$) with $a_b\ll\ell\lambda_{C,1}\lambda_{C,2}$ (formal coefficients), with remainder $O((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\bar M})$. A comparison with the dressed members needs in addition their second-order weights and inertias (06-mark-inertia), which change $w_i/I_i$ at relative order $u$ and $M_{k_i}F_1^2G(0)$, the order of (7.13) or larger, and a common dressed ratio; given it, (7.13)–(7.14) hold with the dressed ratio up to relative corrections of the order of the dressing.
- **(7.15)** (criterion) A member in the branch $+$, at label $0$, cannot decay with one or two quanta at second order iff one of two conditions holds. The first is $\bar M_i>\Omega_{\max}/2$ ($F_2=0$) or $\bar M_i>\Omega_{\max}$ ($F_2\neq0$), with $\Omega_{\max}=\sqrt{m_h^2+4\kappa_h}$ (heavy range N$_h$; it implies $\lambda_{C,i}<r$, resp. $<r/2$). The second is $\bar M_i<\min_q[\omega_q-E_i(q)]$ (light range N$_l$; it implies $\bar M_i<m_h/2$, i.e. $\lambda_{C,i}>2r\ell$, and it is empty for $r>1$ with $(r^2-1)/r\ge1/\ell$). A bound level near labels $0$ is an isolated eigenvalue of the full contract iff $\sigma_1\bar M_1+\sigma_2\bar M_2+\Delta E<m_h-\sqrt{\bar M_1^2+\tau^2}-\sqrt{\bar M_2^2+\tau^2}$ (range I), with a margin large against the second-order shifts of this threshold. (I) lies in N$_l$ and needs $\tau<m_h/2$, i.e. $r<1/(2\ell)$. So no bound level of N$_h$ is isolated, and none of N$_l$ with $r\ge1/(2\ell)$.

## Consistency checks

1. **Held limit.** For $\tau\to0$, $c_i\to1$ and $\Delta_j\to0$, and (7.2)/(7.4) give $-\mu_1\mu_2[F_1^2G+4F_2^2J_2]$, which is (2.11) in (G2) (Step 3). Passed.
2. **Bare bodies.** At first order, (7.1) gives $\varepsilon_i''(0)=\tau^2/(\bar aM_{k_i})$ for $\sigma_i=+$, which is (6.7) with the rest cost $M_k\to\bar aM_k$ of the grip ($\bar a=F_0$ for $\delta F=0$). Passed.
3. **Dimensions.** With $[\text{cost}]=\lambda^{-1}$: $[\mu_i]=[\bar M_i]=[\text{cost}]$, $F$ dimensionless, $[q]=[\text{cost}]^{-1/2}$, $[\kappa_h]=[\text{cost}]^2$, $[c_0]=[\tau]=[\text{cost}]$ (cells per unit $\lambda$), $[F_1^2]=[\text{cost}]$, $[F_2^2]=[\text{cost}]^2$, $[G]=[\text{cost}]^{-2}$, $[J_2]=[\text{cost}]^{-3}$: $[U]=[\text{cost}]$; $[I_i]=[D]=[\text{cost}]^{-1}$, and each term of (7.9) and (7.10) is $[\text{cost}]/[\text{cost}]^2$ or $[\text{cost}]^{-2}[\text{cost}]$. Passed.

## Open issues

- The third-order terms of the elimination ($\propto F_1^2F_2$, $F_2^3$), the fourth-order terms and the single-body self-energies are not computed; they are argued to be of relative order $u$, $r^2|U(0)|/\bar M$ and $\delta F^2$ (assumption (B)), and the code checks only the second-order effective contract.
- Deep bound levels of grips with $\epsilon_J$ not small (two-quantum bond dominant near $r=0$) are excluded, not treated.
- Outside the range (I) of (7.15) the coefficients of items 2 and 3 are formal coefficients: the widths of the levels (fourth or higher order in $\delta F$) are not computed. The isolated levels of (I) need $r<1/(2\ell)$, so $r=1$ is reached only by formal coefficients. Code check (ii) uses members of (N$_h$) only.
- The mixed-branch bound level is treated by the same formulas, with the extra condition $\Delta E<\gamma$ and the exclusion $\bar M_1\approx\bar M_2$.
- The $(-,-)$ bound levels near the labels $\pm\pi/2$ (out of scope by the note of the question) and every bound level outside the continuum regime (C) are not treated.
- Item 3 is answered only for $a_b\ll\ell$ (deep regime); shallow pairs, over which the made field of the source changes by factors $e^{\pm a_b/\ell}$, are not treated. At the excluded cancellation of (7.14), the linear grip at $r=\sqrt2$, the corrections of (7.9) beside the lattice part $-2\langle k^2\rangle$ of (7.13) are not computed, so the fall there is not decided. The fall is compared with the members at first order in the grip; the comparison with the dressed members (their second-order weights and inertias, 06-mark-inertia) is not made.
- Only the ring; the cubic torus (where the static bond falls as $1/|r|$) is a later package.

## Code

- `code/v7/check_mutual_term.py` (role: check; unchanged from v6, rerun). Builds the total contract on a ring of 16 cells in the position basis of the two bodies and the occupation basis of the field modes with at most two quanta, computes the symmetric second-order effective contract by summing over all intermediate states (both branches), keeps the part with one vertex on each body, and compares it with (7.2) for 8 label triples in four descriptions. Output: `code/v7/output/mutual_term_check.json` (maximal relative deviation $3.9\times10^{-14}$); console log of the run in `code/v7/output/run_mutual_term.log`.
- `code/v7/check_bound_pair.py` (role: check; unchanged from v6, rerun; heavy members, $M_{k_1}=5.0$, $M_{k_2}=4.5$). Diagonalizes the effective joint contract (7.2)–(7.3) (without self-energies) in the sectors of the periodic ring, $P=2\pi m/L$ with the basis $p_1=2\pi n/L$, $p_2=P-p_1$ ($n=0,\dots,L-1$), on rings of $L=1024$ cells for $(+,+)$ and $L=3072$ for $(+,-)$ (whose small $|I|=|\bar M_1-\bar M_2|/\tau^2$ needs small $P$). It uses $m=0$ and three values of $m$ with $(I_1/I)m$ an integer (untwisted relative label), near $jhL/(2\pi)$, $j=1,2,3$, with $h=0.015|I|\tau$ for $(+,+)$ and $0.004|I|\tau$ for $(+,-)$. It takes the extreme level (lowest for $(+,+)$, highest for $(+,-)$), fits $E_b(P)-E_b(0)=aP^2+bP^4+cP^6$, and compares $D=1/(2a)-I_1-I_2$ and $E_b(P)$ with (7.9)–(7.10), using $\Delta E$ and $\langle T\rangle$ (relative label measured from the nearest valley) from the same eigenvector; it also records the regime estimates (7.8), (7.11), the valley splitting and the largest $|\Delta_j|/\omega_q$, $|\Delta_j|/\Omega_{ab}$ (at most $0.15$), the no-decay margin of (7.15), the necessary isolation test $2\sum_{\sigma_i=+}\bar M_i+\Delta E<m_h$ implied by (I), the lattice part over $\Delta E/\tau^2$, $D/I$ (at most $0.006$), $\epsilon_J$ (at most $1.7\times10^{-6}$ in the deep cases) and, for one shallow case, the deviation with twisted joint labels on rings of $L$ and $2L$ cells (finite-size demonstration). Parameters and results are listed in Step 8. Output: `code/v7/output/bound_pair_check.json`; console log of the run in `code/v7/output/run_bound_pair.log`.

## Methods used

- Plane-wave and normal-mode diagonalization of quadratic contracts
- Symmetric second-order effective contract (Schrieffer–Wolff/des Cloizeaux)
- Laplace (time) representation of energy denominators; Euclidean isotropy of the continuum propagator
- Expansion in the velocities of the bodies (form factors, retardation)
- Two-body bound levels on a chain: Airy (linear bond) and contact limits; virial theorem
- First-order perturbation theory in the joint label; Hellmann–Feynman theorem
- Exact diagonalization and polynomial fit (code checks)

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