Version 6 · earlier version; the current one is v7 · External review, round 2: minor issues

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

- **Subproject:** 08-ilang-propagation
- **Package:** 07-bond-inertia
- **Version:** v6
- **Mode:** external regeneration
- **Date:** 2026-10-25

## Changes from previous version

- Step 9, (7.13), (7.14), Result, Open issues: the made field of a distant held source on the pinned ring varies on the length $\ell$ at every distance; item 3 is now answered only for pairs with $a_b\ll\ell$ (the deep regime; note (a) of the question), the shallow fall criteria are removed, and the variation over the pair enters (7.13) at $O((a_b/\ell)^2|\Delta E|/\bar M)$ instead of $O((a_b/d)^2)$ (verification I1).
- Assumption (B), Step 7, Step 9, (7.9), (7.13), (7.14): $I_i$ is given at the order that $D$ needs (the first-order inertia; the second-order part $I_i^{(2)}$ cancels from $D$), with the order at which the members' own second-order terms enter $D$; (7.13)–(7.14) are stated as the contribution of the mutual term against the members at first order in the grip, with what a comparison with the dressed members would further need (note (b); I2).
- Step 6 ((7.8)), Step 9, Result: the deep regime requires $\epsilon_J\ll1$, the two-quantum slope small against the one-quantum slope; other grips are excluded from it (note (c); I3).
- Assumption (B), (7.3), Result (7.3), Open issues: the remainder of the elimination is $O(\delta F^3)$ (terms $\propto F_1^2F_2$, $F_2^3$, relative order $u$); $O(\delta F^4)$ only for $F_2=0$ (I4).
- Step 8 (code check (ii)), Code: `check_bound_pair.py` now diagonalizes the exact sectors of the periodic ring, $P=2\pi m/L$ with allowed labels $p_1=2\pi n/L$, $p_2=P-p_1$; rerun, new numbers (I5). `check_mutual_term.py` is rerun unchanged; every cited number is from `code/v6/output/`.

## Response to verification

- I1: Accepted and fixed in Step 9 and Result (7.13)–(7.14). On the pinned ring $G(x)\simeq\ell e^{-|x|/\ell}/(2c_0^2)$, so the made field varies on $\ell$, not on $d$. Item 3 is restricted to $a_b\ll\ell$ (the deep regime), as note (a) of the question asks; the shallow criteria are removed; the variation over the pair enters only through the bond, at relative order $(a_b/\ell)^2$, since the members are weighed at their own places and their term linear in the separation vanishes.
- I2: Accepted and fixed in assumption (B), Step 7 and Step 9, following note (b) of the question. Step 7 gives $I_i=\sigma_i\bar M_i/\tau^2+I_i^{(2)}$, shows that $I_i^{(2)}$ cancels from $D$, and states that the members' own second-order terms enter $D$ at relative order $\delta F^2$. (7.13)–(7.14) are now the contribution of the mutual term, against the members at first order in the grip. Step 9 states what a comparison with the dressed members would need: their second-order weights and inertias, which change $w_i/I_i$ at the order of (7.13) (held limit: $\partial_\phi\Sigma_i/w_i=-2\mu_iF_2G(0)$), and a common dressed ratio.
- I3: Accepted and fixed in Step 6 ((7.8)) and Result (7.8), (7.11), (7.14): the deep regime now requires $\epsilon_J=\frac{2F_2^2}{\pi F_1^2c_0}\ln\frac\ell{a_b}\ll1$ (it also makes the two-quantum constant small); grips with $\epsilon_J$ not small are excluded from it, as note (c) of the question allows.
- I4: Accepted and fixed in assumption (B), (7.3) and Result (7.3): remainder $O(\delta F^3)$, of relative order $u$; $O(\delta F^4)$ only for the linear grip.
- I5: Accepted and fixed in `code/v6/check_bound_pair.py` and Step 8: the sectors are now those of the periodic ring ($P=2\pi m/L$, $p_1$ on the grid, $p_2=P-p_1$), with $m$ chosen so that the relative label is not twisted; the script also shows the finite-size effect of twisted labels and its decrease with $L$; all numbers are from the new run.

## 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/v6/check_mutual_term.py`, output `code/v6/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 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 $c_0^2\to c_0^2-V^2$. Below and above the pinning length, with $U_G(r)=-\mu_1\mu_2F_1^2\ell e^{-|r|/\ell}/(2c_0^2)$:

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

Below $\ell$ the constant and the slope of the bond change with $P$ by different factors; above $\ell$ the bond's range shrinks to $\ell\sqrt{1-V^2/c_0^2}$, so at fixed $|r|\gg\ell$ the moving bond is weaker by $V^2|r|/(2c_0^2\ell)$.

### 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 expansion of $1/E_b''$ below needs $|\Delta E|\ll|\sigma_1\bar M_1+\sigma_2\bar M_2|$: automatic for $(+,+)$ by (B); for mixed branches it excludes the levels with $\bar M_1\approx\bar M_2$ (small $I$, large $|m_{\rm red}|$).

### 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 $|\Delta E|/\bar M$:

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

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 $r\le1$. Higher orders: $I_{\rm pair}-I_1-I_2=D+D^2/I+\dots$, the second term of relative order $|\Delta E|/\bar M$.

### 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/v6/check_bound_pair.py`, output `code/v6/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: with $w_{\rm pair}=w_1+w_2+w_b$, $I_{\rm pair}=I_1+I_2+D$ and (7.10), at first order in $\Delta E/\bar M$,

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

*Variation of the field over the pair.* 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) (midpoint factor $m_{\mathbf r}$). 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 $\phi(x_2)$. 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 $O(\langle\xi^2\rangle/\ell^2)$, and the variation of the field over the pair 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).

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

$$
\text{deep: }w_b=\kappa D,\ \text{i.e. the pair falls like its members at first order in the grip}\iff s=2-r^2\quad(\text{square grip: }r=1;\ \text{linear grip: }r=\sqrt2),
\tag{7.14}
$$

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 $\epsilon_J$. 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 $s=2-r^2$. 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 $-(2+4\tau^2I_1I_2)\langle T\rangle/(\sigma_1\bar M_1+\sigma_2\bar M_2)=O\bigl((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\bar M}\bigr)$.

*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 $O(\ell^{-2},a_b^{-2},\lambda_C^2/a_b^2)$.
- **(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 the 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 $4I_1I_2\langle T\rangle$, at first order in $|\Delta E|/\bar M$, relative corrections $O(|\Delta E|/\bar M,(\lambda_C/a_b)^2,a_b^{-2},\ell^{-2},\delta F^2,u)$. 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 $\kappa=w_i/I_i=F_1\tau^2/\bar a$, at leading order in the grip and first order in $\Delta E/\bar M$, 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); the variation of the field over the pair enters at $O((a_b/\ell)^2|\Delta E|/\bar M)$, at every distance of the source.
- **(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 $s=4F_0F_2/F_1^2=2-r^2$, at leading order in $a_b/\ell$, $a_b/(\ell\lambda_{C,1}\lambda_{C,2})$ and $\epsilon_J$ (square grip: iff $r=1$; linear grip: iff $r=\sqrt2$). In (I) (isolated levels) neither the linear nor the square grip meets it. 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. 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/v6/check_mutual_term.py` (role: check; unchanged from v5, 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/v6/output/mutual_term_check.json` (maximal relative deviation $3.9\times10^{-14}$); console log of the run in `code/v6/output/run_mutual_term.log`.
- `code/v6/check_bound_pair.py` (role: check; changed from v5 only in the choice of sectors and in the recorded diagnostics; 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 $\Delta E/\tau^2$, $\epsilon_J$ (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/v6/output/bound_pair_check.json`; console log of the run in `code/v6/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