Version 4 · earlier version; the current one is v7 · Review of v4

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

- **Subproject:** 08-ilang-propagation
- **Package:** 07-bond-inertia
- **Version:** v4
- **Mode:** internal regeneration
- **Date:** 2026-10-25

## Changes from previous version

- Step 8 ("One speed"), Result (7.12), twin 7.12: the speed statement of the isolated range carries its exception (review I1). For mixed branches the condition of (7.12) has the solutions $\lambda_{C,1}\lambda_{C,2}=4/(1-r^2)$, $r<1$; such levels lie in (I), where (N$_l$) bounds $\lambda_{C,i}$ only from below, and there the deep and the shallow speed coincide and one speed $\tau/\sqrt{2-r^2}$ holds for every binding strength. An example in (I) is given, and the levels that exclude the solution ((N$_l$) with $1-r^2\ge(r\ell)^{-2}$) are named.
- Step 8 ("Size of the lattice part", "One speed"), Result (7.11), (7.12), twin 7.11, 7.12: the parts of (N) now cover all of (N) (review I1). The deep and the shallow speed of (7.11) are stated first for every part of (N), and the third part is (N$_l$) with $r\ell$ not large, which contains (I) and, with formal coefficients, the levels with $r<1/(2\ell)$ outside (I) and those with $r\ell$ of order one; there the lattice part can be of order one and the $r^2$ terms are $O(\ell^{-2})$. The (N$_h$) shallow statement on the dominance of the lattice part is restricted to $(+,+)$, where it was derived.
- Code: both scripts are rerun unchanged into `code/v4/output/`; every cited number is checked against these files.

## 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 neglected fourth-order terms of the elimination vanish for held bodies with $F_2=0$ (a linear term on a quadratic contract is removed exactly by a shift of $q$), so they enter only through the motion of the bodies and through $F_2$; they are smaller by the factors $r^2|U(0)|/\bar M_i$ (recoil of the virtual quanta) and $u=2\max_iM_{k_i}|F_2|G(0)$, both assumed $\ll1$. The single-body self-energies at second order change $\bar M_i$ and $I_i$ at relative order $\delta F^2$; they cancel from $\Delta E$ and $D$ at the order used (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

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

Code check (i) (`code/v4/check_mutual_term.py`, output `code/v4/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 $m_{\rm red}\to-|m_{\rm red}|$; $\Delta E>0$):

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

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 the two-quantum slope ($\propto|r|\ln(\ell/|r|)$), smaller than $\lambda$ by $2F_2^2\ln(\ell/a_b)/(\pi F_1^2c_0)$, is dropped; shallow, relative $O(\ell/a_b)$. 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:* their $p^2$ parts renormalize $I_i$ in $P^2/(2I)$ exactly as for the bodies alone; their $p^4$ parts give $O(\delta F^2\langle k^2\rangle)$, of relative order $\delta F^2$.

With $1/I_{\rm pair}=E_b''(0)=1/I+2I^{-2}[\dots]$, 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 $\Delta E\ll0.045\,m_h$. 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 (the example above lies in (I)); 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/v4/check_bound_pair.py`, output `code/v4/output/bound_pair_check.json`): exact diagonalization of the effective contract (7.2)–(7.3) on a ring of 1024 cells, 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 ($\ell=30$, $a_b\approx93$, $\rho\approx-0.59$; $\ell=8$, $a_b\approx70$, $\rho\approx-0.81$), branches $(+,+)$ with both grips, and $(+,-)$ with the linear grip (the first two). 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 $11.2$, $53.2$, $72.8$ times $|\Delta E|/\tau^2$ in the three regimes and dominates $D$, as stated for (N$_h$) above: for $(+,+)$ at $r=1$, $D/(\Delta E/\tau^2)=-10.22,\,-52.26,\,-71.83$ (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 $\tilde q$. Over a pair of size $a_b$ much smaller than its distance to the source, $\phi$ is uniform up to its gradient (the midpoint factor $m_{\mathbf r}\to1$ of (2.18)), and 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 a member, $w_i=\mu_iF_1$ and $w_i/I_i=F_1\tau^2/\bar a$ for every level and both branches. For the pair, $w=w_1+w_2+\partial\Delta E/\partial\phi$ and, by Hellmann–Feynman on $k^2/(2m_{\rm red})+U$ ($m_{\rm red}\propto\bar a$, $U_G\propto F_1^2$, $U_J\propto F_2^2$): $\partial\Delta E/\partial\phi=\frac{F_1}{\bar a}(s\langle U\rangle-\langle T\rangle)$, at leading order in the grip ($U_J/U_G$ and $F_2w_0/F_0$ dropped, the order of (2.18)). For $s=1$ the static part is (2.18): the held bond is weighed like the rest cost. With (7.10), at first order in $\Delta E/\bar M$:

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

Deep ($\langle T\rangle/\langle U\rangle=O(a_b/\ell)$) and shallow ($\langle U\rangle=-2\langle T\rangle=2\Delta E$):

$$
\text{deep: the pair falls like its members}\iff s=2-r^2\ \ (\text{square grip: }r=1;\ \text{linear grip: }r=\sqrt2);\qquad \text{shallow: }\iff s=-2\tau^2I_1I_2=-\frac{2\sigma_1\sigma_2}{\lambda_{C,1}\lambda_{C,2}},
\tag{7.14}
$$

each in the range (N), at leading order in $a_b/\ell$ and $a_b/(\ell\lambda_{C,1}\lambda_{C,2})$ (deep; the $\langle T\rangle$ term of (7.13) carries $4\tau^2I_1I_2$) or in $\ell/a_b$ (shallow). In (N$_l$) with $r\ell\gg1$ the shallow condition is $s=0$ up to $O((r\ell)^{-2})$: the linear grip, at every such $r$. In (N$_h$), for $(+,+)$, $s=-2/(\lambda_{C,1}\lambda_{C,2})$ needs $F_2<0$, and then $\bar M_i>\Omega_{\max}$ gives $s<-2\Omega_{\max}^2/\tau^2<-8/r^2$: neither the linear nor the square grip falls like its members there. In (I) (isolated levels), the deep condition is $s=2-r^2$ with $r^2<1/(4\ell^2)$, met by neither the linear nor the square grip. The shallow condition is $s=-2\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2})$, with $\lambda_{C,i}$ unrestricted: for $(+,+)$ it needs $F_2<0$.

For the square grip in the shallow regime the binding cost is weighed three times ($\partial\Delta E/\partial\phi=3\frac{F_1}{\bar a}\Delta E$) but enters the inertia with the factor $1-4\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2})$. So for $(+,+)$ the ratio differs by $(2+4\tau^2I_1I_2)\Delta E/(\bar M_1+\bar M_2)$ at every $r$, and by $2\Delta E/(\bar M_1+\bar M_2)$ in (N$_l$) with $r\ell\gg1$. The answer to item 3 is therefore no in general. With the square grip, which weighs the held mark and bond (2.18), a $(+,+)$ pair in the two regimes of (7.14) falls like its members only at $r=1$ and in the deep regime. This holds in (N$_h$) with $a_b\ll\ell\lambda_{C,1}\lambda_{C,2}$, with formal coefficients; for isolated $(+,+)$ levels (I) it holds in neither regime. For mixed branches the square grip also meets the shallow condition at the tuned point $\lambda_{C,1}\lambda_{C,2}=2$. The remainder at $r=1$ in the deep regime is $-(2+4\tau^2I_1I_2)\langle T\rangle/(\bar M_1+\bar M_2)=O\bigl((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\bar M}\bigr)$.

## 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 fourth-order terms and self-energies negligible as in (B); continuum regime (C) and no Cherenkov denominators (D). 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 $O(\delta F^4)$.
- **(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 $1,\lambda_C\ll a_b\ll\ell$ (Airy, relative corrections $O(a_b/\ell)$) 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 $O(|\Delta E|/\bar M,(\lambda_C/a_b)^2,a_b^{-2},\ell^{-2},\delta F^2,u)$.
- **(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: $D=(2-r^2)\Delta E/\tau^2+O((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\tau^2})$; 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 lie in (I)); 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) Relative difference of weight over inertia between the pair and its members, at leading order in the grip and first order in $\Delta E/\bar M$, for a pair small against its distance to a distant held source.
- **(7.14)** (criterion) In the range (N), the pair falls like its members iff $s=4F_0F_2/F_1^2=2-r^2$ in the deep regime, at leading order in $a_b/\ell$ and $a_b/(\ell\lambda_{C,1}\lambda_{C,2})$ (square grip: iff $r=1$), and iff $s=-2\sigma_1\sigma_2/(\lambda_{C,1}\lambda_{C,2})$ in the shallow regime, at leading order in $\ell/a_b$. The shallow condition holds for the linear grip at every $r$ in (N$_l$) with $r\ell\gg1$, up to $O((r\ell)^{-2})$, and for $(+,+)$ in (N$_h$) only for grips with $s<-8/r^2$. In (I) (isolated levels) the deep condition is met by neither the linear nor the square grip. In these two regimes, with the square grip a $(+,+)$ pair falls like its members only at $r=1$ in the deep regime of (N$_h$) (formal coefficients; in neither regime in (I)), with remainder $O((1+\lambda_{C,1}^{-1}\lambda_{C,2}^{-1})\frac{a_b}\ell\frac{|\Delta E|}{\bar M})$. For mixed branches the square grip also falls like its members at $\lambda_{C,1}\lambda_{C,2}=2$ in the shallow regime.
- **(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 fourth-order terms of the elimination and the single-body self-energies are not computed; they are argued to be of relative order $r^2|U(0)|/\bar M$, $u$ and $\delta F^2$ (assumption (B)), and the code checks only the second-order effective contract.
- 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 uses a locally uniform made field ($m_{\mathbf r}\to1$); the corrections of the gradient over the pair, of relative order $(a_b/d)^2$ for a source at distance $d$, are not computed.
- Only the ring; the cubic torus (where the static bond falls as $1/|r|$) is a later package.

## Code

- `code/v4/check_mutual_term.py` (role: check; unchanged from v3, 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/v4/output/mutual_term_check.json` (maximal relative deviation $3.9\times10^{-14}$); console log of the run in `code/v4/output/run_mutual_term.log`.
- `code/v4/check_bound_pair.py` (role: check; unchanged from v3, 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) on a ring of $L=1024$ cells in the sector $P$, for $P=0,h,2h,3h$ with $h=0.015|I|\tau$ for $(+,+)$ and $h=0.004|I|\tau$ for $(+,-)$, where $|I|=|\bar M_1-\bar M_2|/\tau^2$ is small (labels $p_{1,2}=\frac{I_{1,2}}IP\pm k$), 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), and the lattice part over $\Delta E/\tau^2$. Parameters and results are listed in Step 8. Output: `code/v4/output/bound_pair_check.json`; console log of the run in `code/v4/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