Version 1 · earlier version; the current one is v7 · Review of v1

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

- **Subproject:** 08-ilang-propagation
- **Package:** 07-bond-inertia
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-25

## 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).
- (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: $r\min(1,\lambda_{C,i}/2)<1$ for $i=1,2$ (Step 2), so that the elimination holds on the whole band.

## 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/v1/check_mutual_term.py`, 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. 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.* (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 states with field quanta: 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; states reached only through two vertices give a width of order $\delta F^4$, beyond the order used. (iii) 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 is smaller than the members' part by $4\bar M_1\bar M_2/(3\tau^2)=4/(3\lambda_{C,1}\lambda_{C,2})$. 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.

*One speed.* 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. In the continuum part ($\lambda_{C,i}\to\infty$), (7.10) is $D=\Delta E/v^2$ with $\tau^2/v^2=2-r^2+(1-r^2)\rho$. Hence

$$
D=\frac{\Delta E}{v^2}\ \text{with one }v\text{ for every binding strength, level pair and grip}\iff r=1,\ \text{and then }v=\tau=c_0;\qquad \text{at }r=1:\ D=\frac{\Delta E}{\tau^2}+4I_1I_2\langle T\rangle .
\tag{7.12}
$$

For $r\neq1$ the speed depends on the binding: $v=\tau/\sqrt{2-r^2}$ in the deep regime ($r<\sqrt2$), $v=\tau$ in the shallow one; for $r\ge\sqrt2$ the deep leading term vanishes or changes sign and no real $v$ exists. At $r=1$ the lattice part breaks the equality by the relative amount $4|\rho|/(\lambda_{C,1}\lambda_{C,2})$, which is $O(a_b/\ell)$ small in the deep regime and $O(\lambda_C^{-2})$ in the shallow one. The grip enters (7.10) only through $U$, so the linear grip ($F_2=0$) obeys (7.10)–(7.12) unchanged; it is no control for item 2. Controls in the class: every $r\neq1$ (for example deep, $r=\frac12$: $D=\frac74\Delta E/\tau^2$), and $\lambda_{C,i}=O(1)$ at $r=1$ in the shallow regime (lattice part comparable to $\Delta E/\tau^2$). Cancellations, excluded: $r=\sqrt2$ in the deep regime (the leading term vanishes; $D=[-\tau^{-2}+4I_1I_2]\langle T\rangle$); $\lambda_{C,1}\lambda_{C,2}=4$ in the shallow regime (outside (C)); $\bar M_1\approx\bar M_2$ for mixed branches (Step 6 (iii)). For a common grip $U$ has no zero, so no separation is tuned away.

Code check (ii) (`code/v1/check_bound_pair.py`, output `bound_pair_check.json`): exact diagonalization of the effective contract (7.2)–(7.3) on a ring of 1024 cells ($\tau=1$, $M_{k_1}=0.5$, $M_{k_2}=0.4$, so $\lambda_{C,1}\lambda_{C,2}\approx5$; square and linear grip; $r=0.5,1,1.4$), for a deep ($\ell=30$, $a_b\approx10$, $\rho\approx-0.12$) and two shallower bound levels ($\ell=30$, $a_b\approx93$, $\rho\approx-0.59$; $\ell=8$, $a_b\approx70$, $\rho\approx-0.81$), branches $(+,+)$, and for $(+,-)$ with the linear grip. The curvature of the extreme level agrees with (7.10), with $\Delta E$ and $\langle T\rangle$ from the same eigenvector: $|D_{\rm num}-D_{(7.10)}|\le0.18\,|\Delta E|/\tau^2$ in the deep cases ($\le0.072$ for $(+,+)$ after removing the $D^2/I$ term of Step 7), where the expected corrections $|\Delta E|/|\sigma_1\bar M_1+\sigma_2\bar M_2|+(\lambda_C/a_b)^2$ are $\approx0.11$; $\le0.014\,|\Delta E|/\tau^2$ for $\ell=30$, $a_b\approx93$; $\le0.021\,|\Delta E|/\tau^2$ for $\ell=8$, where $O(\ell^{-2})\approx0.016$. $E_b(P)-E_b(0)$ agrees with $P^2/(2I_{\rm pair})$ to $1\%$ of the shift for $(+,+)$ and $0.3\%$ for $(+,-)$. For $(+,+)$ at $r=1$, $D/(\Delta E/\tau^2)=0.88,\,0.53,\,0.34$ in the three regimes ($D$ without the $D^2/I$ term) (formula $0.90,\,0.53,\,0.35$): the departure from $1$ is the lattice part of (7.12) at $\lambda_C\approx2$; at $r=0.5$ in the deep regime it is $1.51$ (formula $1.57$, $2-r^2=1.75$).

### 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 ($\langle U\rangle=-2\langle T\rangle=2\Delta E$):

$$
\text{deep: the pair falls like its members}\iff s=2-r^2\ \ (\text{square grip: }r=1;\ \text{linear grip: }r=\sqrt2);\qquad \text{shallow: }\iff s=-2\tau^2I_1I_2\ \ (\to0:\ \text{linear grip, every }r),
\tag{7.14}
$$

each at leading order in $a_b/\ell$ (deep) or $\ell/a_b$ (shallow).

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 once, so the ratio differs by $(2+4\tau^2I_1I_2)\Delta E/(\bar M_1+\bar M_2)$ at every $r$ ($2\Delta E/(\bar M_1+\bar M_2)$ in the continuum part). The answer to item 3 is therefore no in general: with the square grip, which weighs the held mark and bond (2.18), the pair falls like its members only at $r=1$ and in the deep regime; the remainder there is $-(2+4\tau^2I_1I_2)\langle T\rangle/(\bar M_1+\bar M_2)=O\bigl(\frac{a_b}\ell\frac{|\Delta E|}{\bar M}\bigr)$.

## 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).

- **(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 isolation conditions 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).
- **(7.12)** (criterion) $D=\Delta E/v^2$ with one speed for every binding strength, pair of levels and grip of the class iff $r=1$, then $v=\tau=c_0$, up to the lattice part $4I_1I_2\langle T\rangle$ (relative $4|\langle T\rangle/\Delta E|/(\lambda_{C,1}\lambda_{C,2})$); for $r\neq1$ the speed depends on the binding, from $\tau/\sqrt{2-r^2}$ (deep, $r<\sqrt2$) to $\tau$ (shallow), and no real speed exists in the deep regime for $r\ge\sqrt2$. Excluded cancellations: $r=\sqrt2$ (deep), $\lambda_{C,1}\lambda_{C,2}=4$ (shallow), $\bar M_1\approx\bar M_2$ (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) 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 $a_b/\ell$ (square grip: iff $r=1$), and iff $s=-2\tau^2I_1I_2\to0$ in the shallow regime, at leading order in $\ell/a_b$ (linear grip, every $r$); with the square grip the answer to item 3 is no except at $r=1$ in the deep regime, with remainder $O(\frac{a_b}\ell\frac{|\Delta E|}{\bar M})$.

## 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.
- The mixed-branch bound level is treated by the same formulas, with the extra isolation condition $\Delta E<\gamma$ and the exclusion $\bar M_1\approx\bar M_2$; its width from two-vertex processes (order $\delta F^4$) is not computed.
- The $(-,-)$ bound levels near the labels $\pm\pi/2$ 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/v1/check_mutual_term.py` (role: check). 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/v1/output/mutual_term_check.json` (maximal relative deviation $3.9\times10^{-14}$).
- `code/v1/check_bound_pair.py` (role: check). 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 $h=0.015|I|\tau$ (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 $0.93$). Parameters and results are listed in Step 8. Output: `code/v1/output/bound_pair_check.json`.

## 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