Version 1 · earlier version; the current one is v3 · External review, round 1: major errors

# Two free bodies under the field they make

- **Subproject:** 07-ilang-weight
- **Package:** 02-two-free-bodies
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-20

## Setup and assumptions

- **Field** (assumption, M8): $N_f\ge1$ identical field objects ($c=+1$) on the ring $\mathbb Z_L$, $L\ge5$, no pair terms among them, own term $C_1=-t_f\sum_x(\lvert x\rangle\langle x+1\rvert+\mathrm{h.c.})$, $t_f>0$; $\Phi_x$ the number of field objects at $x$, $n:=N_f/L$.
- **Bodies** (assumption): Model D bodies $b_1,b_2$ on the same ring of position labels, in clock levels $k_i$ with half-gaps $M_i:=M^{(i)}_{k_i}>0$, motion amplitudes $\tau_i$, each coupled by (a) or (c) with a real constant $\gamma_i$ (either sign); no pair term between $b_1$ and $b_2$; records at $\kappa\to0$ (medium terms dropped). Write $S_i:=\sigma_3\otimes(m_i\mathbb 1+D_i)$, $\Gamma_i$ the coupling term of $b_i$, $A_i'$ its motion term ($A_i$ for (a); $A_i$ plus its $\gamma_i$-part for (c)), and
$$g_i:=\sigma_i\gamma_iM_i ,\qquad \sigma_i\in\{+1,-1\}\ \text{the branch of } b_i.$$
- **Held bodies** (A15): $\tau_i=0$ at fixed $x_1,x_2$. **Field state** (M9): lowest; highest where stated.
- **Inputs** (quoted in question.md): (4.2), (4.4) of 06-ilang-source/04-profile ($M$, $M^{(2)}=-\frac{2ng_1g_2}{t_f}\Lambda^+(x_1,x_2)$, $\Lambda^+$ the pseudo-inverse of the ring Laplacian $\Delta$); (6.10) of 06-ilang-source/06-own-mark (dressed curvature $E''_{k,\sigma}(0)$, error $O(\gamma^3)$); (7.1), (7.2), (7.7) of 06-ilang-source/07-dressed-fall.
- **Regime of item 3** (assumption, named R): **R1** (isolated band) in each sector of the conserved labels of Step 1 the $\tau=0$ group $\mathcal G=\{e_{jl}\}$, $e_{jl}:=\lvert j\,\sigma_1k_1,\ l\,\sigma_2k_2\rangle\otimes\Omega_{jl}$ ($\Omega_{jl}$ the lowest, resp. highest, field state of the held problem with $b_1$ at $j$, $b_2$ at $l$) is separated from all other $\tau=0$ levels by a gap $\delta>0$, and $\tau_i\ll\delta$; **R2** (order) accelerations at order $\tau_i^2$, relative errors $O(\tau^2/\delta^2)$; **R3** (starts) $\Psi(0)=\mathcal U\psi$, $\psi=\sum_{j,l}\phi^1_j\phi^2_l\,e_{jl}$ with real normalized envelopes, $\mathcal U$ the band transformation of Step 4. *First order in the place dependence* means: linear in quantities that vanish when the other body is uncoupled (the mutual term $M$ and the change of a body's hopping caused by the other body).
- **Gauges.** $\Omega_{jl}$ is real, positive for the lowest state (Perron–Frobenius) and $(-1)^y$-positive for the highest (on every object). With $U_b=\mathrm{diag}(1,i)$ on each branch label, $U_b\sigma_1U_b^\dagger=\sigma_2$, so $C$ is a real matrix and a real start has $\langle i[C,x_i]\rangle=0$: it starts from rest.

## Derivation

### Step 1. Conserved quantities (item 1)

Let $\mathcal T:=T^{(b_1)}T^{(b_2)}\prod_\alpha T^{(\alpha)}$, $T\lvert x\rangle=\lvert x+1\rangle$ on every position label. $A_i$, $B_i$ and $C_1$ are built from $\sum_x\lvert x+1\rangle\langle x\rvert$ and $\sum_x\lvert x\rangle\langle x\rvert$; under $\mathcal T$ the couplings go to $\sum_xP^{(i)}_{x+1}\otimes S_i\otimes\Phi_{x+1}$ and the (c) bond factors $\tfrac12(\Phi_x+\Phi_{x+1})$ to those of the shifted bond. $\mathcal T$ commutes with the swaps of field objects. Hence
$$[\mathcal T,C]=0,\qquad \mathcal T^L=\mathbb 1,\qquad \mathcal T=e^{-i\hat P},\quad P=p_1+p_2+\textstyle\sum_\alpha q_\alpha\in\tfrac{2\pi}{L}\mathbb Z_L , \tag{2.1}$$
the total plane-wave label of bodies and field is conserved. Further: $C$ acts on the clock label of $b_i$ only through $\mathbb 1$ and $D_i$, so $[D_i,C]=0$ and $k_i$ is conserved; the joint reflection $\mathcal R$ ($x\to-x$ on all labels, times $\sigma_3$ on both branch labels) commutes with $C$ (the reversal flips the sign of the $\sigma_1$ terms, $\sigma_3$ flips it back) and $\mathcal R\mathcal T\mathcal R^\dagger=\mathcal T^{-1}$, so every level satisfies $E(P)=E(-P)$; $C$ itself and $\sum_x\Phi_x=N_f$ are conserved. **Bodies alone.** For $\mathcal T_b:=T^{(b_1)}T^{(b_2)}$, which commutes with $C_{\rm D}^{(i)}$ and $C_1$,
$$\mathcal T_bC\mathcal T_b^\dagger-C=\sum_i\gamma_i\sum_xP^{(i)}_x\otimes S_i\otimes(\Phi_{x-1}-\Phi_x)\;+\;(\text{(c): bond terms, off-diagonal in }x_i)\neq0 \tag{2.2}$$
whenever $(\gamma_1,\gamma_2)\ne0$: $S_i$ is invertible ($m_i>0$), $\Phi_{x-1}\neq\Phi_x$, and terms on different position labels, or diagonal and off-diagonal ones, cannot cancel. So the plane-wave label of the bodies is not conserved (the field takes the recoil); the bodies alone conserve their clock levels $k_1,k_2$, and, when held, also $x_i$ and $\sigma_i$ ($C$ is then diagonal in them).

### Step 2. Held bodies: exact form (item 2)

At $\tau_i=0$ couplings (a) and (c) coincide (the (c) addition is proportional to $\tau_i$). In the sector $(x_i,\sigma_i,k_i)$, $S_i=\sigma_iM_i$ and $C=\sigma_1M_1+\sigma_2M_2+\sum_\alpha h^{(\alpha)}$ with $h=C_1+g_1P_{x_1}+g_2P_{x_2}$, i.e. (7.1) with $g_b\to g_1$, $g_s\to g_2$. Without pair terms the field eigenvalues are $\sum_jn_j\varepsilon_j(h)$, $\sum_jn_j=N_f$. By Perron–Frobenius ($-h$ irreducible with non-negative off-diagonal entries) $\varepsilon_0(h)$ is simple for all real $g_i$, hence analytic and connected to $\gamma_i=0$:
$$E(x_1,x_2)=\sigma_1M_1+\sigma_2M_2+N_f\,\varepsilon_0(x_1,x_2)=\mathrm{const}+M(x_1,x_2), \tag{2.3}$$
with $M$ of (4.2); the one-body terms do not depend on the positions because $C_1$ is translation invariant. The ring maps $x\to x+a$ and $x\to x_1+x_2-x$ leave $C_1$ invariant, the second exchanges the two places. Hence, **exactly** (every order in $g_i$):
$$M(x_1,x_2)=\mathcal M(\lvert x_2-x_1\rvert_L),\qquad M(g_1@x_1,g_2@x_2)=M(g_1@x_2,g_2@x_1),\qquad F_1=-F_2 , \tag{2.4}$$
where $F_i:=-\tfrac12[E(x_i+1)-E(x_i-1)]$ (other label fixed): for $E=f(x_2-x_1)$, $F_1=-\tfrac12[f(d-1)-f(d+1)]=-F_2$. The force with which $b_1$ acts on $b_2$ is opposite and equal to that with which $b_2$ acts on $b_1$. **Highest state.** For even $L$, $U_f:=\prod_\alpha(-1)^{y_\alpha}$ gives $U_fC_1U_f=-C_1$, $U_f\Phi_xU_f=\Phi_x$, and $\sigma_2$ anticommutes with $\sigma_1,\sigma_3$; so $\mathcal W:=U_f\,\sigma_2^{(1)}\sigma_2^{(2)}$ satisfies
$$\mathcal WC\mathcal W^\dagger=-C,\qquad M_{\rm top}(g_1,g_2)=-M_{\rm low}(-g_1,-g_2)\quad(L\ \text{even}). \tag{2.5}$$
The $(\sigma_1,\sigma_2,\text{highest})$ problem is the $(-\sigma_1,-\sigma_2,\text{lowest})$ problem with $C\to-C$; since $e^{-iC\lambda}\mathcal W\Psi=\mathcal We^{iC\lambda}\Psi$, accelerations from rest are equal. Odd $L$: Open issues.

### Step 3. The mutual term order by order (item 2)

Lowest state (highest: by (2.5)). $C_1+2t_f=t_f\Delta$, the lowest one-object state is uniform $u$, the gap is $4t_f\sin^2(\pi/L)$, and the reduced resolvent is $R=-\Lambda^+/t_f$. Rayleigh–Schrödinger perturbation theory in $V=g_1P_{x_1}+g_2P_{x_2}$ ($\lVert V\rVert\le\lvert g_1\rvert+\lvert g_2\rvert$) converges for $\lvert g_1\rvert+\lvert g_2\rvert<2t_f\sin^2(\pi/L)$ (Kato). On the ring ($D=1$), with $u:=d(L-d)$, $d\in\{0,\dots,L-1\}$:
$$\Lambda^+(d)=\frac{L^2-1}{12L}-\frac{u}{2L},\qquad (\Lambda^+)^2(d)=\frac{(L^2-1)(L^2+11)}{720L}-\frac{u}{12L}-\frac{u^2}{24L}. \tag{2.6}$$
Proof: the second differences of $u$ and $u^2$ are $-2$ and $2L^2+2-12u$ (use $(L-2d)^2=L^2-4u$) for $d\neq0$, so $\Delta$ maps the right-hand sides to $\delta_{d0}-\frac1L$ and to $\Lambda^+(d)$; the constants give zero mean ($\sum_du=L(L^2-1)/6$, $\sum_du^2=L(L^4-1)/30$). With (4.4):
$$M^{(2)}(d)=-\frac{2n}{t_f}g_1g_2\Lambda^+(d)=\frac{n\,g_1g_2}{6t_fL}\bigl[6u-(L^2-1)\bigr]. \tag{2.7}$$
Third order: $E^{(3)}=\langle u\vert VRVRV\vert u\rangle-E^{(1)}\langle u\vert VR^2V\vert u\rangle$, $V\lvert u\rangle=L^{-1/2}\sum_ig_i\lvert x_i\rangle$, $E^{(1)}=(g_1+g_2)/L$. Keeping the terms containing both $g_1$ and $g_2$ and using (2.6) ($\Lambda_d:=\Lambda^+(d)$, $(\Lambda^2)_d:=(\Lambda^+)^2(d)$):
$$M^{(3)}(d)=\frac{n}{t_f^2}g_1g_2(g_1+g_2)\Bigl[\Lambda_d^2+2\Lambda_0\Lambda_d-\tfrac{(\Lambda^2)_0+2(\Lambda^2)_d}{L}\Bigr]=\frac{n\,g_1g_2(g_1+g_2)}{60\,t_f^2L^2}\bigl[20u^2-10(L^2-2)u+(L^2-1)(L^2-4)\bigr]. \tag{2.8}$$
**Product form.** At second order $M^{(2)}=g_1\cdot g_2\cdot F_2(d)$, $F_2=-2n\Lambda^+/t_f$: one number per body ($g_i$, the same for acting and for being acted upon) times one function of $d$, for every pair of the class. At third order it fails:
$$\nexists\ p_1(g_1),\,p_2(g_2),\,F(d):\quad M=p_1p_2F(d)+O(g^4)\ \text{for all }g_1,g_2\text{ near }0 . \tag{2.9}$$
Proof: such a form makes $M(d)/M(0)=F(d)/F(0)$ independent of $g_i$ ($M^{(2)}(0)\ne0$ since $\Lambda_0>0$). Expanding, the $O(g_1+g_2)$ term of $M(d)/M(0)$ vanishes only if $B_3(d)\Lambda_0=B_3(0)\Lambda_d$ for all $d$, with $B_3$ the bracket of (2.8). $B_3$ is quadratic in $u$ with leading coefficient $1/(3L^2)\neq0$, $\Lambda$ is linear in $u$, and $u$ takes the three distinct values $0,L-1,2L-4$ at $d=0,1,2$ ($L\ge5$): impossible. (Among pairs with $g_1g_2\neq0$, $M^{(3)}\equiv0$ only for $g_1+g_2=0$.)
**Controls.** (2.4) fails in the wider class with a non-uniform on-site field term $\sum_xv_x\Phi_x$: the one-body terms then depend on $x_i$ and $F_1+F_2\neq0$. The second-order product form has no control in the class ((2.7) holds for all its members); its failure at third order is (2.9).

### Step 4. Band dynamics of the free bodies (item 3)

By R1 there is a Schrieffer–Wolff unitary $\mathcal U=e^{S}$, $S=-S^\dagger=O(\tau/\delta)$ off-diagonal with respect to the projector $\mathcal P_0$ on $\mathcal G$, such that $H:=\mathcal U^\dagger C\mathcal U$ is block diagonal; the band is $\mathcal U\,\mathrm{Ran}\,\mathcal P_0$ (standard). In the basis $e_{jl}$:
$$H_{\rm eff}=\mathcal P_0H\mathcal P_0=W+K_1+K_2+O(\tau^4),\qquad W=\mathrm{diag}\,E(j,l)\ \text{of (2.3)}. \tag{2.10}$$
$\mathcal P_0A_i'\mathcal P_0=0$ because $A_i'\propto\sigma_1$ flips $\sigma_i$; so the first-order part is $W$, and $K_i$, the second-order term of $A_i'$, moves $b_i$ by $r\in\{0,\pm2\}$ and leaves the other body fixed (a product $A_1'A_2'$ flips both branches and leaves $\mathcal G$). In the real gauge $K_1$ is real symmetric; let $\kappa(j,l):=\langle e_{j+1,l}\vert K_1\vert e_{j-1,l}\rangle=\bar\kappa+\delta\kappa(j,l)$, where $\bar\kappa$ is the same amplitude for $b_1$ alone with the field and $\delta\kappa\to0$ as $g_2\to0$. For $b_1$ alone the Bloch sums $\sum_je^{iPj}e_j$ have total label $P$ (2.1), so its dressed level is $E_{k_1,\sigma_1}(P)=\sum_r\bar\kappa_re^{-iPr}$ with $r\in\{0,\pm2\}$ at order $\tau^2$, and
$$E''_{k_1,\sigma_1}(0)=-\textstyle\sum_rr^2\bar\kappa_r=-8\bar\kappa , \tag{2.11}$$
the curvature (6.10). The same holds for $b_2$. For the highest state ($L$ even) the gauge gives $\mathcal Te_j=(-1)^{N_f}e_{j+1}$, which shifts $P$ by $\pi N_f$ but does not change (2.11) since $r$ is even, and by (2.5) $E''_{\rm top,\sigma}=-E''_{{\rm low},-\sigma}$, of sign $\sigma$.

### Step 5. Accelerations from rest (item 3)

$a_1:=\frac{\mathrm d^2}{\mathrm d\lambda^2}\langle x_1\rangle\big|_0=-\langle\Psi\vert[C,[C,x_1]]\vert\Psi\rangle=-\langle\psi\vert[H_{\rm eff},[H_{\rm eff},X_1]]\vert\psi\rangle$, $X_1:=\mathcal P_0\mathcal U^\dagger x_1\mathcal U\mathcal P_0$, because $H$ is block diagonal and $\psi\in\mathrm{Ran}\,\mathcal P_0$. Write $S=S_++S_-+S_2$ (parts moving $b_1$ by $\pm1$, and moving $b_2$). Then $[x_1,S_\pm]=\pm S_\pm$, $\mathcal P_0[x_1,S]\mathcal P_0=0$, and $\tfrac12\mathcal P_0[[x_1,S],S]\mathcal P_0=\mathcal P_0[S_+,S_-]\mathcal P_0$ moves nobody: $X_1=x_1+\xi_1$ with $\xi_1$ diagonal in $e_{jl}$ and $O(\tau^2/\delta^2)$. Since $W,x_1,\xi_1$ are diagonal and $[K_2,x_1]=0$, $a_1=-\langle\psi\vert[W,[K_1,x_1]]\vert\psi\rangle+O(\tau^4)$. With $\langle e_{j\pm1,l}\vert[K_1,x_1]\vert e_{j\mp1,l}\rangle=\mp2\kappa(j,l)$ this gives $a_1=8\sum_{j,l}\kappa\,\phi^1_{j+1}\phi^1_{j-1}(\phi^2_l)^2\,\partial_1W$. At first order in the place dependence $\kappa\to\bar\kappa$, and by (2.11):
$$a_1=-E''_{k_1,\sigma_1}(0)\sum_{j,l}\phi^1_{j+1}\phi^1_{j-1}(\phi^2_l)^2\,\partial_1W(j,l),\qquad a_2=-E''_{k_2,\sigma_2}(0)\sum_{j,l}(\phi^1_j)^2\phi^2_{l+1}\phi^2_{l-1}\,\partial_2W(j,l), \tag{2.12}$$
with $\partial_1W(j,l):=\tfrac12[W(j+1,l)-W(j-1,l)]$, $\partial_2$ likewise, $W$ the exact held eigenvalue (2.3) (lowest or highest), at order $\tau_i^2$ with relative errors $O(\tau^2/\delta^2)$, and $\partial_2W=-\partial_1W=\mathcal M'(l-j):=\tfrac12[\mathcal M(d+1)-\mathcal M(d-1)]$ by (2.4).

### Step 6. Ratio, inertias, rest costs and directions (item 3)

For real Gaussian envelopes $\phi^i_j\propto e^{-(j-\bar x_i)^2/4w_i^2}$ (width $w_i\ll L$), $\phi^i_{j+1}\phi^i_{j-1}=e^{-1/2w_i^2}(\phi^i_j)^2$ exactly on $\mathbb Z$, so both sums in (2.12) are $\mp e^{-1/2w_i^2}\sum_{j,l}(\phi^1_j\phi^2_l)^2\mathcal M'(l-j)$ and
$$\frac{a_1}{a_2}=-\frac{E''_{k_1,\sigma_1}(0)}{E''_{k_2,\sigma_2}(0)}\,e^{(w_2^{-2}-w_1^{-2})/2},\qquad w_1=w_2:\ \ \frac{a_1}{a_2}=-\frac{\mu_2}{\mu_1},\quad \mu_1a_1+\mu_2a_2=0,\quad \mu_i:=\frac1{E''_{k_i,\sigma_i}(0)} . \tag{2.13}$$
This holds for the exact $M$ (every order in $g_i$) and to the order to which $E''$ is known ((6.10): $O(\gamma^3)$). For other envelopes the factor is $\langle\mathcal M'\rangle_1/\langle\mathcal M'\rangle_2$ with the weights of (2.12). **Rest costs.** Let $R_i$ be half the difference of the $\tau=0$ eigenvalues of $b_i$ alone in its two branches (field lowest), a difference of eigenvalues (M6): $R_i=M_i+\tfrac{N_f}2[\varepsilon_0(\gamma_iM_i)-\varepsilon_0(-\gamma_iM_i)]=M_i(1+\gamma_in)+O(\gamma_i^3)$, since the odd part of $\varepsilon_0(g)$ is $g/L+O(g^3)$. From (6.10), $\frac1{1-x+x^2}=1+x+O(x^3)$ and $\Delta''=O(\gamma^2)$:
$$\mu_i=\sigma_i\frac{R_i}{\tau_i^2}+O(\gamma_i^2)\ \ \text{(a)},\qquad \mu_i=\sigma_i\frac{R_i}{\tau_i^2(1+\gamma_in)^2}+O(\gamma_i^2)\ \ \text{(c)}, \tag{2.14}$$
so for $w_1=w_2$, $a_1/a_2=-\sigma_1\sigma_2\,(R_2/\tau_2^2)/(R_1/\tau_1^2)\,[1+O(\gamma^2)]$ for (a), with $\tau_i\to\tau_i(1+\gamma_in)$ for (c). **Directions.** By (2.7), for $1\le d\le L-1$, $\mathcal M'(d)=\frac{ng_1g_2}{t_fL}(L-2d)$ at second order (exact central difference of a quadratic), $\mathcal M'(0)=0$. Take $\bar x_1<\bar x_2$ with the separations concentrated in $0<d<L/2$ (shorter arc). Then $a_1=E''_1\langle\mathcal M'\rangle_1$ and $a_2=-E''_2\langle\mathcal M'\rangle_2$ with $\langle\mathcal M'\rangle_i$ of the sign of $g_1g_2=\sigma_1\sigma_2\gamma_1\gamma_2M_1M_2$, and $\operatorname{sgn}E''_i=\sigma_i$ for small $\gamma_in$ (6.10). Hence, at second order in the couplings:
$$\text{lowest state: towards each other}\iff\sigma_1=\sigma_2=\operatorname{sgn}(\gamma_1\gamma_2);\quad\text{apart}\iff\sigma_1=\sigma_2=-\operatorname{sgn}(\gamma_1\gamma_2);\quad \sigma_1\neq\sigma_2:\ \operatorname{sgn}a_1=\operatorname{sgn}a_2 , \tag{2.15}$$
and for the highest state ($L$ even) the same with $\sigma_i\to-\sigma_i$, by (2.5).

### Step 7. Where the ratio deviates from the ratio of inertias (item 3)

At first order in the place dependence (2.13) holds for equal widths. The first coupling-dependent deviation is $\delta\kappa$, the change of $b_1$'s hopping caused by $b_2$; it multiplies $\partial W$ and is thus of second order in the place dependence. Leading term, coupling (a): in $\kappa$ the intermediate states are $b_1$ at $j$ in branch $-\sigma_1$ with field states $\chi$; expanding the symmetrized denominators $2\sigma_1M_1+(\varepsilon_{\Omega}-\varepsilon_\chi)$ around $2\sigma_1M_1$ and using $\sum_\chi\lvert\chi\rangle\langle\chi\rvert(\varepsilon_{\Omega_{j\mp1,l}}-\varepsilon_\chi)\Omega_{j\mp1,l}=g_1(\Phi_{j\mp1}+\Phi_j)\Omega_{j\mp1,l}$ gives $\kappa\propto\langle\Omega_{j+1,l}\vert\Omega_{j-1,l}\rangle-\frac{\gamma_1}{4}\langle\Omega_{j+1,l}\vert\Phi_{j-1}+2\Phi_j+\Phi_{j+1}\vert\Omega_{j-1,l}\rangle+O(\gamma_1^2)$. At leading order in $g_1$ the matrix element is the field density $n+\delta\rho_l(y)$ made by $b_2$ alone; the overlap depends on $l$ only at $O(g_1^2g_2)$. For (c) the bond factors add $+2\gamma_1\times$ the same average. Hence
$$\frac{\delta\kappa(j,l)}{\bar\kappa}=\mp\gamma_1\,\overline{\delta\rho}_l(j)\,[1+O(\gamma_1)],\qquad \overline{\delta\rho}_l(j):=\tfrac14[\delta\rho_l(j-1)+2\delta\rho_l(j)+\delta\rho_l(j+1)],\qquad \gamma_1\delta\rho_l(y)=\frac{\sigma_1M^{(2)}(y,l)}{M_1}+O(g_2^2), \tag{2.16}$$
upper sign (a), lower (c) (the last identity: $\delta\rho_l(y)=-\frac{2ng_2}{t_f}\Lambda^+(y-l)$ at first order, so $g_1\delta\rho_l=M^{(2)}$; highest state likewise with $M^{(2)}_{\rm top}$). Each $a_i$ of (2.12) acquires the factor $1\mp\gamma_i\overline{\delta\rho}$ inside its sum, and for broad equal-width starts the ratio becomes $-\frac{\mu_2}{\mu_1}\bigl[1\mp\bar M^{(2)}(\bar d)\bigl(\frac{\sigma_1}{M_1}-\frac{\sigma_2}{M_2}\bigr)\bigr]$: a deviation of relative order $\gamma_1\gamma_2$, which vanishes for $\sigma_1=\sigma_2$, $M_1=M_2$. Each body's inertia is changed in place by the other's mark. **Controls** of (2.13): unequal widths (factor $e^{(w_2^{-2}-w_1^{-2})/2}$), and (2.16).

## Result

- **(2.1)–(2.2)** [class: both couplings, all $N_f\ge1$, $L\ge5$; exact] Conserved: the total plane-wave label $P$ of bodies and field (joint translation $\mathcal T$), the clock levels $k_1,k_2$ ($D_1,D_2$), $C$, $\sum_x\Phi_x$; the joint reflection maps $P\to-P$. The plane-wave label of the bodies alone is not conserved for $(\gamma_1,\gamma_2)\ne0$; held bodies also conserve $x_i,\sigma_i$.
- **(2.3)–(2.4)** [held bodies, both couplings, lowest (likewise highest) state; exact, every order in $g_i$] The eigenvalue is $\sigma_1M_1+\sigma_2M_2+N_f\varepsilon_0$; it depends on the positions only through the ring distance, is unchanged when the two bodies exchange places, and $F_1=-F_2$ (action equals reaction).
- **(2.5)** [$L$ even, exact] $\mathcal WC\mathcal W^\dagger=-C$: the highest field state with branches $(\sigma_1,\sigma_2)$ is the lowest with $(-\sigma_1,-\sigma_2)$ and $C\to-C$; $M_{\rm top}(g_1,g_2)=-M_{\rm low}(-g_1,-g_2)$.
- **(2.6)–(2.8)** [ring, lowest state, $\lvert g_1\rvert+\lvert g_2\rvert<2t_f\sin^2(\pi/L)$] Closed forms of $\Lambda^+$, $(\Lambda^+)^2$, $M^{(2)}$ and $M^{(3)}$.
- **(2.9)** [same class] At second order the mutual term is a product $g_1g_2F_2(d)$ (yes); at third order no product $p_1(g_1)p_2(g_2)F(d)$ exists (no); $M^{(3)}=0$ for pairs with $g_1+g_2=0$.
- **(2.11)–(2.12)** [regime R1–R3, both couplings, lowest or (even $L$) highest state; order $\tau_i^2$, first order in the place dependence, relative errors $O(\tau^2/\delta^2)$] The drift accelerations from rest are $a_i=-E''_{k_i,\sigma_i}(0)\times$ the envelope-weighted symmetric gradient of the exact held eigenvalue, with $E''$ the curvature (6.10) of the isolated dressed body.
- **(2.13)–(2.14)** [same, real Gaussian envelopes of widths $w_i\ll L$, $\sum_{j,l}(\phi^1_j\phi^2_l)^2\mathcal M'(l-j)\neq0$] $a_1/a_2=-(E''_1/E''_2)e^{(w_2^{-2}-w_1^{-2})/2}$; for equal widths $a_1/a_2=-\mu_2/\mu_1$ and $\mu_1a_1+\mu_2a_2=0$, for every order in $g_i$ of the mutual term and to the order of (6.10); $\mu_i=\sigma_iR_i/\tau_i^2+O(\gamma_i^2)$ for (a), $\mu_i=\sigma_iR_i/[\tau_i(1+\gamma_in)]^2+O(\gamma_i^2)$ for (c), $R_i$ the rest cost.
- **(2.15)** [same, second order in the couplings, small $\gamma_in$, separations on the shorter arc] Lowest state: towards each other iff $\sigma_1=\sigma_2=\operatorname{sgn}(\gamma_1\gamma_2)$, apart iff $\sigma_1=\sigma_2=-\operatorname{sgn}(\gamma_1\gamma_2)$, same direction iff $\sigma_1\ne\sigma_2$; highest state ($L$ even): $\sigma_i\to-\sigma_i$.
- **(2.16)** [same; second order in the place dependence, leading order in $\gamma_1$] The first coupling-dependent deviation of the ratio from $-\mu_2/\mu_1$: the hopping of each body is changed by the other's mark by the relative amount $\mp\gamma_i\overline{\delta\rho}$, giving a relative deviation of order $\gamma_1\gamma_2$, $\mp\bar M^{(2)}(\sigma_1/M_1-\sigma_2/M_2)$ for broad equal-width starts.

## Consistency checks

1. **Dimensions.** $C,\tau_i,M_i,t_f,g_i\sim\lambda^{-1}$; $\gamma_i,n,\Lambda^+$ dimensionless; $M^{(2)}\sim g^2/t_f$, $M^{(3)}\sim g^3/t_f^2\sim\lambda^{-1}$; $E''\sim\tau^2/M\sim\lambda^{-1}$, so $a_i\sim\lambda^{-2}$ as $\mathrm d^2\langle x\rangle/\mathrm d\lambda^2$ with dimensionless $x$.
2. **Held source and test body.** For $\tau_2\to0$ and $\phi^2_l=\delta_{l,x_s}$, (2.12) is (7.7) with $W$ of (7.2), $g_s=g_2$. For $\gamma_1\to0$: $a_1=-(\sigma_1\tau_1^2/M_1)\langle\partial_1M^{(2)}\rangle=-\tau_1^2\gamma_1\langle\partial_1\delta\rho\rangle$, independent of $\sigma_1$ and $k_1$: the common local scale of 06-ilang-source at leading order.
3. **Equal bodies.** Same type, coupling, $\gamma$, branch and widths, mirror-symmetric start: $a_1=-a_2$ by the exchange symmetry, and (2.13) gives $-\mu_2/\mu_1=-1$.

## Open issues

- Odd $L$, highest state: the highest one-object level of $C_1$ is twofold degenerate; the mutual term then starts at first order in $g$ and is not treated.
- The coefficient of (2.16) beyond its leading order in $\gamma_i$ is not derived; the code finds corrections of relative first order in $\gamma_i$ that are large for coupling (c).
- R1 is assumed: for large $N_f$ the isolation from multi-object field excitations at distance $2M_i$ is not derived.
- Whether (2.13) holds beyond order $\tau^2$ for broad starts is not derived.

## Code

- `code/v1/check_held_pair.py` (check of Steps 2–3): $N_f=1$, $L=8$ and $L=13$, $t_f=1$, unequal $(g_1,g_2)=(0.04,-0.07),(0.05,-0.03),(0.03,0.06)$; exact $M$ from the lowest eigenvalue with two held point terms. Output `code/v1/output/held_pair.json`, `held_pair.log`. Results: (2.6) to $6\cdot10^{-15}$; translation, reflection, exchange, $F_1+F_2=0$ and the map (2.5) to $4\cdot10^{-15}$; $\max\lvert M-M^{(2)}-M^{(3)}\rvert$ falls by $15.9$–$16.4$ when $g$ is halved ($O(g^4)$); the odd part of $M$ in $g$ matches $M^{(3)}$ to $O(g^5)$; (odd part)$/M^{(2)}$ varies with $d$ (e.g. $L=13$: $-0.003$ to $-0.056$), as (2.9) requires.
- `code/v1/check_mutual_fall.py` (check of Steps 4–7): $N_f=1$, $L=14$, $t_f=0.5$, two Model D bodies and the field object, diagonalized in the $L$ sectors of (2.1); start (R3) by projecting $\psi$ on the band; exact $a_i=i\langle[C,v_i]\rangle$ and the slope of $\langle v_i\rangle(\lambda)$ from the joint evolution; prediction (2.12) with $E''_i$ from the band of the single body with the field. Output `code/v1/output/mutual_fall.json`, `mutual_fall.log`. Cases (R1 holds: $2M_i-4t_f\ge1.2$): A (a), $\sigma=(+,+)$, lowest; B (c), $(+,-)$, lowest; C (a), $(-,-)$, highest; D (c), $(+,+)$, widths $1.1,1.6$; E (c), $(+,+)$; A and E also with $\gamma_i/2$; $M_i\in[1.6,3]$, $\gamma_i\in[0.01,0.03]$, $\tau_i=s\,(0.08\text{–}0.15)$, $s=1,\tfrac12,\tfrac14$; centres $3,8$; equal widths $1.3$. Results: $v_i(0)\le10^{-17}$ (from rest); evolution slope = exact $a_i$ to $2.4\cdot10^{-7}$; (2.12) matches the exact $a_i$ to $2.0\cdot10^{-3}$ at $s=1$ and $1.2\cdot10^{-4}$ at $s=\tfrac14$ (the $\tau^2$ decrease of R2); $a_1/a_2$ matches $-E''_1/E''_2$ for equal widths to $3.0\cdot10^{-4}$ ($s=1$), $5\cdot10^{-5}$ ($s=\tfrac14$); D: $a_1/a_2=-0.53682$ against $-E''_1/E''_2=-0.66719$, factor $0.8046$ against $e^{(w_2^{-2}-w_1^{-2})/2}=0.8042$ (ring wrap); C reproduces A to all printed digits ((2.5)); A, C, D, E move towards each other, B in the same direction ((2.15)). The $\tau^2$ coefficient of $a_i/(2.12)-1$ (fit in $s^2,s^4$) against the leading term of (2.16): A $(-4.76,-3.61)$ vs $(-4.80,-3.29)$, A at $\gamma/2$ $(-1.13,-0.80)$ vs $(-1.16,-0.78)$, E $(3.28,1.71)$ vs $(4.80,3.29)$, E at $\gamma/2$ $(0.97,0.60)$ vs $(1.16,0.78)$, all $\times10^{-5}$: the deviation scales as $\gamma_1\gamma_2$ and the leading term is approached linearly in $\gamma$; ratio deviations: A $-1.15$ vs $-1.51$, E $1.57$ vs $1.51$, B $-6.39$ vs $-6.81$ ($\times10^{-5}$).

## Methods used

- translation, reflection and sublattice symmetries; Perron–Frobenius
- Rayleigh–Schrödinger perturbation theory; lattice Green function of the ring
- Schrieffer–Wolff (quasi-degenerate) perturbation theory; Heisenberg equations for mean positions
- Gaussian envelope identities; numerical diagonalization by sectors

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