Version 2 · earlier version; the current one is v3 · External review, round 2: minor issues

# Two free bodies under the field they make

- **Subproject:** 07-ilang-weight
- **Package:** 02-two-free-bodies
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-20

## Changes from previous version

- Setup: item 3 for bodies of different types and the highest state for even $L$ only (dated note of the question); R1 restated with the gaps $\delta_1,\delta$ and the single-body isolation; R3 with chart centres (I1, I2, I3, I6).
- Step 1: conserved sublattice-branch operators $Q_i$ (2.17); label-free conserved quantities for bodies of one type (I2, I4).
- Step 2: held bodies of one type: the eigenvalue as a function of the set of the two bodies (I2); highest state for even $L$ (I3).
- Step 4: Schrieffer–Wolff construction under the restated R1 (I6); hops by even numbers of places by (2.17).
- Step 5: the mean position is a chart mean position (2.18); the twist derivative replaces $[x_1,S_\pm]=\pm S_\pm$; (2.12) now carries the chart weights $\tfrac12\Delta\chi_i$ and the free-packet term $a_i^{(0)}$ (I1).
- Step 6: (2.13) for the mutual parts $a_i-a_i^{(0)}$, with the corrections from the places next to the cut (I1).
- Step 7: the ratio deviation is the force-weighted average (2.19), for different couplings of the two bodies (I5).
- Consistency check 3: two types with equal parameters (I2).
- Code: `check_mutual_fall.py` measures the chart mean position, the cut term, the free-packet term, the gaps of R1 and $[Q_i,C]$; both scripts re-run in `code/v2/` (I1, I4, I6).

## Response to verification

- **I1:** Accepted and fixed in Step 5 and Step 6. The mean position is now the chart mean position $\langle\hat x_i\rangle$ (2.18); its second derivative is the acceleration of the winding current plus an exact cut term; the band evaluation uses the twist derivative, which counts every hop as $\pm1$ by construction, and gives (2.12) with the chart weights and the free-packet term. The code evolves $\langle\hat x_i\rangle$ itself.
- **I2:** Accepted and fixed in Setup, Step 1, Step 2 and check 3. By the question's note (i), item 3 is asked for bodies of different types only, for which the labels are data; for one type, items 1 and 2 are restated with symmetric conserved quantities and with the eigenvalue as a function of the set of the two held bodies.
- **I3:** Accepted; resolved by the question's note (ii), which restricts the highest state to even $L$. Setup, Step 2 and the Result state this class; odd $L$ is listed as out of scope.
- **I4:** Accepted and fixed in Step 1, (2.17), with its consequence in Step 4.
- **I5:** Accepted and fixed in Step 7, (2.19): the force-weighted average is kept; no evaluation at the mean separation.
- **I6:** Accepted and fixed in Setup (R1, R2), Step 4 and the Code section: the single-body isolation is assumed explicitly, and the code reports all $\tau=0$ gaps, the exact band gaps and $\tau'/\delta_1$, $\tau'^2/(\delta_1\delta)$ for every run.

## 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). Items 1–2: of different types, or of one type (then with equal contract terms). Item 3: of different types (question note (i)), so the labels $1,2$ are data. 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 for even $L$ only (question note (ii)).
- **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 $O(\gamma^3)$, under the isolation condition (6.4) there); (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 $l$), has distance $\delta_1$ from the $\tau=0$ levels reached by one motion term (one branch flipped) and $\delta\le\delta_1$ from all other $\tau=0$ levels (same branches with an excited field; both branches flipped). Assume $\tau'\ll\delta_1$ and $\tau'^2/\delta_1\ll\delta$, with $\tau':=\max_i\tau_i(1+\lvert\gamma_i\rvert N_f)$ for (c) and $\max_i\tau_i$ for (a). Each body alone with the field satisfies the same with its own gaps (the isolation (6.4) of 06-ilang-source/06-own-mark, under which (6.10) holds).
  - **R2** (order). Accelerations at order $\tau_i^2$, plus the leading free-packet term of the chart (Step 5); relative errors $O(\tau'^2/\delta_1^2)+O(\tau'^2/(\delta_1\delta))$.
  - **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; a chart centre $c_i$ for each body (Step 5).
  - *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 $\langle i[C,\hat x_i]\rangle=0$ ($\hat x_i$ of Step 5): 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). **Sublattice-branch label** ($L$ even). $\Pi_i:=\sum_x(-1)^xP^{(i)}_x$ anticommutes with every hop of $b_i$, the hop across the closing bond included, and $\sigma_3^{(i)}$ anticommutes with $\sigma_1^{(i)}$; so $\Pi_i\sigma_3^{(i)}$ commutes with every term of $A_i'$ (the (c) bond factors act on the field). Every other term of $C$ is diagonal in $x_i$ and carries $\mathbb 1$ or $\sigma_3^{(i)}$ on the branch label of $b_i$. Hence
$$Q_i:=\Pi_i\,\sigma_3^{(i)},\qquad [Q_i,C]=0\quad(L\ \text{even}),\qquad Q_1e_{jl}=\sigma_1(-1)^je_{jl},\quad Q_2e_{jl}=\sigma_2(-1)^le_{jl}. \tag{2.17}$$
A change of $x_i$ by an odd number of places goes with a flip of $\sigma_i$. For odd $L$ the closing bond preserves $(-1)^x$ and $[Q_i,C]\neq0$. The bodies alone thus conserve $k_1,k_2$ and, for even $L$, $Q_1,Q_2$; held bodies also conserve $x_i,\sigma_i$ ($C$ is then diagonal in them). **One type.** For two bodies of one type, $[P_{12},C]=0$ (equal contract terms), and the labeled $x_i,D_i,Q_i$ are not observables (Law 5, M6); the conserved quantities of the bodies alone are their symmetric functions $D_1+D_2$, $D_1D_2$ and ($L$ even) $Q_1+Q_2$, $Q_1Q_2$, which commute with $P_{12}$ and $C$; $\mathcal T$ and $\mathcal R$ commute with $P_{12}$.

### 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 $b_1$. **One type.** For held bodies of one type $T$ at $(x_1,\sigma_1,k_1)\neq(x_2,\sigma_2,k_2)$, $P_{12}$ commutes with $C$ and leaves $h$ unchanged (it swaps the two point terms together with the bodies), so the admissible eigenvector is $2^{-1/2}(1+c_TP_{12})\lvert x_1\sigma_1k_1,x_2\sigma_2k_2\rangle\otimes\Omega$ with the eigenvalue (2.3) (for coinciding entries: the product itself if $c_T=+1$, no admissible state if $c_T=-1$). So $E$ is a function of the set $\{(x_1,\sigma_1,k_1),(x_2,\sigma_2,k_2)\}$; (2.4) holds with $F_i$ the change when the member $(x_i,\sigma_i,k_i)$ of the set is moved, and for equal internal states it reduces to the dependence on $\lvert x_2-x_1\rvert_L$: which body acts is then not data. **Highest state** ($L$ even). $U_f:=\prod_\alpha(-1)^{y_\alpha}$ 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 $e^{-iC\lambda}\mathcal W\Psi=\mathcal We^{iC\lambda}\Psi$ and $\mathcal W$ commutes with every $\hat x_i$, accelerations from rest are equal.

### 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 $\mathcal U=e^{S}$, $S=-S^\dagger$, built from $C$, such that $H:=\mathcal U^\dagger C\mathcal U$ is block diagonal with respect to the projector $\mathcal P_0$ on $\mathcal G$; the band is $\mathcal U\,\mathrm{Ran}\,\mathcal P_0$ (standard). Its first-order part, $O(\tau'/\delta_1)$, connects $\mathcal G$ to the one-flip levels; the levels at distance $\ge\delta$ are reached only at second order, with couplings $O(\tau'^2/\delta_1)\ll\delta$, and their elimination is a relative correction $O(\tau'^2/(\delta_1\delta))$ (R2). $\mathcal U$ commutes with $\mathcal T$ and, for even $L$, with $Q_i$. 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 $\mathcal G$); for even $L$, by (2.17), every term of $H_{\rm eff}$ moves each body by an even number of places. 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), which applies by the single-body part of R1. 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. Mean positions and accelerations from rest (item 3)

**Mean position.** For body $i$ let $\chi_i(x)$ be the representative of $x\in\mathbb Z_L$ in the window $\{c_i-\lfloor L/2\rfloor,\dots,c_i-\lfloor L/2\rfloor+L-1\}$ around its chart centre $c_i$ (R3), and let the *cut bond* $\beta_i$ join the last place $x_i^*$ of the window to $x_i^*+1$. The mean position is $\langle\hat x_i\rangle$, $\hat x_i:=\sum_x\chi_i(x)P^{(i)}_x$: a diagonal self-adjoint operator on the places of $b_i$, well defined for distinguishable bodies and independent of the common phase. Let $C(\theta)$ be $C$ with every term that moves $b_i$ one place forward multiplied by $e^{-i\theta}$ (backward: $e^{i\theta}$), $v_i:=\partial_\theta C(\theta)\vert_{0}$ the *winding current* of $b_i$, and $v_{i*}$ its part on $\beta_i$. Since $\chi_i(x+1)-\chi_i(x)=1$ except $1-L$ on $\beta_i$, and $\frac{\mathrm d}{\mathrm d\lambda}\langle O\rangle=\langle i[C,O]\rangle$ by (1.9), exactly:
$$i[C,\hat x_i]=v_i-L\,v_{i*},\qquad \frac{\mathrm d^2}{\mathrm d\lambda^2}\langle\hat x_i\rangle=\langle i[C,v_i]\rangle-L\,\langle i[C,v_{i*}]\rangle . \tag{2.18}$$
$[C,v_{i*}]$ acts on $b_i$ only at the four places within one place of $\beta_i$, so the cut term vanishes for states without weight there. We write $a_i:=\frac{\mathrm d^2}{\mathrm d\lambda^2}\langle\hat x_i\rangle\vert_0$; by the gauge remark of the Setup the start is at rest.

**Band evaluation.** $C(\theta)$ has the same $\tau=0$ part as $C$, so Step 4 applies to it: $C(\theta)=\mathcal U(\theta)H(\theta)\mathcal U(\theta)^\dagger$, and with $Z:=\mathcal U^\dagger\partial_\theta\mathcal U$, $v_i=\mathcal U(\partial_\theta H+[Z,H])\mathcal U^\dagger$. As $H$ is block diagonal and $\psi\in\mathrm{Ran}\,\mathcal P_0$, $\langle i[C,v_1]\rangle=\langle\psi\vert i[H_{\rm eff},\partial_\theta H_{\rm eff}]\vert\psi\rangle+\langle\psi\vert i[H_{\rm eff},[\zeta,H_{\rm eff}]]\vert\psi\rangle$ with $\zeta:=\mathcal P_0Z\mathcal P_0$. In $H_{\rm eff}(\theta)$ a term that moves $b_1$ by $r$ places, every hop counted, carries $e^{-ir\theta}$, so $\partial_\theta H_{\rm eff}=-i\sum_rrK_{1,r}$. The first-order part of $S(\theta)$ is $e^{-i\theta}S_++e^{i\theta}S_-+S_2$ ($S_\pm$ moving $b_1$ by $\pm1$, $S_2$ moving $b_2$): $\mathcal P_0\partial_\theta S\mathcal P_0=0$, and $\mathcal P_0[S,\partial_\theta S]\mathcal P_0\propto\mathcal P_0[S_+,S_-]\mathcal P_0$ moves nobody ($[S_\pm,S_2]$ flips both branches), so $\zeta$ is diagonal in $e_{jl}$ and $O(\tau^2/\delta_1^2)$, and its term is a relative $O(\tau'^2/\delta_1^2)$ correction. The cut term is the same with the twist on $\beta_1$ alone, i.e. with $r$ replaced by the signed number of crossings of $\beta_1$. Together, a hop of $K_1$ from $j-1$ to $j+1$ enters with the chart displacement $\Delta\chi_1(j):=\chi_1(j+1)-\chi_1(j-1)$, which is $2$, except $2-L$ for the two places $j\in\{x_1^*,x_1^*+1\}$ next to the cut. The $W$-part gives $4\sum_{j,l}\Delta\chi_1(j)\,\kappa\,\phi^1_{j+1}\phi^1_{j-1}(\phi^2_l)^2\,\partial_1W$, and at first order in the place dependence $\kappa\to\bar\kappa=-E''_1/8$ (2.11). The $K$-part vanishes for the winding current at this order ($[K_1,\sum_rrK_{1,r}]=0$ for $\kappa=\bar\kappa$), but not for the cut: with $K_1=\bar\kappa(T_1^2+T_1^{-2})+\mathrm{const}$ and $T_{1*}^{2}$ the part of $T_1^2$ that crosses $\beta_1$, it is $-L\bar\kappa^2\langle\psi\vert[T_1^2+T_1^{-2},T_{1*}^2-T_{1*}^{-2}]\vert\psi\rangle$, evaluated term by term. Hence
$$a_1=a_1^{(0)}-E''_{k_1,\sigma_1}(0)\sum_{j,l}\tfrac12\Delta\chi_1(j)\,\phi^1_{j+1}\phi^1_{j-1}(\phi^2_l)^2\,\partial_1W(j,l),\qquad a_1^{(0)}=-\frac{L}{64}\bigl(E''_{k_1,\sigma_1}(0)\bigr)^2\,\Xi_1 , \tag{2.12}$$
with $\Xi_1:=2\bigl[\phi_{a-1}\phi_{a+3}+\phi_a\phi_{a+4}-\phi_{a-3}\phi_{a+1}-\phi_{a-2}\phi_{a+2}+\phi_{a-1}^2+\phi_a^2-\phi_{a+1}^2-\phi_{a+2}^2\bigr]$ at $\phi=\phi^1$, $a=x_1^*$, and $a_2$ likewise (with $\phi^1\leftrightarrow\phi^2$, $\partial_2W$, $\Delta\chi_2$, $\Xi_2$). Here $\partial_1W(j,l):=\tfrac12[W(j+1,l)-W(j-1,l)]$, $\partial_2$ likewise, $W$ the exact held eigenvalue (2.3) (lowest or, $L$ even, highest), and $\partial_2W=-\partial_1W=\mathcal M'(l-j):=\tfrac12[\mathcal M(d+1)-\mathcal M(d-1)]$ by (2.4). The sum is of order $\tau_i^2$ with the relative errors of R2. $a_i^{(0)}$ does not involve the other body: it is the zeroth-order (free-packet) part, of order $\tau_i^4$, from the spreading of the packet across $\beta_i$; we call $a_i-a_i^{(0)}$ the *mutual part*. If $\phi^i$ vanishes on the four places within one place of $\beta_i$, then $a_i^{(0)}=0$, $\tfrac12\Delta\chi_i=1$ on every non-zero summand, and (2.12) coincides with the formula for the winding-current acceleration $\langle i[C,v_i]\rangle$.

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

For real Gaussian envelopes $\phi^i_j\propto e^{-(\chi_i(j)-c_i)^2/4w_i^2}$ (width $w_i\ll L$), $\phi^i_{j+1}\phi^i_{j-1}=e^{-1/2w_i^2}(\phi^i_j)^2$ for every $j$ except the two places next to $\beta_i$. So the mutual parts in (2.12) are $\mp e^{-1/2w_i^2}\sum_{j,l}(\phi^1_j\phi^2_l)^2\mathcal M'(l-j)$ up to the terms of those places, and
$$\frac{a_1-a_1^{(0)}}{a_2-a_2^{(0)}}=-\frac{E''_{k_1,\sigma_1}(0)}{E''_{k_2,\sigma_2}(0)}\,e^{(w_2^{-2}-w_1^{-2})/2},\qquad w_1=w_2:\ \ \frac{a_1-a_1^{(0)}}{a_2-a_2^{(0)}}=-\frac{\mu_2}{\mu_1},\quad \mu_i:=\frac1{E''_{k_i,\sigma_i}(0)} . \tag{2.13}$$
This holds for the exact $M$ (every order in $g_i$) and to the order to which $E''$ is known ((6.10): $O(\gamma^3)$), up to relative corrections at most of order $L\varpi_i\max\lvert\mathcal M'\rvert/\lvert\sum_{j,l}(\phi^1_j\phi^2_l)^2\mathcal M'\rvert$, $\varpi_i:=\sum_{j\in\{x_i^*,x_i^*+1\}}\lvert\phi^i_{j+1}\phi^i_{j-1}\rvert$, from the places next to the cuts (the summands there carry $\tfrac12\Delta\chi_i=1-L/2$ and violate the Gaussian identity). With $a_i^{(0)}=0$ (no envelope weight next to the cuts) it reads $\mu_1a_1+\mu_2a_2=0$. For other envelopes the factor is the ratio of the weighted sums 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 $w_1=w_2$ the ratio of the mutual parts is $-\sigma_1\sigma_2\,(R_2/\tau_2^2)/(R_1/\tau_1^2)\,[1+O(\gamma^2)]$ 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 $c_1<c_2$ with the separations concentrated in $0<d<L/2$ (shorter arc). Then the mutual parts are $E''_1\langle\mathcal M'\rangle_1$ and $-E''_2\langle\mathcal M'\rangle_2$ 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, for the mutual parts:
$$\text{lowest state: towards each other}\iff\sigma_1=\sigma_2=\operatorname{sgn}(\gamma_1\gamma_2);\quad\text{apart}\iff\sigma_1=\sigma_2=-\operatorname{sgn}(\gamma_1\gamma_2);\quad \sigma_1\neq\sigma_2:\ \text{same direction} , \tag{2.15}$$
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 $M^{(2)}_{\rm top}$). With $s_i:=+1$ for (a) and $-1$ for (c), the summand of the mutual part of $a_i$ in (2.12) acquires the factor $1-s_i\frac{\sigma_i}{M_i}\bar M^{(2)}(l-j)$, $\bar M^{(2)}(d):=\tfrac14[M^{(2)}(d-1)+2M^{(2)}(d)+M^{(2)}(d+1)]$. For equal-width Gaussian envelopes the common envelope factor cancels, and with $p_{jl}:=(\phi^1_j\phi^2_l)^2$:
$$\frac{a_1-a_1^{(0)}}{a_2-a_2^{(0)}}=-\frac{\mu_2}{\mu_1}\Bigl[1-\langle\!\langle\bar M^{(2)}\rangle\!\rangle\Bigl(\frac{s_1\sigma_1}{M_1}-\frac{s_2\sigma_2}{M_2}\Bigr)\Bigr],\qquad \langle\!\langle f\rangle\!\rangle:=\frac{\sum_{j,l}p_{jl}\,f(l-j)\,\mathcal M'(l-j)}{\sum_{j,l}p_{jl}\,\mathcal M'(l-j)} , \tag{2.19}$$
at leading order in $\gamma_i$ of $\delta\kappa$, up to $O\bigl((\bar M^{(2)}/M_i)^2\bigr)$ and the cut corrections of (2.13). The force-weighted average $\langle\!\langle\bar M^{(2)}\rangle\!\rangle$ is not in general the value at the mean separation. The deviation is of relative order $\gamma_1\gamma_2$; its leading term vanishes iff $s_1\sigma_1/M_1=s_2\sigma_2/M_2$ or $\langle\!\langle\bar M^{(2)}\rangle\!\rangle=0$. 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 (2.19).

## Result

- **(2.1), (2.2), (2.17)** [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 $k_1,k_2$ ($D_1,D_2$), $C$, $\sum_x\Phi_x$, and for even $L$ the sublattice-branch labels $Q_i=(\sum_x(-1)^xP^{(i)}_x)\sigma_3^{(i)}$; 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 $x_i,\sigma_i$. For bodies of one type the conserved quantities of the bodies alone are $D_1+D_2$, $D_1D_2$ and (even $L$) $Q_1+Q_2$, $Q_1Q_2$.
- **(2.3)–(2.4)** [held bodies, both couplings, lowest state, and highest state for even $L$; 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). For bodies of one type it is a function of the set of the two held bodies, and (2.4) holds for the members of that set.
- **(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 $g_1+g_2=0$.
- **(2.18)** [both couplings, bodies of different types, every state; exact] The chart mean position $\langle\hat x_i\rangle$ has the second derivative $\langle i[C,v_i]\rangle-L\langle i[C,v_{i*}]\rangle$: the acceleration of the winding current plus a cut term that acts only on the four places within one place of the cut bond.
- **(2.11)–(2.12)** [regime R1–R3, both couplings, bodies of different types, lowest or (even $L$) highest state; mutual part at order $\tau_i^2$ and first order in the place dependence, relative errors $O(\tau'^2/\delta_1^2)+O(\tau'^2/(\delta_1\delta))$] The drift acceleration from rest of the chart mean position is $a_i=a_i^{(0)}-E''_{k_i,\sigma_i}(0)\times$ the envelope-weighted symmetric gradient of the exact held eigenvalue with the chart weights $\tfrac12\Delta\chi_i$, with $E''$ the curvature (6.10) of the isolated dressed body; the free-packet part $a_i^{(0)}=-\frac{L}{64}E''^2_i\Xi_i$ (leading order, $\tau_i^4$) vanishes for envelopes without weight on the four places next to the cut, and then $\tfrac12\Delta\chi_i=1$ on every non-zero summand.
- **(2.13)–(2.14)** [same, real Gaussian envelopes of widths $w_i\ll L$, $\sum_{j,l}(\phi^1_j\phi^2_l)^2\mathcal M'(l-j)\neq0$; up to relative corrections at most of order $L\varpi_i\max\lvert\mathcal M'\rvert/\lvert\sum p\mathcal M'\rvert$ from the places next to the cuts] $(a_1-a_1^{(0)})/(a_2-a_2^{(0)})=-(E''_1/E''_2)e^{(w_2^{-2}-w_1^{-2})/2}$; for equal widths it is $-\mu_2/\mu_1$, 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; mutual parts] 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), (2.19)** [same; second order in the place dependence, leading order in $\gamma_i$ of $\delta\kappa$, equal-width Gaussian envelopes] The hopping of each body is changed by the other's mark by the relative amount $\mp\gamma_i\overline{\delta\rho}$ (upper sign (a), lower (c)); the ratio of the mutual parts deviates from $-\mu_2/\mu_1$ by the relative amount $-\langle\!\langle\bar M^{(2)}\rangle\!\rangle(s_1\sigma_1/M_1-s_2\sigma_2/M_2)$, of order $\gamma_1\gamma_2$, with the force-weighted average $\langle\!\langle\cdot\rangle\!\rangle$.

## Consistency checks

1. **Dimensions.** $C,\tau_i,M_i,t_f,g_i\sim\lambda^{-1}$; $\gamma_i,n,\Lambda^+,\chi_i,\Xi_i$ 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 both terms of $a_i$ in (2.12) are $\sim\lambda^{-2}$, as $\mathrm d^2\langle\hat x_i\rangle/\mathrm d\lambda^2$.
2. **Held source and test body.** For $\tau_2\to0$, $\phi^2_l=\delta_{l,x_s}$ and $\phi^1$ without weight next to the cut, (2.12) is (7.7) with $W$ of (7.2), $g_s=g_2$. For $\gamma_1\to0$: the mutual part is $-(\sigma_1\tau_1^2/M_1)\langle\partial_1M^{(2)}\rangle=-\tau_1^2\gamma_1\langle\partial_1\delta\rho\rangle$, independent of $\sigma_1$ and $k_1$: the common local scale of 06-ilang-source at leading order.
3. **Equal parameters.** Two types with equal parameters, coupling, branch and widths, envelopes mirror-symmetric about the midpoint and without weight next to the cuts: $\partial_2W=-\partial_1W$ and the mirror maps one sum of (2.12) to minus the other, so $a_1=-a_2$, and (2.13) gives $-\mu_2/\mu_1=-1$.

## Open issues

- Out of scope by the question's note: the highest state on odd rings (degenerate highest one-object level), and item 3 for two bodies of one type.
- The coefficient of (2.16), (2.19) 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; the free-packet part $a_i^{(0)}$ is given only at its leading order $\tau_i^4$.
- The list of conserved operators of the bodies alone is not shown to be complete.

## Code

- `code/v2/check_held_pair.py` (check of Steps 2–3; the v1 script, re-run): $N_f=1$, $L=8$ 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. Outputs `code/v2/output/held_pair.json`, `code/v2/output/held_pair.log`. Results: (2.6) to $6\cdot10^{-15}$; translation, reflection, exchange, $F_1+F_2=0$ and the map (2.5) ($L=8$) 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 $d$ ($L=13$: $-0.003$ to $-0.056$ for $(0.05,-0.03)$, $-0.012$ to $-0.253$ for $(0.03,0.06)$), as (2.9) requires.
- `code/v2/check_mutual_fall.py` (check of Steps 1 and 4–7): $N_f=1$, $L=14$, $t_f=0.5$, two Model D bodies of different types and the field object, diagonalized in the $L$ sectors of (2.1); start (R3) by projecting $\psi$ on the band; the joint state is rebuilt on the full space from the sectors. It computes the chart mean positions $\langle\hat x_i\rangle$ of (2.18) (chart centres $c_i$ = start centres $3,8$), the exact $a_i=-\langle[C,[C,\hat x_i]]\rangle$, the winding-current acceleration $\langle i[C,v_i]\rangle$, $\langle\hat x_i\rangle(\lambda)$ from the joint evolution on $[0,60]$, the exact zeroth-order part $a_i^{(0)}$ (same start, other body uncoupled), the prediction (2.12) with $E''_i$ from the band of the single body with the field, the $\tau=0$ gaps and exact band gaps, and $\lVert[Q_i,C]\rVert$ on body and field. Outputs `code/v2/output/mutual_fall.json`, `code/v2/output/mutual_fall.log`. Cases: A (a), $\sigma=(+,+)$, lowest; B (c), $(+,-)$, lowest; C (a), $(-,-)$, highest; D (c), $(+,+)$, widths $1.1,1.6$; E (c), $(+,+)$; A and E also with $\gamma_i/2$; $M_i\in[1.6,3]$, $\gamma_i\in[0.01,0.03]$, $\tau_i=s\,(0.08\text{–}0.15)$, $s=1,\tfrac12,\tfrac14$; equal widths $1.3$. Results:
  - (2.17): $\lVert[Q_i,C]\rVert=0$ for $L=14$ (both couplings), $\approx\tau_i$ ($0.10$–$0.15$) for $L=13$.
  - R1: $\delta_1=2.00$ (A, C, D, E), $3.00$ (B); $\delta=0.092$ (same branches, excited field; B also has both branches flipped at $0.094$); single bodies $\delta_1\ge2.00$, $\delta\ge0.095$; exact band gaps $0.088$–$0.095$; weight of the band outside $\mathcal G$ $\le1.2\cdot10^{-3}$; $\tau'/\delta_1\le0.076$; $\tau'^2/(\delta_1\delta)=0.05$–$0.13$ at $s=1$ (marginal for R1), $\le0.032$ at $s=\tfrac12$, $\le0.008$ at $s=\tfrac14$.
  - From rest: $\lvert\tfrac{\mathrm d}{\mathrm d\lambda}\langle\hat x_i\rangle\rvert\le6\cdot10^{-15}$; the second derivative of $\langle\hat x_i\rangle(\lambda)$ from the evolution equals the exact $a_i$ to $3\cdot10^{-8}$ (relative); weight on the four places next to the cut $1.2\cdot10^{-5}$–$2.4\cdot10^{-3}$, at most $2.5\cdot10^{-3}$ over the run.
  - (2.18): the cut term is $4\cdot10^{-5}$ to $0.11$ of $a_i$ (largest for D, body 2, $w_2=1.6$, $s=1$); $a_i^{(0)}$ of (2.12) matches the exact zeroth-order part to $1.2\%$.
  - (2.12) matches the exact $a_i$ to $2.0\cdot10^{-3}$ at $s=1$, $4.6\cdot10^{-4}$ at $s=\tfrac12$, $1.2\cdot10^{-4}$ at $s=\tfrac14$ (the $\tau^2$ decrease of R2).
  - (2.13): the ratio of the mutual parts differs from $-E''_1/E''_2$ by $-3.4\cdot10^{-4},-1.3\cdot10^{-4},-6.5\cdot10^{-5}$ (A, $s=1,\tfrac12,\tfrac14$), $1.0\cdot10^{-4},-3.9\cdot10^{-5},-7.8\cdot10^{-5}$ (B), $-3.0\cdot10^{-4},-9.9\cdot10^{-5},-3.7\cdot10^{-5}$ (E); the cut correction is $-3.2\cdot10^{-5}$ (A, C, E), $-2.8\cdot10^{-5}$ (B). D: factor $0.7992$ against the Gaussian factor $e^{(w_2^{-2}-w_1^{-2})/2}=0.8042$; (2.12) gives $0.7992$, the difference is the cut correction for $w_2=1.6$. Without subtracting $a_i^{(0)}$ the ratio $a_1/a_2$ is off by $2.3\%$ (A, $s=1$), $0.15\%$ ($s=\tfrac14$).
  - C reproduces A to all printed digits ((2.5)). A, C, D, E move towards each other and B in one direction, for $a_i$ and for the mutual parts ((2.15)).
  - (2.16), (2.19): the $\tau\to0$ extrapolation (fit in $s^2,s^4$) of $a_i/(2.12)-1$ against the leading term of (2.16): A $(-4.76,-3.61)$ vs $(-4.79,-3.29)$, A at $\gamma/2$ $(-1.13,-0.81)$ vs $(-1.16,-0.78)$, E $(3.28,1.71)$ vs $(4.79,3.29)$, E at $\gamma/2$ $(0.97,0.58)$ 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 against (2.19): A $-1.15$ vs $-1.51$, B $-6.39$ vs $-6.81$, D $1.11$ vs $1.28$, E $1.57$ vs $1.51$ ($\times10^{-5}$).

## Methods used

- translation, reflection, sublattice-branch and swap symmetries; Perron–Frobenius
- Rayleigh–Schrödinger perturbation theory; lattice Green function of the ring
- Schrieffer–Wolff (quasi-degenerate) perturbation theory; local chart and twist (winding-current) derivative; Heisenberg equations for mean positions
- Gaussian envelope identities; numerical diagonalization by sectors; reconstruction of the joint state from the sectors

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