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

# Bound pair of two Model D bodies: bound level, curvature, cost and clock rates

- **Subproject:** 05-ilang-free-fall
- **Package:** 11-bound-pair
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## Setup and assumptions

- **Setting.** As in `question.md`: two Model D bodies $b_1$, $b_2$ of different types, one medium $c$, $C=C_{\rm pair}+C_{\rm rec}$, $C_{\rm pair}=C^{\rm D}_1+C^{\rm D}_2+U$, the binding $U$ of the class $\mathcal W$, fixed clock levels $k_1,k_2$ with $M_i=M^{(i)}_{k_i}$, the fibre contract $H(P)$, a bound level $E_b(P)$ with the eigenvector $\psi_P$, the curvature inertia $\mathfrak m_b=1/E_b''(0)$, $\mathfrak m_i=M_i/\tau_i^2$, and the regime $\mathsf R$ ($\tau_i=\epsilon\hat\tau_i$, $w_{s_1s_2}=\epsilon^2\hat w_{s_1s_2}$, $\epsilon\to0$). The bodies, the class $\mathcal W$, the diagonal on-site binding, $\mathsf R$ and every start are assumptions (M8, M9); a bound level is a distinguished set of states (A11). No field acts.
- **Inputs used.**
  - (5.5), (5.6) of 04-ilang-time/05-moving-clock: on a plane wave $\lvert p\rangle$ and a level $k$ the Model D contract acts as $-\tau\sin p\,\sigma_1+\mu_k\sigma_3$, with the eigenvalues $\pm\omega_k(p)$, $\omega_k=\sqrt{\mu_k^2+\tau^2\sin^2p}$. (5.11) there: $v_{\max}=\tau\kappa\sigma_X$. (5.1), (5.21) there: $P_N(a;\vartheta)$ and $q(a;\lambda)=P_N(a;\Gamma(p_0)\lambda/\lambda_0)+\mathcal R$, with $\Gamma(0)=1$.
  - (6.7), (6.20) of 04-ilang-time/06-inertia-internal-energy: the gap $2M_k$, the upper branch is the eigenvector $\lvert1\rangle$ of $\sigma_3$, and $1/\omega_k''(0)=M_k/\tau^2=I^{\rm lab}_k$.
  - (17.15) of 03-ilang-space/17-motion and (16.11) of 03-ilang-space/16-common-space: the position of either body is $\kappa\,\bar m_i\,e$ with $\bar m_i$ the mean position label, and $r=x_1-x_2$ is the separation of corresponding points.
  - (21.2), (21.12), (21.15), (21.16) of 03-ilang-space/21-composite-body, as quoted in the question.
- **Notation.** $\omega_i(p):=\sqrt{M_i^2+\tau_i^2\sin^2p}$, $\Omega_i:=\sqrt{M_i^2+\tau_i^2}$. For item 3: $a_i:=\tau_i^2/(4M_i)=1/(4\mathfrak m_i)$, $a:=a_1+a_2$, $j(P):=\sqrt{a_1^2+a_2^2+2a_1a_2\cos2P}$ (so $j(0)=a$), $w:=w_{11}$, $\varsigma:=\operatorname{sgn}w$, $u:=\lvert w\rvert/a=\lvert\hat w_{11}\rvert/\hat a$ and $\Lambda:=1+\varsigma/\sqrt{1+u^2}\in(0,2)$. In $\mathsf R$ the quantities $a_i,a,j,w$ are of order $\epsilon^2$ and $u,\Lambda$ of order one.
- **Standard facts used.** Weyl's theorem (a finite-range change does not change the bands); exponential decay of the eigenvector of an isolated eigenvalue of a finite-range operator; analytic perturbation theory of an isolated eigenvalue (first and second order); degenerate perturbation theory for an isolated group of eigenvalues (Schrieffer–Wolff), valid for $\tau_1+\tau_2+\max\lvert w_{s_1s_2}\rvert\ll2\min(M_1,M_2)$.
- **Simple level.** A bound level is taken simple (multiplicity one), which is the case in Step 7. For a degenerate level, (11.5)–(11.9) hold for each analytic branch of the varied parameter with its eigenvector.
- **Status words.** *Exact*: an identity for every member of the stated class. *Leading order in $\mathsf R$*: the stated remainder is neglected.

## Derivation

### Step 1. Fibre contract and two-body bands (item 1a)

On the levels $k_1,k_2$ a vector with the label $P$ of the joint shift is $\Psi(x_1,x_2)=e^{iP(\eta x_1+(1-\eta)x_2)}\psi(x_1-x_2)$ with $\psi(r)\in\mathbb C^2\otimes\mathbb C^2$. The real number $\eta$ is the phase convention. Changing it is the unitary map $\psi(r)\to e^{i(\eta-\eta')Pr}\psi(r)$, so the spectrum and every $\langle O\rangle_P$ below are independent of it. With $(A_i\Psi)(x_i)=\frac{\tau_i}2\bigl[-i\sigma_1\Psi(x_i-1)+i\sigma_1\Psi(x_i+1)\bigr]$ and $(S\psi)(r):=\psi(r-1)$:

$$
H(P)=\underbrace{\tfrac{i\tau_1}{2}\,\sigma^{(1)}_1\bigl(e^{i\eta P}S^{\dagger}-e^{-i\eta P}S\bigr)}_{A_1}+\underbrace{\tfrac{i\tau_2}{2}\,\sigma^{(2)}_1\bigl(e^{i(1-\eta)P}S-e^{-i(1-\eta)P}S^{\dagger}\bigr)}_{A_2}+M_1\sigma^{(1)}_3+M_2\sigma^{(2)}_3+\underbrace{W(\hat r)}_{U}.
\tag{11.1}
$$

$A_i$, $U$, $\sigma^{(i)}_3$ denote from here on these fibres. On $\psi(r)=e^{iqr}\beta$ the part without $W$, $H^0(P)$, acts as $h_P(q)=-\tau_1\sin p_1\,\sigma^{(1)}_1-\tau_2\sin p_2\,\sigma^{(2)}_1+M_1\sigma^{(1)}_3+M_2\sigma^{(2)}_3$ with $p_1=\eta P+q$, $p_2=(1-\eta)P-q$, $p_1+p_2=P$: each body as in (5.5).

For $\lvert r\rvert>R$, $H(P)$ acts as $H^0(P)$, so an eigenvector that is bounded and not square-summable is there a combination of plane waves with real $q$. By (5.6) their eigenvalues are $\sigma_1\omega_1(p_1)+\sigma_2\omega_2(p_2)$, $\sigma_i=\pm$. By Weyl's theorem this set is the same for every $W\in\mathcal W$. With $v=\sin^2q\in[0,1]$ at $P=0$ and $f(v):=\sqrt{M_1^2+\tau_1^2v}-\sqrt{M_2^2+\tau_2^2v}$:

$$
\mathcal B_{\sigma_1\sigma_2}(P)=\bigl\{\sigma_1\omega_1(p_1)+\sigma_2\omega_2(P-p_1)\bigr\}_{p_1\in(-\pi,\pi]},\qquad \mathcal B_{++}(0)=[\,M_1+M_2,\ \Omega_1+\Omega_2\,],\quad \mathcal B_{+-}(0)=[\,\min f,\ \max f\,],\quad \mathcal B_{--}=-\mathcal B_{++},\quad \mathcal B_{-+}=-\mathcal B_{+-}.
\tag{11.2}
$$

- The extremes of $f$ are among $f(0)=M_1-M_2$, $f(1)=\Omega_1-\Omega_2$ and, if $\tau_1\neq\tau_2$ and $v^{*}:=\frac{\tau_2^4M_1^2-\tau_1^4M_2^2}{\tau_1^2\tau_2^2(\tau_1^2-\tau_2^2)}\in(0,1)$, the stationary value $f(v^{*})=\operatorname{sgn}(\tau_1-\tau_2)\sqrt{(\tau_1^2-\tau_2^2)(\tau_2^2M_1^2-\tau_1^2M_2^2)}/(\tau_1\tau_2)$.
- The edge $M_1+M_2$ is attained at $q=0$ and at $q=\pi$.
- Exceptional members: for equal data ($M_1=M_2$, $\tau_1=\tau_2$) $\mathcal B_{\pm\mp}(0)=\{0\}$, a point of infinite multiplicity that has square-summable eigenvectors; for $\tau_i/M_i$ of order one the bands can overlap. In $\mathsf R$ the four bands have widths $O(\epsilon^2)$ around $\pm M_1\pm M_2$.

### Step 2. Replacements and well-defined quantities (item 1b)

Expand a change of the four listed terms with zero sum in products of operators on $b_1$, $b_2$, $c$. A component that is non-trivial on two objects occurs in one listed term only and must vanish. Hence, as in (21.2) of 03-ilang-space/21-composite-body, with single-object operators in place of the numbers there (the terms here are not diagonal in the places): with $T_1,\alpha$ self-adjoint on $\mathcal H_{b_1}$ and $T_2,\beta$ on $\mathcal H_{b_2}$,

$$
\delta C^{\rm D}_1=T_1,\qquad \delta C^{\rm D}_2=T_2,\qquad \delta C_{\rm rec}=\alpha+\beta,\qquad \delta U=-(T_1+\alpha)-(T_2+\beta),
\tag{11.3}
$$

and inside $C_{\rm rec}$ an operator $Z$ on $\mathcal H_c$ can move between its two pair terms.

- **What the range condition fixes.** If $U$ and $U+\delta U$ are in $\mathcal W$, then $\delta W(x_1-x_2)=-\theta_1(x_1)\otimes\mathbb 1-\mathbb 1\otimes\theta_2(x_2)$ with $\theta_i(x)$ on the branch label of $b_i$, and $\delta W=0$ for $\lvert x_1-x_2\rvert$ beyond the larger range. Every $x_1$ has such $x_2$ and conversely, so $\theta_1=t\,\mathbb 1$, $\theta_2=-t\,\mathbb 1$ with one real $t$, and $\delta U=0$. The range condition fixes $U$ completely. Without it, single-object terms can be listed in $U$, for example $W(r)\to W(r)+M_1\sigma^{(1)}_3$ with $B_1\to0$.
- **What remains.** $\delta C_{\rm pair}=-(\alpha+\beta)=-\delta C_{\rm rec}$ and the constant $t$ between the two bodies. For $C_{\rm rec}$ to stay a sum of record terms, $\alpha=\sum_x\alpha_x\lvert x\rangle\langle x\rvert$ with real numbers $\alpha_x$, and $\beta$ likewise: place-dependent terms that move between a body and its records (M6). The fibres need $C_{\rm pair}$ to commute with the joint shift, which gives $\alpha_x=c_1+gx$, $\beta_y=c_2-gy$, and the one operator $X$ in both record terms gives $g=0$. Within the stated forms only real constants remain: $\delta C^{\rm D}_1=t-c_1$, $\delta C^{\rm D}_2=-t-c_2$, $\delta C_{\rm pair}=-(c_1+c_2)$. The conventions $\eta$ (Step 1) and the choice $s$ of the correspondence ($r\to r+s$ relabels $W$) change no quantity below.

$$
\begin{array}{ll}
W(r) & \text{yes, by the range condition (no without it)}\\
M_i & \text{yes as half the gap } 2M_i \text{ of (6.7), the coefficient of } \sigma^{(i)}_3;\ \text{no as a cost of } b_i \text{ (branch costs } \pm M_i+t-c_i)\\
E_b(P) & \text{no: } \delta E_b=-(c_1+c_2)\\
E_b(P)-(M_1+M_2) & \text{yes, as the distance from the edge of } \mathcal B_{++}(0),\ \text{which is } M_1+M_2 \text{ in the listed form and shifts with } E_b\\
E_b(P)-E_b(0),\ E_b''(0) & \text{yes}\\
\langle\sigma^{(i)}_3\rangle_P & \text{yes: a constant does not change } \psi_P
\end{array}
\tag{11.4}
$$

### Step 3. Derivatives with respect to the data (item 2a)

$H(P)$ is linear in $M_i$, in $\tau_i$ (through $A_i$) and in a factor $\theta$ of $W$, and these terms are bounded. For a simple isolated eigenvalue, first-order perturbation theory (Hellmann–Feynman) gives, for every $W\in\mathcal W$ and every $P\in\mathcal P$:

$$
\frac{\partial E_b}{\partial M_i}=\langle\sigma^{(i)}_3\rangle_P,\qquad
\tau_i\frac{\partial E_b}{\partial\tau_i}=\langle A_i\rangle_P,\qquad
\frac{\partial E_b}{\partial\theta}\Big|_{\theta=1}=\langle U\rangle_P\qquad(i=1,2).
\tag{11.5}
$$

$H(P)$ at the data $(s\tau_i,sM_i,sW)$ is $s$ times $H(P)$ at $(\tau_i,M_i,W)$, with the same eigenvectors. So $E_b$ is homogeneous of degree one in $(\tau_1,\tau_2,M_1,M_2,\theta)$, and Euler's relation with (11.5) reads

$$
E_b(P)=M_1\langle\sigma^{(1)}_3\rangle_P+M_2\langle\sigma^{(2)}_3\rangle_P+\langle A_1\rangle_P+\langle A_2\rangle_P+\langle U\rangle_P .
\tag{11.6}
$$

### Step 4. First derivative in $P$ and the velocities (item 2b)

Let $V_i:=i[C_{\rm pair},\hat x_i]=i[A_i,\hat x_i]$; $B_i$, $U$ and $C_{\rm rec}$ commute with $\hat x_i$, so $\frac{\mathrm d}{\mathrm d\lambda}\langle\hat x_i\rangle=\langle V_i\rangle$ holds for the full $C$. $V_i$ commutes with the joint shift, with the fibres $V_1=-\frac{\tau_1}2\sigma^{(1)}_1(e^{i\eta P}S^\dagger+e^{-i\eta P}S)$ and $V_2=-\frac{\tau_2}2\sigma^{(2)}_1(e^{i(1-\eta)P}S+e^{-i(1-\eta)P}S^\dagger)$ (on plane waves $-\tau_i\cos p_i\,\sigma^{(i)}_1$). From (11.1), $\partial_PH=\eta V_1+(1-\eta)V_2$ and $V_1-V_2=i[H(P),\hat r]$. The eigenvector $\psi_P$ decays exponentially, so $\hat r\psi_P$ is square-summable and $\langle[H(P),\hat r]\rangle_P=0$. With Hellmann–Feynman in $P$:

$$
E_b'(P)=\langle V_1\rangle_P=\langle V_2\rangle_P .
\tag{11.7}
$$

A state of the pair built from the bound level is $\Phi=\int\mathrm dP\,g(P)\,\lvert P;\psi_P\rangle$ with $\int\lvert g\rvert^2=1$. Under $C_{\rm pair}$, $g(P)\to g(P)e^{-iE_b(P)\lambda}$, and $V_i$ has no elements between different $P$. Hence for every $\lambda$, with $P_0$ the mean and $\delta P$ the width of $\lvert g\rvert^2$:

$$
\frac{\mathrm d\bar m_1}{\mathrm d\lambda}=\frac{\mathrm d\bar m_2}{\mathrm d\lambda}=\int\mathrm dP\,\lvert g(P)\rvert^2E_b'(P)=E_b'(P_0)+O(\delta P^2).
\tag{11.8}
$$

The two bodies have the same mean velocity: yes, exactly under $C_{\rm pair}$, for every $W\in\mathcal W$, every bound level and every $g$. By (17.15) both velocities are $\kappa\,e\,\dot{\bar m}_i$. The records change the state, and (11.8), at $O(\kappa)$ for fixed $\lambda$.

### Step 5. Second derivative in $P$ (item 2c)

From (11.1), $\partial_P^2H=-\eta^2A_1-(1-\eta)^2A_2$ (a lattice property: $\partial_p^2\sin p=-\sin p$). Second-order perturbation theory, with $V_\eta:=\eta V_1+(1-\eta)V_2$, $Q_P:=\mathbb 1-\lvert\psi_P\rangle\langle\psi_P\rvert$ and $\mu$ the spectral measure of $H(P)$ in the vector $V_\eta\psi_P$:

$$
E_b''(P)=-\eta^2\langle A_1\rangle_P-(1-\eta)^2\langle A_2\rangle_P-2\,\mathcal S_\eta(P),\qquad
\mathcal S_\eta(P):=\bigl\langle V_\eta\psi_P\big\vert Q_P\bigl(H(P)-E_b\bigr)^{-1}Q_P\big\vert V_\eta\psi_P\bigr\rangle=\int_{\operatorname{spec}H(P)\setminus\{E_b\}}\frac{\mathrm d\mu(E)}{E-E_b}.
\tag{11.9}
$$

The sum runs over the bands and the other bound levels. Each of the two parts depends on the convention $\eta$; their sum does not (M6).

### Step 6. Equation for the bound levels of a diagonal on-site binding (item 3a)

$U=\lvert0\rangle\langle0\rvert\otimes W_0$, with $\lvert0\rangle$ the place $r=0$. For $E$ outside the bands, $(H^0+U-E)\psi=0$ is equivalent to $\psi=-(H^0-E)^{-1}\lvert0\rangle W_0\psi(0)$, hence to $[\mathbb 1_4+G\,W_0]\psi(0)=0$ with $G(E,P):=\langle0\rvert(H^0(P)-E)^{-1}\lvert0\rangle=\int\frac{\mathrm dq}{2\pi}(h_P(q)-E)^{-1}$; a solution gives $\psi(r)=-\int\frac{\mathrm dq}{2\pi}e^{iqr}(h_P(q)-E)^{-1}W_0\psi(0)$, which is square-summable. The two parts of $h_P(q)$ commute, so $(h_P(q)-E)^{-1}=\sum_{\sigma_1,\sigma_2}\Pi^{(1)}_{\sigma_1}\Pi^{(2)}_{\sigma_2}/(\sigma_1\omega_1+\sigma_2\omega_2-E)$ with $\Pi^{(i)}_\sigma=\frac12\bigl(\mathbb 1+\sigma(M_i\sigma^{(i)}_3-\tau_i\sin p_i\,\sigma^{(i)}_1)/\omega_i\bigr)$ and $\omega_i=\omega_i(p_i)$. Under $q\to q+\pi$ both $\sin p_i$ change sign and the $\omega_i$ do not, so the terms with one factor $\sin p_i$ integrate to zero. $G$ has the diagonal elements $G_{s_1s_2}$ and the only other elements $\langle11\vert G\vert22\rangle=\langle12\vert G\vert21\rangle=g_\times$ (and their transposes). The bound levels of $H(P)$ are the solutions $E$ outside the bands of

$$
\begin{aligned}
&F_+(E,P)\,F_-(E,P)=0,\qquad F_+:=(1+w_{11}G_{11})(1+w_{22}G_{22})-w_{11}w_{22}\,g_\times^2,\quad F_-:=(1+w_{12}G_{12})(1+w_{21}G_{21})-w_{12}w_{21}\,g_\times^2,\\
&G_{s_1s_2}=\int_{-\pi}^{\pi}\frac{\mathrm dq}{2\pi}\sum_{\sigma_1,\sigma_2=\pm}\frac{\bigl(1+(-1)^{s_1-1}\sigma_1\frac{M_1}{\omega_1}\bigr)\bigl(1+(-1)^{s_2-1}\sigma_2\frac{M_2}{\omega_2}\bigr)}{4\,(\sigma_1\omega_1+\sigma_2\omega_2-E)},\qquad
g_\times=\int_{-\pi}^{\pi}\frac{\mathrm dq}{2\pi}\sum_{\sigma_1,\sigma_2=\pm}\frac{\sigma_1\sigma_2\,\tau_1\tau_2\sin p_1\sin p_2}{4\,\omega_1\omega_2\,(\sigma_1\omega_1+\sigma_2\omega_2-E)} .
\end{aligned}
\tag{11.10}
$$

The multiplicity of a level is $\dim\ker(\mathbb 1_4+GW_0)$. The two factors belong to the two sectors of the exact symmetry $\Xi:=(-1)^{\hat r}\sigma^{(1)}_3\sigma^{(2)}_3$, $[\Xi,H(P)]=0$, which holds for every diagonal on-site binding because each $A_i$ changes $r$ by one and flips one branch. $F_+$ ($\Xi=+1$: equal branches on even $r$, different branches on odd $r$) contains only $w_{11},w_{22}$; $F_-$ ($\Xi=-1$) only $w_{12},w_{21}$. Class: every diagonal on-site binding, all data; exact.

### Step 7. Regime $\mathsf R$: the bound level at the band $\mathcal B_{++}$ (item 3b)

In $\mathsf R$, $H(P)=H_0+\epsilon H_1+\epsilon^2H_2$ with $H_0=M_1\sigma^{(1)}_3+M_2\sigma^{(2)}_3$, $\epsilon H_1=A_1+A_2$, $\epsilon^2H_2=U$. The eigenvalue $M_1+M_2$ of $H_0$ (projector $\Pi$ onto the branches $\lvert11\rangle$, all $r$) is separated from the other three by at least $2\min(M_1,M_2)$. All terms are bounded uniformly in $r$, so degenerate perturbation theory applies: the part of $H(P)$ with spectrum near $M_1+M_2$ is unitarily equivalent, by a unitary that differs from $\mathbb 1$ by $O(\epsilon)$, to

$$
H_{\rm eff}(P)=M_1+M_2+\Pi\,U\,\Pi+\Pi\Bigl(\frac{A_1^2}{2M_1}+\frac{A_2^2}{2M_2}\Bigr)\Pi+O(\epsilon^4)\quad\text{in norm}.
$$

Here $\Pi A_i\Pi=0$; $A_i$ leads to the branch value $M_1+M_2-2M_i$ of $H_0$; $\Pi A_1A_2\Pi=0$; and the odd orders vanish because $\sigma^{(1)}_3\sigma^{(2)}_3$ maps $H$ at $\epsilon$ to $H$ at $-\epsilon$. With $A_i^2=\tau_i^2\sin^2p_i=2M_ia_i(1-\cos2p_i)$, on the coefficient $\psi(r)$ of $\lvert11\rangle$:

$$
(H_{\rm eff}\psi)(r)=\bigl(M_1+M_2+a+w\,\delta_{r,0}\bigr)\psi(r)-\tfrac12\bigl[J\,\psi(r+2)+J^{*}\psi(r-2)\bigr],\qquad J(P):=a_1e^{2i\eta P}+a_2e^{-2i(1-\eta)P}=j(P)\,e^{i\alpha(P)} .
\tag{11.11}
$$

Of the four numbers only $w_{11}$ enters. Even and odd $r$ decouple, and the binding acts on even $r$ only. With $\psi(2n)=e^{-i\alpha n}\phi(n)$ this is a chain with the hopping $-\frac j2$ and one changed site: its band is $M_1+M_2+a+[-j,j]$ (the band $\mathcal B_{++}(P)$ at this order), and for $w\neq0$ it has exactly one further eigenvector, $\phi(n)\propto\rho^{\lvert n\rvert}$ with $\lvert\rho\rvert<1$. The equations at $n\neq0$ and at $n=0$ are $E-(M_1+M_2+a)=-\frac j2(\rho+\rho^{-1})$ and $\frac j2(\rho-\rho^{-1})=w$, hence $\rho=-\varsigma\bigl(\sqrt{w^2+j^2}-\lvert w\rvert\bigr)/j$ and

$$
E_b(P)=M_1+M_2+a+\varsigma\sqrt{w^2+j(P)^2}+O(\epsilon^4),\qquad j(P)^2=a_1^2+a_2^2+2a_1a_2\cos2P ,
\tag{11.12}
$$

$$
\psi_P(r)=\mathcal N_P\,e^{-i\alpha(P)\,r/2}\,\rho(P)^{\lvert r\rvert/2}\,\lvert11\rangle\ \ (r\ \text{even}),\qquad \psi_P(r)=0\ \ (r\ \text{odd}),\qquad \mathcal N_P^2=\frac{\lvert w\rvert}{\sqrt{w^2+j^2}},\qquad\text{up to } O(\epsilon).
\tag{11.13}
$$

The $O(\epsilon)$ part is $\bigl(\frac{A_1}{2M_1}+\frac{A_2}{2M_2}\bigr)\psi_P$: the branches $\lvert21\rangle$, $\lvert12\rangle$ on odd $r$. The odd orders of $E_b$ vanish, so the remainder in (11.12) is $O(\epsilon^4)$ against terms of order $\epsilon^2$.

- **Existence and number.** The level (11.12) is at the distance $\sqrt{w^2+j^2}-j$ from the band, of order $\epsilon^2$, and the remainder of $H_{\rm eff}$ is $O(\epsilon^4)$ in norm. So for small $\epsilon$: $H(P)$ has a bound level at a distance of order $\epsilon^2$ from $\mathcal B_{++}(P)$ if and only if $\hat w_{11}\neq0$. It is then exactly one, simple, for every $P\in(-\pi,\pi]$: below the band for $\hat w_{11}<0$, above the band for $\hat w_{11}>0$.
- **What the order leaves out.** Every other point of the spectrum near $M_1+M_2$ lies within $O(\epsilon^4)$ of the band. For $\hat w_{11}=0$, and in the sector $\Xi=-1$ (the branches $\lvert11\rangle$ on odd $r$, where $w_{12},w_{21}$ act at order $\epsilon^4$), a bound level, if any, is at such a distance and is not seen at leading order.
- **Other numbers.** $w_{22}$ enters $E_b$ at order $\epsilon^4$ (through $g_\times$ in $F_+$); $w_{12},w_{21}$ do not enter this level at any order.

### Step 8. Leading terms and the two limits (items 3c, 3d)

$E_b$ is analytic in the data with a remainder bounded uniformly near real data, so (11.12) can be differentiated and the remainders stay $O(\epsilon^4)$. From (11.5), (11.12), with $(j^2)'(0)=0$, $(j^2)''(0)=-8a_1a_2$, $\partial_{M_i}a=-a_i/M_i$, $\tau_i\partial_{\tau_i}a=2a_i$:

$$
E_{\rm bind}=a+\varsigma\sqrt{w^2+a^2},\quad E_b''(0)=-\varsigma\frac{4a_1a_2}{\sqrt{w^2+a^2}},\quad \langle\sigma^{(i)}_3\rangle_0=1-\frac{a_i}{M_i}\Lambda,\quad \langle A_i\rangle_0=2a_i\Lambda,\quad \langle U\rangle_0=\varsigma\frac{w^2}{\sqrt{w^2+a^2}},\qquad \Lambda=1+\frac{\varsigma\,a}{\sqrt{w^2+a^2}} .
\tag{11.14}
$$

Each is given up to $O(\epsilon^4)$; the terms shown (for $\langle\sigma^{(i)}_3\rangle_0$ the deviation from one) are of order $\epsilon^2$. The combination of the binding that enters is $u=\lvert w_{11}\rvert/a$. Limits within $\mathsf R$, with relative corrections $O(u^2)$ (wide) and $O(u^{-2})$ (tight):

$$
\begin{array}{l|ccc}
 & \text{wide } (u\to0),\ w<0 & \text{wide } (u\to0),\ w>0 & \text{tight } (u\to\infty)\\ \hline
E_{\rm bind} & -\dfrac{w^2}{2a} & 2a+\dfrac{w^2}{2a} & w+a+\dfrac{a^2}{2w}\\
E_b''(0) & \dfrac{4a_1a_2}{a}\bigl(1-\tfrac{u^2}2\bigr) & -\dfrac{4a_1a_2}{a}\bigl(1-\tfrac{u^2}2\bigr) & -\dfrac{4a_1a_2}{w}\\
1-\langle\sigma^{(i)}_3\rangle_0 & \dfrac{a_i}{M_i}\dfrac{u^2}2 & \dfrac{2a_i}{M_i} & \dfrac{a_i}{M_i}\bigl(1+\tfrac aw\bigr)\\
\langle A_i\rangle_0 & a_iu^2 & 4a_i & 2a_i\bigl(1+\tfrac aw\bigr)\\
\langle U\rangle_0 & -\dfrac{w^2}{a} & \dfrac{w^2}{a} & w-\dfrac{a^2}{2w}
\end{array}
\tag{11.15}
$$

### Step 9. Curvature inertia, inertias of the parts and cost (item 4)

**(a)** With $4a_1a_2/a=1/(\mathfrak m_1+\mathfrak m_2)$, $1/(4a_1a_2)=4\mathfrak m_1\mathfrak m_2$ and $\varsigma\sqrt{w^2+a^2}=E_{\rm bind}-a$, (11.14) gives, at leading order in $\mathsf R$ (relative remainder $O(\epsilon^2)$), for both signs of $w_{11}$:

$$
\mathfrak m_b=-\varsigma\,(\mathfrak m_1+\mathfrak m_2)\sqrt{1+u^2}=\mathfrak m_1+\mathfrak m_2-4\,\mathfrak m_1\mathfrak m_2\,E_{\rm bind} .
\tag{11.16}
$$

The difference $\mathfrak m_b-(\mathfrak m_1+\mathfrak m_2)$ has the sign of $-w_{11}$, which is the sign of $-E_{\rm bind}$. For $w_{11}<0$ it is $(\mathfrak m_1+\mathfrak m_2)(\sqrt{1+u^2}-1)>0$, which grows from $(\mathfrak m_1+\mathfrak m_2)u^2/2$ (wide) to $(\mathfrak m_1+\mathfrak m_2)u$ (tight). For $w_{11}>0$, $\mathfrak m_b<0$ and the difference is $-(\mathfrak m_1+\mathfrak m_2)(\sqrt{1+u^2}+1)$. Through $E_{\rm bind}$ it is $-4\mathfrak m_1\mathfrak m_2E_{\rm bind}$, for a wide pair and for every $u$. All terms of (11.16) are of order $\epsilon^{-2}$.

**(b)** Let $\tau_1=\tau_2=\tau$, so $\tau^2(\mathfrak m_1+\mathfrak m_2)=M_1+M_2$ and $4\tau^2\mathfrak m_1\mathfrak m_2=4M_1M_2/\tau^2$. From (11.16) and (11.14):

$$
\tau^2\mathfrak m_b=M_1+M_2-\frac{4M_1M_2}{\tau^2}E_{\rm bind}+O(\epsilon^2),\qquad M_1+M_2+E_{\rm bind}=M_1+M_2+O(\epsilon^2),\qquad \sum_iM_i\langle\sigma^{(i)}_3\rangle_0=M_1+M_2-a\Lambda+O(\epsilon^4).
\tag{11.17}
$$

- $\tau^2\mathfrak m_b$ against the cost $M_1+M_2+E_{\rm bind}$: not at all. The leading difference is $-\frac{4M_1M_2}{\tau^2}E_{\rm bind}=(M_1+M_2)\bigl(-\varsigma\sqrt{1+u^2}-1\bigr)$, of order $\epsilon^0$, while the binding changes the cost by $E_{\rm bind}=O(\epsilon^2)$ with the opposite sign. The two agree only in the limit $u\to0$ with $w_{11}<0$, and exactly without a binding (d).
- $\tau^2\mathfrak m_b$ against $\sum_iM_i\langle\sigma^{(i)}_3\rangle_0$: not at all, with the same leading difference.
- The cost against $\sum_iM_i\langle\sigma^{(i)}_3\rangle_0$: they agree at order $\epsilon^0$ and differ at order $\epsilon^2$ by $\langle A_1\rangle_0+\langle A_2\rangle_0+\langle U\rangle_0=2a\Lambda+\varsigma w^2/\sqrt{w^2+a^2}$; this difference is exact by (11.6).

**(c)** Comparison with (21.15), (21.16) of 03-ilang-space/21-composite-body, in label units ($\mathfrak m\leftrightarrow\sigma_X^2I$).
- Common: the same function, $\mathfrak m_b/(\mathfrak m_1+\mathfrak m_2)=\sqrt{1+u^2}$ against $I_c/2I_1=\sqrt{1+U^2/16t^2}$, where in both cases the binding is divided by the half-width of the two-body band at total label zero ($a$ here, $4t$ there); an attractive binding raises the inertia, and the excess is linear in $E_{\rm bind}$ for every strength.
- Different, the coefficient: there $I_c-2I_1=-E_{\rm bind}/v_{\max}^2$, which is $-\mathfrak m_1\mathfrak m_2E_{\rm bind}$; here $-4\mathfrak m_1\mathfrak m_2E_{\rm bind}$. The factor $4$ arises because the upper branch moves by two labels per step (11.11), so $a=\frac14(1/\mathfrak m_1+1/\mathfrak m_2)$. With (5.11) the coefficient here is $4M_1M_2/\tau^4$ for $\tau_1=\tau_2$: it contains the rest costs and is not $1/v_{\max}^2$.
- Different, the cost: there, without rest terms, $I_cv_{\max}^2=\sqrt{U^2+16t^2}=-E_{\min}(t,U)$ by (21.12), for every $U$. Here a rest cost exists, and by (11.17) $\tau^2\mathfrak m_b$ does not follow the cost of the pair.
- Different, the notion: (21.15) is an inertia from a response; $\mathfrak m_b$ is a curvature.

**(d) Control, no binding.** For $W=0$ there is no bound level; the lower edge of $\mathcal B_{++}(P)$ is $e(P)=\min_{p_1}[\omega_1(p_1)+\omega_2(P-p_1)]$. At the minimum $\omega_1'(p_1)=\omega_2'(p_2)$, and for a minimum over a split of $P$ the inverse curvatures add. With (6.7), exactly and for all data:

$$
\frac1{e''(0)}=\mathfrak m_1+\mathfrak m_2=\frac{M_1}{\tau_1^2}+\frac{M_2}{\tau_2^2},\qquad e(0)=M_1+M_2,\qquad \langle\sigma^{(i)}_3\rangle=1 .
\tag{11.18}
$$

For $\tau_1=\tau_2=\tau$ the three quantities of (b) agree exactly: $\tau^2/e''(0)=M_1+M_2$. For $\tau_1\neq\tau_2$ they do not: $\tau_1^2/e''(0)-(M_1+M_2)=M_2(\tau_1^2/\tau_2^2-1)\neq0$, and the ratio $1/[e''(0)(M_1+M_2)]$ lies strictly between $\tau_1^{-2}$ and $\tau_2^{-2}$ and depends on $M_1/M_2$, hence on the clock levels. So the difference in (b) is caused by the binding, and for $\tau_1\neq\tau_2$ the comparison with a cost fails already without one. (11.16) holds for $\tau_1\neq\tau_2$ unchanged.

### Step 10. Clocks of the parts (item 5)

**(a)** Start: the clock label of the part $i$ in $\lvert0\rangle=N_i^{-1/2}\sum_k\lvert f_k\rangle$, the other part on its level $k_j$, the medium in $\chi$, and the positions and branches in $\Phi$ of Step 4, built from the bound eigenvectors of the levels $(0,k_j)$ with $\lvert g\rvert^2$ of width $\delta P$ around $0$. The levels are conserved, so the component on $\lvert f_k\rangle$ evolves with $M_i\to m_i+d^{(i)}_k$. Let $g_0$ be the distance of $E_b(0)$ from the rest of the spectrum and $\delta_i:=d^{(i)}_{N_i-1}/g_0$. By (11.5) and perturbation theory in $d^{(i)}_k$, the bound eigenvector changes by $O(\delta_i)$ and $E^{[k]}_b(P)=E^{[0]}_b(P)+d^{(i)}_k\langle\sigma^{(i)}_3\rangle_P+O\bigl((d^{(i)}_k)^2/g_0\bigr)$. For a diagonal on-site binding $\psi(r)\to\sigma^{(1)}_3\sigma^{(2)}_3\psi(-r)$ maps $H(P)$ to $H(-P)$, so $E_b$ and $\langle\sigma^{(i)}_3\rangle_P$ are even in $P$: the pair is at rest by (11.8), and $\langle\sigma^{(i)}_3\rangle_P=\langle\sigma^{(i)}_3\rangle_0+O(P^2)$. The reading of the clock label of $i$ (the partition by $a_i$, (1.3)) has the statistics $N_i^{-2}\sum_{k,k'}e^{2\pi i(k-k')a/N_i}\langle\Psi_{k'}(\lambda)\vert\Psi_k(\lambda)\rangle$ with $\langle\Psi_{k'}\vert\Psi_k\rangle=e^{-i(d_k-d_{k'})\langle\sigma_3^{(i)}\rangle_0\lambda}$ up to the remainders. With $d^{(i)}_k=\frac{2\pi k}{N_i\lambda_{0,i}}$ and (5.1), in the form of (5.21):

$$
q_i(a;\lambda)=P_{N_i}\Bigl(a;\Gamma_i\frac{\lambda}{\lambda_{0,i}}\Bigr)+\mathcal R_i,\qquad \Gamma_i=\langle\sigma^{(i)}_3\rangle_0=\frac{\partial E_b(0)}{\partial M_i},\qquad \mathcal R_i=O\bigl(\delta_i^2(1+g_0\lambda)\bigr)+O\bigl(\delta P^2(1+\lambda/\lambda_{0,i})\bigr)+O(\kappa).
\tag{11.19}
$$

The rate is $\Gamma_i/\lambda_{0,i}$. Its ratio to the rate $1/\lambda_{0,i}$ of the free part at rest ((5.21) with $\Gamma(0)=1$) is $\Gamma_i=\langle\sigma^{(i)}_3\rangle_0\le1$, taken on the levels $(0,k_j)$. Class: every diagonal on-site binding and every simple bound level at $P=0$; status: first order in the level spacing, remainder $\mathcal R_i$.

**(b)** With (11.14), at leading order in $\mathsf R$ (remainder $O(\epsilon^4)$), with $M_i=m_i$ and $M_j=M^{(j)}_{k_j}$:

$$
\Gamma_i=1-\frac{\tau_i^2}{4M_i^2}\,\Lambda,\qquad \Lambda=1+\frac{\varsigma}{\sqrt{1+u^2}},\qquad \frac{1-\Gamma_1}{1-\Gamma_2}=\frac{\tau_1^2/M_1^2}{\tau_2^2/M_2^2}.
\tag{11.20}
$$

Both clocks are slowed. They are not slowed by the same factor: $\Lambda$ is common, and the factors agree at this order if and only if $\tau_1/M_1=\tau_2/M_2$ (for example equal data). For a wide pair with $w_{11}<0$, $1-\Gamma_i=\frac{\tau_i^2}{8M_i^2}u^2$; for a tight pair $1-\Gamma_i=\frac{\tau_i^2}{4M_i^2}(1+\frac aw)$. Without a binding $\Gamma_i=1$.

## Result

Classes, regimes and statuses. Items 1 and 2 hold for every $W\in\mathcal W$, all data, every simple bound level and every $P\in\mathcal P$; they are exact. Item 3(a) and the control 4(d) are exact, for every diagonal on-site binding (4(d): no binding) and all data. Items 3(b)–(d), 4(a)–(c) and 5(b) hold for every diagonal on-site binding in $\mathsf R$ at leading order, with the remainders stated. Item 5(a) holds for every diagonal on-site binding and all data, at first order in the level spacing of the clock.

1. **Structure.** $H(P)$ is (11.1), with the convention $\eta$. The two-body bands and their edges at $P=0$ are (11.2), for every $W$. The replacements are (11.3); the range condition fixes $U$, and only constants remain. The seven answers are (11.4): $E_b(P)$ alone is not well defined, the other six are (for $M_i$: as half the gap).
2. **Exact relations.** The five derivatives are (11.5) and the scaling relation is (11.6). $E_b'(P)=\langle V_1\rangle_P=\langle V_2\rangle_P$ (11.7), and both mean position labels move at the rate $\int\lvert g\rvert^2E_b'$ (11.8): the two bodies have the same mean velocity. $E_b''(P)$ is (11.9).
3. **Diagonal on-site binding.** The equation is (11.10), exact for all data. In $\mathsf R$: a bound level at a distance of order $\epsilon^2$ from $\mathcal B_{++}$ exists iff $\hat w_{11}\neq0$, it is one, with $E_b(P)$ of (11.12) (remainder $O(\epsilon^4)$) and $\psi_P$ of (11.13) (up to $O(\epsilon)$), from the effective contract (11.11); only $w_{11}$ enters. The five leading terms are (11.14), the limits (11.15).
4. **Curvature, inertias, cost.** $\mathfrak m_b=\mathfrak m_1+\mathfrak m_2-4\mathfrak m_1\mathfrak m_2E_{\rm bind}$ (11.16), relative remainder $O(\epsilon^2)$. For $\tau_1=\tau_2$, $\tau^2\mathfrak m_b$ differs from the cost of the pair and from $\sum_iM_i\langle\sigma^{(i)}_3\rangle_0$ at order $\epsilon^0$, by $-\frac{4M_1M_2}{\tau^2}E_{\rm bind}$ (11.17). The comparison with 03-ilang-space/21-composite-body is Step 9(c). The control without a binding is (11.18), exact.
5. **Clocks.** The statistics are (11.19) with the ratio $\Gamma_i=\langle\sigma^{(i)}_3\rangle_0$. In $\mathsf R$, $\Gamma_i$ is (11.20): the two clocks are not slowed by the same factor unless $\tau_1/M_1=\tau_2/M_2$.

Controls (M9). (11.5)–(11.9) are proven for the whole class, so no member violates them. The leading-order results fail outside $\mathsf R$: for $\tau_i/M_i$ of order one the closed forms (11.14) are off by $8$ per cent ($E_b''(0)$) to $23$ per cent ($\langle U\rangle_0$) (set E under "Code"). Further controls: no binding (11.18), $\tau_1\neq\tau_2$ (Step 9d), wide against tight (11.15).

## Consistency checks

1. **Dimensions.** $\tau_i$, $M_i$, $w$, $a_i$, $j$, $E_b$, $E_b''$ are costs ($P$ and $q$ are numbers); $u$, $\rho$, $\Lambda$, $\langle\sigma^{(i)}_3\rangle$, $\Gamma_i$ are numbers; $\mathfrak m$ is an inverse cost, and so is $\mathfrak m_1\mathfrak m_2E_{\rm bind}$ in (11.16); $\Gamma_i\lambda/\lambda_{0,i}$ in (11.19) is a number. Passed.
2. **No binding.** For $w\to0^-$, (11.12) tends to $M_1+M_2+a-j(P)$, the lower edge of $\mathcal B_{++}(P)$ at this order, and (11.14) gives $E_b''(0)\to4a_1a_2/a=1/(\mathfrak m_1+\mathfrak m_2)$, which is (11.18), with $\langle\sigma^{(i)}_3\rangle_0\to1$, $\langle A_i\rangle_0\to0$, $\langle U\rangle_0\to0$, $\Gamma_i\to1$. Passed.
3. **Scaling relation on the closed form.** With (11.14) the right side of (11.6) at $P=0$ is $M_1+M_2-a\Lambda+2a\Lambda+\varsigma\frac{w^2}{\sqrt{w^2+a^2}}=M_1+M_2+a+\varsigma\frac{a^2+w^2}{\sqrt{w^2+a^2}}=E_b(0)$ of (11.12). Passed.

## Open issues

- Whether $\mathfrak m_b$ is the inertia of the pair in a response to a gradient is not derived here (not asked; (11.17) shows that it is not the cost divided by $\tau^2$).
- Outside the order asked in item 3(b): bound levels closer than $O(\epsilon^4)$ to the band (for $\hat w_{11}=0$, and in the sector $\Xi=-1$) are neither established nor excluded.
- The coefficients of the $O(\epsilon^4)$ remainders are not derived; the code only confirms their order.
- (11.19) is of first order in the level spacing and needs $d^{(i)}_{N_i-1}\ll g_0$; in $\mathsf R$ the isolation $g_0$ is of order $\epsilon^2$.

## Code

`code/v1/check_bound_pair.py` (role: check; outputs `code/v1/output/check_bound_pair.txt`, a log, and `code/v1/output/check_bound_pair.json`, all numbers). It builds $H(P)$ of (11.1) with $\eta=\frac12$ on $r=-L,\dots,L$ with open ends and diagonalizes it: densely, once per set and $P$ (all eigenvectors, needed for (11.9)), and with a banded shift-and-invert solver for the finite differences and the doubled range. The bound level is the eigenvector of largest weight on $(r=0,\lvert11\rangle)$. Derivatives of $E_b$ are five-point differences (step $0.01$ in $P$, relative step $10^{-3}$ in $M_i$, $\tau_i$, $\theta$). Nothing in the derivation rests on the code.

- **Range of $r$ and convergence.** $L=120$ ($964$ components). Convergence was checked in two ways: $E_b$ at $L=240$ differs by less than $10^{-13}$, and the weight of $\psi_P$ at $\lvert r\rvert>60$ is at most $5\times10^{-8}$ (set D) and below $10^{-19}$ otherwise. $E_b$ at $\eta=0,0.3,1$ and from the two solvers agrees to $10^{-13}$.
- **Parameter sets** ($M_1,M_2;\ \tau_1,\tau_2;\ w_{11}$; the other three $w_{s_1s_2}$ are non-zero and of the same order), each at $P\in\{0,\pm0.15,\pm0.30\}$: A $(1,1;\ 0.1,0.1;\ -0.004)$, equal data; B $(1,1.7;\ 0.2,0.12;\ -0.01)$; C $(1,0.8;\ 0.1,0.15;\ +0.02)$, level above the band; D $(1.3,1;\ 0.15,0.2;\ -0.004)$, wide; E $(1,1.3;\ 1,0.8;\ -0.6)$, $\tau_i/M_i$ of order one. In all sets the level is outside the bands and has $\langle\Xi\rangle_P=+1$ (`Gamma` in the output files).
- **Exact relations, all five sets and all $P$** (largest absolute deviation): (11.5) derivatives against expectation values, below $10^{-10}$; (11.6), below $10^{-14}$; (11.7) $E_b'$ against $\langle V_1\rangle_P$ and $\langle V_2\rangle_P$, below $2\times10^{-10}$ (below $10^{-11}$ in A–D), and $\langle V_1\rangle_P-\langle V_2\rangle_P$ below $10^{-15}$; (11.9) against the difference quotient, below $10^{-9}$; (11.10): the coefficients against a direct inversion of $h_P(q)-E$, relative deviation below $10^{-12}$, and the root of $F_+$ against $E_b$, below $10^{-14}$. These are the sizes of the rounding and truncation errors of the differences.
- **Closed forms, sets in $\mathsf R$.**

| set | $\max_i\tau_i/M_i$ | $u$ | $\max_P\lvert E_b-(11.12)\rvert$ | the same over $(\max\tau_i/M_i)^4$ | rel. deviation of $E_b''(0)$ | the same over $(\max\tau_i/M_i)^2$ |
| --- | --- | --- | --- | --- | --- | --- |
| A | 0.10 | 0.80 | $2.1\times10^{-6}$ | 0.021 | $3.2\times10^{-4}$ | 0.032 |
| B | 0.20 | 0.83 | $8.3\times10^{-6}$ | 0.005 | $5.4\times10^{-4}$ | 0.014 |
| C | 0.19 | 2.10 | $2.0\times10^{-4}$ | 0.164 | $1.7\times10^{-2}$ | 0.47 |
| D | 0.20 | 0.28 | $7.7\times10^{-7}$ | 0.0005 | $3.6\times10^{-5}$ | 0.001 |
| E (control) | 1.0 | 1.61 | $4.6\times10^{-2}$ | 0.046 | $7.7\times10^{-2}$ | 0.077 |

  The expectation values of (11.14) agree with the numerical ones to $2.1\times10^{-5}$ (A, B, D) and $5.3\times10^{-4}$ (C), a fraction of $(\tau_i/M_i)^2$ of their leading terms. (11.16), $\mathfrak m_b-(\mathfrak m_1+\mathfrak m_2)$ against $-4\mathfrak m_1\mathfrak m_2E_{\rm bind}$ with the numerical $E_{\rm bind}$: A $56.043$ against $56.045$; B $42.321$ against $42.326$; C $-455.96$ against $-447.76$ ($\mathfrak m_b<0$); D $3.169$ against $3.161$.
- **Next order in $\epsilon$** (set B with $\tau_i\to\epsilon'\tau_i$, $w_{s_1s_2}\to\epsilon'^2w_{s_1s_2}$, $\epsilon'=1,\frac12,\frac14,\frac18$): each halving of $\epsilon'$ divides $E_b-(11.12)$ by $16.00$ (at $P=0$ and at $P=0.3$), $\langle\sigma^{(i)}_3\rangle_0-(11.14)$ by $15.6$ to $16.0$, and the relative deviation of $E_b''(0)$ by $3.97$ to $4.00$. The remainders are of order $\epsilon^4$, and of relative order $\epsilon^2$ for $E_b''(0)$, as stated.
- **Control E.** Outside $\mathsf R$ the exact relations hold as above, while $E_{\rm bind}=-0.288$ against $-0.333$ from (11.14) and $E_b''(0)$ is off by $7.7$ per cent.

No check failed and no step of the derivation was changed after a check.

## Methods used

- Fibre decomposition of a shift-invariant two-body contract (total plane-wave label)
- Weyl's theorem on the essential spectrum
- Hellmann–Feynman theorem; second-order perturbation theory; Euler's relation for homogeneous functions
- Resolvent (Green function) equation for a rank-four on-site change
- Degenerate perturbation theory (Schrieffer–Wolff) and the one-defect chain
- Commutator identity for the relative coordinate in a bound state
- Numerical diagonalization, finite differences (check)

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