Version 3 · current

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 11-bound-pair
- **Version:** v3
- **Mode:** external regeneration
- **Date:** 2026-10-11

## Changes from previous version

Changes against v2, caused by the issues I1–I4 of `verification.r2.md` (external round 2). Everything that these issues do not touch is kept word for word: the steps, the tags (11.1)–(11.26), the displays (three are restated with a condition), the three consistency checks (one parenthesis is added to check 2).

- **I1 (Step 2, Result item 1).** Three questions on the expression $E_b(P)-(M_1+M_2)$ are kept apart: (i) formed from the listed terms: no; (ii) formed from $C_{\rm bod}$, under every replacement (11.3) at fixed $C$: yes, and it is then (11.22); (iii) a constant added to $C$: not a replacement, it changes the expression (ii) and not (11.22).
  - **(11.4) restated**: the row of this expression. From $C_{\rm bod}$: "yes" (v2: "yes as $E_{\rm bind}(P)$"); from the listed terms: "no" (v2 added "the literal expression is not well defined", without saying from which quantities).
  - **(11.22) restated**: the equality $e_{++}(0)=M_1+M_2$ carries its condition (the $C$ of the question).
- **I2 (Setup, new Step 1a, Step 6, Result, Open issues).** The claim of v2 that every bound level lies outside the bands is withdrawn: it does not follow from the definition, and it is false for a member that is given. New (11.27): a band is a single point iff the data are equal and $P\in\{0,\pi\}$. So a bound level lies outside the bands, with finite multiplicity, at every $P$ of its interval except possibly such a point with the value $0$; new (11.28) bounds the level near such a point. Steps 3–5a are unchanged; their class is now stated.
- **I3 (Steps 8, 9, 10, Result).** Every wide-pair and tight-pair form is stated as an ordered limit: first $\epsilon\to0$ at fixed $u$, then $u\to0$ or $u\to\infty$ in the leading coefficient. New (11.30): sufficient inequalities for a joint range, for the level $E_b(P)$ only. For the other quantities no joint limit is derived.
- **I4 (Steps 4, 5a, 10, Result).** The class $\mathcal G(\mathcal P_0)$ of packets is defined. **(11.8) restated** with this class and a bound on its remainder. New (11.29): the statement that holds for every normalized $g$ (expectation values of the velocity operators). Steps 5a and 10 name their class of $g$.
- **Tags.** (11.1)–(11.26) are kept. New: (11.27)–(11.30). Withdrawn: none.
- **Code.** `code/v3/check_bound_pair.py` is the script of v2 with the output folder of v3 (Parts 1 and 2, same log) and a new Part 3 for I2 and I3.

## Response to verification

- **I1: Accepted and fixed** in Step 2: the row of (11.4), the bullet "$E_b(P)-(M_1+M_2)$" with the three questions (i)–(iii), and (11.22) with its text; Result item 1 follows.
- **I2: Accepted and fixed** in the Setup ("Multiplicity and location") and in the new Step 1a: (11.27), (11.28), a member with a flat value, and the list of what is proven where. The claim of v2 is not proven; it is withdrawn. What remains open is named in "Open issues".
- **I3: Accepted and fixed** in Step 8 ("Order of the limits", "Joint range", (11.30)), with the order of the limits stated after (11.16), (11.17), (11.20) and (11.26) and in the Result.
- **I4: Accepted and fixed** in Step 4 (the class $\mathcal G(\mathcal P_0)$, (11.29), the restated (11.8)), in Step 5a and in Step 10(a).

## 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.
- **Operator on the two bodies.** The listed $C_{\rm pair}$ is a convention of the listing (A5, M6). Every result below refers to the operator $C_{\rm bod}$ of (11.21) in Step 2, which is defined by the total contract $C$ and the start $\chi$ of the medium alone. Assumption (M8): $\langle\chi\vert X\vert\chi\rangle=0$. Under it $C_{\rm bod}=C^{\rm D}_1+C^{\rm D}_2+U$, and from here on $C_{\rm pair}$, $H(P)$, $E_b$, $\psi_P$, $\tau_i$, $M_i$ and $W$ denote this operator, its fibres and its data.
- **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 $\tau_1+\tau_2+\max\lvert w_{s_1s_2}\rvert\ll2\min(M_1,M_2)$; Rellich's theorem (analytic branches of an isolated eigenvalue of finite multiplicity in one real parameter) with degenerate perturbation theory to second order. Added in v3: an isolated point of the spectrum outside the essential spectrum is an eigenvalue of finite multiplicity; a simple eigenvalue at the distance $g$ from the rest of the spectrum stays simple, and moves by at most $\varrho$, under a self-adjoint change of norm $\varrho<g/2$.
- **Multiplicity and location.** By Step 1a, at every $P\in\mathcal P$ a bound level lies outside the bands (11.2), which are the essential spectrum of $H(P)$, and has a finite multiplicity $n$, with one possible exception: for equal data ($M_1=M_2$, $\tau_1=\tau_2$) at $P\in\{0,\pi\}$ the value $0$, the point $\mathcal B_{\pm\mp}=\{0\}$ of infinite multiplicity (a *flat value*). The definition of the question does not exclude this value, and a member with it exists (Step 1a). Steps 3–5a are derived at the points of $\mathcal P$ where the level lies outside the bands; Step 1a says what holds at a flat value. Steps 3–5 treat a simple level ($n=1$), which is the case in Step 7. Step 5a treats $n\ge2$.
- **Status words.** *Exact*: an identity for every member of the stated class. *Leading order in $\mathsf R$*: the stated remainder is neglected, at fixed hatted data. *Ordered limit*: first $\epsilon\to0$ at fixed hatted data, then a limit in $u$ of the leading coefficient (Step 8).

## 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 1a. Where a bound level lies, and its multiplicity (class of items 1 and 2)

The definition of the question is pointwise in $P$: $E_b(P)$ is an eigenvalue of $H(P)$ with a square-summable eigenvector, isolated from the rest of the spectrum of $H(P)$. It does not say that $E_b(P)$ lies outside the bands, and it does not ask for continuity in $P$. The interval $\mathcal P$ is taken of positive length.

**Flat bands.** The essential spectrum of $H(P)$ is the union of the four bands (11.2) (Weyl). Each band is the range of a continuous function of $p_1$ on the circle: a closed interval of positive length or a single point, and every point of an interval of positive length is a limit of other points of it. So an isolated point of the spectrum lies either outside the bands, where it is an eigenvalue of finite multiplicity, or in a band that is a single point. Let $\sigma_1\omega_1(p_1)+\sigma_2\omega_2(P-p_1)=c$ for all $p_1$.
- If $c\neq0$: squaring $\sigma_2\omega_2=c-\sigma_1\omega_1$ gives $\omega_1(p_1)=\sigma_1\bigl[c^2+\omega_1(p_1)^2-\omega_2(P-p_1)^2\bigr]/(2c)$, a trigonometric polynomial $T$ of degree at most one in $e^{2ip_1}$. Then $T^2$ has the degree $0$ or $2$, while $\omega_1^2=M_1^2+\frac{\tau_1^2}2(1-\cos2p_1)$ has the degree one ($\tau_1>0$). This is impossible.
- If $c=0$: $\sigma_1=-\sigma_2$ and $\omega_1(p_1)^2=\omega_2(P-p_1)^2$ for all $p_1$. The coefficients of $\sin2p_1$, $\cos2p_1$ and $1$ give $\tau_2^2\sin2P=0$, $\tau_1^2=\tau_2^2\cos2P$ and $M_1^2+\frac{\tau_1^2}2=M_2^2+\frac{\tau_2^2}2$.

Hence, for all data and every $P$:

$$
\text{a band of (11.2) is a single point}\iff M_1=M_2,\ \ \tau_1=\tau_2\ \ \text{and}\ \ P\in\{0,\pi\};\qquad\text{then}\quad \mathcal B_{+-}(P)=\mathcal B_{-+}(P)=\{0\}.
\tag{11.27}
$$

**What the definition gives.** Let $E_b$ be a bound level on $\mathcal P$.
- **Rule.** At every $P\in\mathcal P$ at which $E_b(P)$ is not the single point of a band, $E_b(P)$ lies outside the bands and has a finite multiplicity $n(P)$. By (11.27) this is every $P\in\mathcal P$ with one possible exception: for equal data, a point $P_*\in\{0,\pi\}\cap\mathcal P$ with $E_b(P_*)=0$, a *flat value*. There the multiplicity is infinite.
- **The pointwise conditions do not exclude a flat value.** For equal data and every $W\in\mathcal W$, $0$ is an eigenvalue of $H(P_*)$ with square-summable eigenvectors, isolated from the rest of the spectrum: $U$ has finite rank, so it vanishes on a subspace of finite codimension of the kernel of $H^0(P_*)$, and only finitely many eigenvalues lie in the gap around $0$.
- **What the interval adds.** For equal data write $M$, $\tau$, $\omega$ and $\Omega=\sqrt{M^2+\tau^2}$. The two mixed bands are $[-\Delta(P),\Delta(P)]$ with $\Delta(P)=\max_{p_1}[\omega(p_1)-\omega(P-p_1)]$, because $p_1\to P-p_1$ changes the sign. With $x:=P-P_*$ and $p_1=q+\frac x2$ ($\omega$ is even and has the period $\pi$), $\omega(p_1)-\omega(P-p_1)=\omega(q+\frac x2)-\omega(q-\frac x2)=x\,\omega'(q)+O(x^3)$, and $\max_p\lvert\omega'(p)\rvert=\max_p\tau^2\lvert\sin p\cos p\rvert/\omega(p)=\Omega-M$, attained at $\sin^2p=M/(\Omega+M)$. At every other point of $\mathcal P$ the level lies outside these bands, so:

$$
E_b(P_*)=0\ \Longrightarrow\ \lvert E_b(P)-E_b(P_*)\rvert>\Delta(P)\ \ \text{for every }P\in\mathcal P\setminus\{0,\pi\},\qquad \Delta(P)=(\Omega-M)\,\lvert P-P_*\rvert+O\bigl(\lvert P-P_*\rvert^3\bigr)\ \ (P\to P_*).
\tag{11.28}
$$

  Consequences. A bound level that is differentiable at $P_*$ with $\lvert E_b'(P_*)\rvert<\Omega-M$, in particular with $E_b'(P_*)=0$, has no flat value at $P_*$: it lies outside the bands there as well. A bound level with a flat value at an interior point $P_*$ of $\mathcal P$ that is even about $P_*$ is not differentiable at $P_*$.
- **A continuous member with a flat value.** Equal data and the on-site binding $W(r)=\delta_{r,0}\,w\,\lvert v\rangle\langle v\rvert$ with $v:=(\lvert11\rangle-\lvert22\rangle)/\sqrt2$ and $w>0$. It belongs to $\mathcal W$ and is not diagonal. By the first lines of Step 6, which hold for every on-site binding, the eigenvalues outside the bands are the roots of $1+w\,g_v(E,P)=0$ with $g_v:=\langle v\vert G(E,P)\vert v\rangle$, and they are simple.
  - $P=0$. With $n_1:=(M\sigma^{(1)}_3-\tau\sin q\,\sigma^{(1)}_1)/\omega$ and $n_2:=(M\sigma^{(2)}_3+\tau\sin q\,\sigma^{(2)}_1)/\omega$, $h_0(q)=\omega\,(n_1+n_2)$, and its kernel is the eigenspace $n_1n_2=-1$. From $\sigma^{(1)}_3\sigma^{(2)}_3v=v$, $\sigma^{(1)}_1\sigma^{(2)}_1v=-v$ and $(\sigma^{(1)}_3\sigma^{(2)}_1-\sigma^{(1)}_1\sigma^{(2)}_3)v=0$ follows $n_1n_2v=v$ for every $q$. So $U$ vanishes on the kernel of $H^0(0)$, and $0$ is an eigenvalue of $H(0)$ of infinite multiplicity. On $v$ the branches $++$ and $--$ have the weight $\frac12$ each, so $g_v(E,0)=\int\frac{\mathrm dq}{2\pi}\frac{E}{4\omega^2-E^2}$ for $\lvert E\rvert<2M$, and $1+w\,g_v(E,0)$ has no root near $E=0$: the value $0$ is isolated.
  - $P\neq0$, small. On the gap above the mixed bands $g_v(\cdot,P)$ increases with $E$ ($\partial_EG=\int\frac{\mathrm dq}{2\pi}(h_P-E)^{-2}>0$). For $E\downarrow\Delta(P)$ it tends to $-\infty$: the branch $\omega_1-\omega_2$ has its maximum $\Delta(P)$ at some $q^*$, where $\Delta-(\omega_1-\omega_2)\le c\,(q-q^*)^2$, and the weight of $v$ in this branch is $\frac12(\sqrt\alpha\mp\sqrt\beta)^2$ with $\alpha,\beta=\frac14(1\pm\frac M{\omega_1})(1\mp\frac M{\omega_2})$, which vanishes only where $\omega_1=\omega_2$. For a fixed $E_1\in(0,2M)$, $g_v(E_1,P)\to g_v(E_1,0)>0$ as $P\to0$. So for every small $P\neq0$ there is exactly one root $E_b(P)\in(\Delta(P),E_1)$: a simple eigenvalue outside the bands, and $E_b(P)\to0$ as $P\to0$.
  - The function equal to this root for $0<\lvert P\rvert<P_1$ and to $0$ at $P=0$ is continuous and meets the definition of the question on $\mathcal P=(-P_1,P_1)$. Its value at $P=0$ lies in the bands and has infinite multiplicity. The map $\psi(r)\to\sigma^{(1)}_3\sigma^{(2)}_3\psi(-r)$ takes $H(P)$ to $H(-P)$ for this binding, so $E_b$ is even, and by (11.28) $E_b(P)>(\Omega-M)\lvert P\rvert+O(\lvert P\rvert^3)$ on both sides: $E_b'(0)$ does not exist. Part 3 of the code finds this eigenvalue (see "Code").
- **What is proven where.**
  - Steps 3–5a use an isolated eigenvalue of finite multiplicity. They hold at every $P\in\mathcal P$ at which the level lies outside the bands: by the rule, at every $P\in\mathcal P$ except a flat value.
  - At a flat value: (11.6) holds for every square-summable eigenvector, as the expectation value of (11.1). (11.5), (11.7), (11.9), (11.23) and (11.24) are not derived there. A derivative with respect to $M_i$ or $\tau_i$ of one body is not defined there: by (11.27) such a change turns the point into an interval of the bands. In the member above (11.7) has no meaning at $P=0$.
  - Steps 6–10 are not affected: a root of (11.10) lies outside the bands, the level of Step 7 is near $M_1+M_2\neq0$, and Step 10 assumes a simple level.
  - Open: whether a bound level can have a flat value and be differentiable at $P_*$. By (11.28) it would have $\lvert E_b'(P_*)\rvert\ge\Omega-M$ and would lie above the mixed bands on one side of $P_*$ and below them on the other.

### 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.** With $\delta U=0$, (11.3) reads $\delta C^{\rm D}_1=t-\alpha$, $\delta C^{\rm D}_2=-t-\beta$, $\delta C_{\rm rec}=\alpha+\beta$, so $\delta C_{\rm pair}=-(\alpha+\beta)=-\delta C_{\rm rec}$, with every self-adjoint $\alpha$ on $\mathcal H_{b_1}$ and $\beta$ on $\mathcal H_{b_2}$ (A5: a single-object term may be listed in the pair term of the body with the medium).
  - Examples: a real constant; $\alpha=h\,\sigma^{(1)}_3$, after which the listed coefficient of $\sigma^{(1)}_3$ is $M_1-h$; a part of the motion term $A_1$; $\alpha=g\,\hat x_1$. None of them changes $C$, the evolution or the record vectors $(K-\langle K\rangle)\chi$.
  - The stated forms of the question select among them. 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. 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: $\delta C^{\rm D}_1=t-c_1$, $\delta C^{\rm D}_2=-t-c_2$, $\delta C_{\rm pair}=-(c_1+c_2)$.
  - M6 asks for invariance under every replacement (11.3), not only under these. So no quantity may be defined through the listed $C_{\rm pair}$. The conventions $\eta$ (Step 1) and the choice $s$ of the correspondence ($r\to r+s$ relabels $W$) change no quantity below.

**The operator on the two bodies.** Let $\langle\chi\vert\,\cdot\,\vert\chi\rangle_c$ be the partial expectation value over $\mathcal H_c$:

$$
C_{\rm bod}:=\langle\chi\vert C\vert\chi\rangle_c=\operatorname{Tr}_c\bigl[C\,(\mathbb 1\otimes\lvert\chi\rangle\langle\chi\rvert)\bigr]=C^{\rm D}_1+C^{\rm D}_2+U+\kappa\,\langle\chi\vert X\vert\chi\rangle\,(\hat x_1+\hat x_2).
\tag{11.21}
$$

The first two expressions use $C$ and $\chi$ only. So $C_{\rm bod}$ is the same operator on $\mathcal H_{b_1}\otimes\mathcal H_{b_2}$ in every listing; in a listing changed by $\alpha,\beta$ the listed pair term changes by $-\alpha-\beta$ and $C_{\rm bod}$ does not. The last expression is its value in the listing of the question.

- **Why this operator.** $L(\phi\otimes\chi)=\langle\phi\vert C_{\rm bod}\vert\phi\rangle$ is the total cost (1.8) of a product start. The view of $\{b_1,b_2\}$ of such a start evolves by $C_{\rm bod}$ up to second order in $C-C_{\rm bod}\otimes\mathbb 1=\kappa(\hat x_1+\hat x_2)\otimes(X-\langle X\rangle)$, because the mean of this operator in $\chi$ vanishes. This is the weak-record regime.
- **Assumption** $\langle\chi\vert X\vert\chi\rangle=0$ (M8). Then $C_{\rm bod}$ is the listed $C_{\rm pair}$ of the question and commutes with the joint shift. Without the assumption $C_{\rm bod}$ contains the uniform gradient $\kappa\langle X\rangle(\hat x_1+\hat x_2)$, which does not commute with the joint shift and is of order $\kappa$ on a fixed chain; the results below then hold at leading order in $\kappa$ only.
- **Data.** By the argument of the range condition, the split $C_{\rm bod}=K_1\otimes\mathbb 1+\mathbb 1\otimes K_2+U$ with $U\in\mathcal W$ is unique up to one real constant ($K_1+t$, $K_2-t$). So $U$, and $K_1$, $K_2$ up to a constant, are fixed by $C$ and $\chi$. $\tau_i$, the gaps $2M^{(i)}_k$ and the bands (11.2) are read from $K_i$, which is $C^{\rm D}_i$ up to the constant.

The seven quantities, under every replacement (11.3):

$$
\begin{array}{l|l|l}
 & \text{computed from } C_{\rm bod} \text{ of (11.21)} & \text{computed from the listed terms}\\ \hline
W(r) & \text{yes, by the range condition (no without it)} & \text{yes, the same, if the listed } U\in\mathcal W\\
M_i & \text{yes as half the gap } 2M_i;\ \text{no as a cost of } b_i & \text{no (the listed coefficient of } \sigma^{(i)}_3)\\
E_b(P) & \text{yes as a total cost (1.8) only; no as a cost of the pair} & \text{no}\\
E_b(P)-(M_1+M_2) & \text{yes (eigenvalue of the fibre and half gaps of } C_{\rm bod});\ \text{it is } E_{\rm bind}(P) \text{ of (11.22)} & \text{no (listed pair contract, listed coefficients)}\\
E_b(P)-E_b(0),\ E_b''(0) & \text{yes} & \text{no (unchanged under constants only)}\\
\langle\sigma^{(i)}_3\rangle_P & \text{yes (for } n\ge2 \text{ it depends on the eigenvector)} & \text{no (unchanged under constants only)}
\end{array}
\tag{11.4}
$$

- **$W(r)$.** $\delta U=0$ in every replacement that keeps $U\in\mathcal W$. $U$ is the only element of $\mathcal W$ for which $C_{\rm bod}-U$ is a sum of single-object operators.
- **$M_i$.** Half the difference of the two eigenvalues of $K_i$ on the level $k_i$ at the plane-wave label $0$, (6.7). The listed coefficient of $\sigma^{(i)}_3$ changes under $\alpha=h\,\sigma^{(i)}_3$. The branch costs $\pm M_i$ of $b_i$ change by the constant $t$.
- **$E_b(P)$.** It is an eigenvalue of the fibre of $C_{\rm bod}$. For $\Phi$ of Step 4, $L(\Phi\otimes\chi)=\int\lvert g\rvert^2E_b$ is a total cost, which does not depend on the listing (A5). It is not a cost of the pair apart from the medium: which contract carries a constant or a single-object term is a convention, and the eigenvalue of the listed $C_{\rm pair}$ changes ($\delta E_b=-(c_1+c_2)$ under constants). Its zero enters no reading. Where a later step speaks of the cost of the pair, it means $E_b$ in this sense.
- **$E_b(P)-(M_1+M_2)$.** Three questions, each with its own answer.
  - (i) *Formed from the listed terms*: the eigenvalue $E^{\rm list}_b(P)$ of the fibre of the listed pair contract, minus the listed coefficients of $\sigma^{(1)}_3$ and $\sigma^{(2)}_3$. This expression is not invariant under (11.3), so it is not well defined: constants $c_1+c_2$ listed in the record terms shift $E^{\rm list}_b$ by $-(c_1+c_2)$ and no coefficient, and $\alpha=h\,\sigma^{(1)}_3$ replaces $M_1$ by $M_1-h$ in both. The same holds with the half gaps of the listed pair contract in place of the listed coefficients: the constants shift $E^{\rm list}_b$ and no half gap.
  - (ii) *Formed from $C_{\rm bod}$*: the eigenvalue $E_b(P)$ of its fibre, minus the half gaps $M_1$, $M_2$ of $K_1$, $K_2$. A replacement (11.3) leaves $C$ unchanged, and $C_{\rm bod}$, $K_1+K_2$ and the gaps are fixed by $C$ and $\chi$. So this expression is invariant under every replacement (11.3): it is well defined. For the $C$ of the question it is the distance $E_{\rm bind}(P)$ of the level from the band edge, (11.22) below.
  - (iii) *A constant added to the total contract*, $C\to C+c\,\mathbb 1$. This is not a replacement (11.3): it changes $C$. It multiplies the weighting by the common phase $e^{-ic\lambda}$ (1.2), changes no reading, and changes the cost (1.8) of every state by $c$. Then $C_{\rm bod}\to C_{\rm bod}+c$, $E_b(P)\to E_b(P)+c$ and $e_{++}(P)\to e_{++}(P)+c$, and the half gaps do not change. So the expression (ii) changes by $c$, and $E_{\rm bind}(P)$ of (11.22) does not; for $C+c$, $e_{++}(0)=M_1+M_2+c$. The form (11.22) is invariant under (11.3) and under the constant; the expression (ii) is invariant under (11.3) only, and so is $E_b(P)$ alone.

  The distance from the band edge:

$$
E_{\rm bind}(P):=E_b(P)-e_{++}(0),\qquad e_{++}(P):=\min\mathcal B_{++}(P),\qquad e_{++}(0)=M_1+M_2\ \ (\text{for the } C \text{ of the question}),\qquad E_{\rm bind}:=E_{\rm bind}(0).
\tag{11.22}
$$

  $e_{++}(P)$ is the least sum of the upper-branch costs of $K_1$ and $K_2$ with the total label $P$; the constant $t$ cancels in it. $E_{\rm bind}(P)$ is a difference of two total costs of the same $C$. For the $C$ of the question $C_{\rm bod}$ has no constant term, so $e_{++}(0)=M_1+M_2$ by (11.2), and the expression (ii) equals $E_{\rm bind}(P)$.
- **$E_b(P)-E_b(0)$, $E_b''(0)$.** Differences of total costs. Computed from the listed $C_{\rm pair}$ they change under $\alpha=h\,\sigma^{(1)}_3$ (in (11.12): $M_1\to M_1-h$).
- **$\langle\sigma^{(i)}_3\rangle_P$.** $\sigma^{(i)}_3$ is diagonal in the places of $b_i$: it is the reading of the branch label, (1.3), the probability of $s_i=1$ minus that of $s_i=2$. $\psi_P$ is an eigenvector of the fibre of $C_{\rm bod}$. With $\psi_P$ of the listed $C_{\rm pair}$ it changes under $\alpha=h\,\sigma^{(1)}_3$.

### 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 $P$:

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

**Packets and the class of $g$.** Let $\mathcal P_0\subseteq\mathcal P$ be an open interval on which the bound level is simple and lies outside the bands (Step 1a). On $\mathcal P_0$, $E_b$ is analytic and $\psi_P$ can be chosen analytic in $P$ (analytic perturbation theory), and $\lVert\hat r\psi_P\rVert$ and $\lVert\partial_P\psi_P\rVert$ are bounded on every compact subset. A state of the pair built from the bound level is $\Phi=\int\mathrm dP\,g(P)\,\lvert P;\psi_P\rangle$ with $g$ supported in $\mathcal P_0$ and $\int\lvert g\rvert^2=1$. Under $C_{\rm pair}$, $g(P)\to g(P)e^{-iE_b(P)\lambda}$, and $V_i$ has no elements between different $P$.

- **Every normalized $g$ supported in $\mathcal P_0$.** $V_i$ is bounded, so its expectation value exists, and by (11.7), for every $\lambda$:

$$
\langle\Phi(\lambda)\vert V_1\vert\Phi(\lambda)\rangle=\langle\Phi(\lambda)\vert V_2\vert\Phi(\lambda)\rangle=\int\mathrm dP\,\lvert g(P)\rvert^2E_b'(P).
\tag{11.29}
$$

  This says nothing on mean position labels: on the infinite chain they need not exist. Example: for $g$ constant on a subinterval of $\mathcal P_0$ and zero outside (a sharp cutoff), $\lvert\Phi\rvert^2$ summed over $r$ falls off as the inverse square of the centre label $\eta x_1+(1-\eta)x_2$, and $\sum\lvert x_i\rvert\,\lvert\Phi\rvert^2$ diverges.
- **The class $\mathcal G(\mathcal P_0)$** (sufficient): $g$ continuously differentiable with compact support in $\mathcal P_0$, and $\int\lvert g\rvert^2=1$. On the fibres $\eta\hat x_1+(1-\eta)\hat x_2$ acts as $i\partial_P$ and $\hat x_1-\hat x_2$ as $\hat r$, so $\hat x_i\Phi(\lambda)$ is square-summable for every $\lambda$ ($g\,e^{-iE_b\lambda}$ belongs to the class again). Then the mean position labels $\bar m_i(\lambda)=\langle\Phi(\lambda)\vert\hat x_i\vert\Phi(\lambda)\rangle$ exist and are differentiable with $\dot{\bar m}_i=\langle V_i\rangle$, and (11.29) gives the first two equalities below. Let $P_0:=\int P\,\lvert g\rvert^2$, $\delta P^2:=\int(P-P_0)^2\lvert g\rvert^2$, and let $I_g\subset\mathcal P_0$ be the smallest closed interval that contains the support of $g$. Taylor's formula for $E_b'$ around $P_0$ on $I_g$ gives the third equality:

$$
\frac{\mathrm d\bar m_1}{\mathrm d\lambda}=\frac{\mathrm d\bar m_2}{\mathrm d\lambda}=\int\mathrm dP\,\lvert g(P)\rvert^2E_b'(P)=E_b'(P_0)+O(\delta P^2)\qquad\text{for } g\in\mathcal G(\mathcal P_0)\ \text{and every }\lambda,\qquad \bigl\lvert O(\delta P^2)\bigr\rvert\le\tfrac12\max_{I_g}\lvert E_b'''\rvert\;\delta P^2 .
\tag{11.8}
$$

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 such $\mathcal P_0$, in two senses. For every normalized $g$ supported in $\mathcal P_0$ the two velocity operators have the same expectation value (11.29). For $g\in\mathcal G(\mathcal P_0)$ the two mean position labels exist and change at the same rate (11.8), so $\bar m_1-\bar m_2$ is constant. 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 $\lambda$ and a fixed $g\in\mathcal G(\mathcal P_0)$.

### 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 5a. Degenerate bound levels (items 2a–2c)

Let $E_b$ be an eigenvalue of $H(P)$ of multiplicity $n\ge2$ outside the bands, at given data and $P$. Let $\mathcal E_b$ be its eigenspace, $\Pi_b$ the projector on it, $\bar Q:=\mathbb 1-\Pi_b$ and $\hat S:=\bar Q\,(H(P)-E_b)^{-1}\bar Q$ the reduced resolvent, which is bounded. Every vector of $\mathcal E_b$ decays exponentially.

Let $\xi$ be one of the six real parameters $M_1,M_2,\tau_1,\tau_2,\theta,P$, the others fixed. $H$ is analytic in $\xi$, with $\partial_{M_i}H=\sigma^{(i)}_3$, $\tau_i\partial_{\tau_i}H=A_i$, $\partial_\theta H=U$, $\partial_PH=V_\eta$. By Rellich's theorem the eigenvalues near $E_b$ are $n$ analytic functions $E^{(\nu)}(\xi)$, the *branches of $\xi$*, with analytic orthonormal eigenvectors; at the given point these are the *branch vectors* $\psi^{(\nu)}_\xi\in\mathcal E_b$. A bound level that is differentiable in $\xi$ is one branch. First-order degenerate perturbation theory gives

$$
\Pi_b\,(\partial_\xi H)\,\Pi_b\;\psi^{(\nu)}_\xi=\partial_\xi E^{(\nu)}\;\psi^{(\nu)}_\xi ,\qquad
\langle\psi\vert\partial_\xi H\vert\psi\rangle=\sum_{\nu=1}^{n}\bigl\lvert\langle\psi^{(\nu)}_\xi\vert\psi\rangle\bigr\rvert^2\,\partial_\xi E^{(\nu)}\quad\text{for every unit vector }\psi\in\mathcal E_b .
\tag{11.23}
$$

- **For every eigenvector of the level** (every unit vector $\psi\in\mathcal E_b$):
  - (11.6), as the expectation value of (11.1) in an eigenvector;
  - $\langle V_1\rangle=\langle V_2\rangle$, and more: $\Pi_bV_1\Pi_b=\Pi_bV_2\Pi_b$, because $\langle\psi\vert[H(P),\hat r]\vert\psi'\rangle=0$ for $\psi,\psi'\in\mathcal E_b$;
  - the second relation of (11.23): each of the six expectation values is a mean of the $n$ branch derivatives. It is a derivative only if $\psi$ is an eigenvector of the matrix $\Pi_b(\partial_\xi H)\Pi_b$.
- **For the branch vectors of the parameter that is varied.** (11.5) holds for $M_i$, $\tau_i$, $\theta$ and (11.7) for $P$, each with the eigenvectors of its own matrix $\Pi_b(\partial_\xi H)\Pi_b$. The six matrices need not commute, so in general no eigenvector gives all the derivatives. If $\xi$ does not split the level (its $n$ branches coincide), its matrix is $\partial_\xi E_b\,\Pi_b$, and its identity holds for every eigenvector.
- **(11.8) and (11.29).** Let $\mathcal P_0\subseteq\mathcal P$ be an open interval on which the level lies outside the bands, with the branch vectors chosen analytic in $P$ (Rellich). A state built from one branch of $P$ obeys (11.29) for every normalized $g$ supported in $\mathcal P_0$, and (11.8) for $g\in\mathcal G(\mathcal P_0)$. So does a state $\Phi=\int\mathrm dP\sum_\nu g_\nu(P)\lvert P;\psi^{(\nu)}_P\rangle$ built from a level that is $n$-fold on $\mathcal P_0$, with $\sum_\nu\lvert g_\nu\rvert^2$ in place of $\lvert g\rvert^2$ and every $g_\nu$ in the class named: the two bodies have the same mean velocity. Control: where two branches have different values, $\langle\psi^{(\nu)}\vert(V_1-V_2)\vert\psi^{(\nu')}\rangle=i\bigl(E^{(\nu)}-E^{(\nu')}\bigr)\langle\psi^{(\nu)}\vert\hat r\vert\psi^{(\nu')}\rangle$, and in a superposition of the two the mean velocities differ by a term that oscillates as $e^{\pm i(E^{(\nu)}-E^{(\nu')})\lambda}$. Such a state is built from two bound levels.
- **What replaces (11.9).** Second-order degenerate perturbation theory in $P$ gives, for a branch vector $\psi^{(\nu)}_P$ and with $\langle\cdot\rangle_\nu$ its expectation value:

$$
E^{(\nu)\prime\prime}(P)=-\eta^2\langle A_1\rangle_\nu-(1-\eta)^2\langle A_2\rangle_\nu-2\,\bigl\langle V_\eta\psi^{(\nu)}_P\big\vert\hat S\big\vert V_\eta\psi^{(\nu)}_P\bigr\rangle,\qquad \hat S=\bar Q\bigl(H(P)-E_b\bigr)^{-1}\bar Q .
\tag{11.24}
$$

  The sum over the rest of the spectrum leaves out the whole eigenspace $\mathcal E_b$. If several branches have the same first derivative, with $\Pi'$ the projector on their branch vectors, their second derivatives are the eigenvalues of $\Pi'\bigl[\partial_P^2H-2V_\eta\hat SV_\eta\bigr]\Pi'$, and their branch vectors are its eigenvectors. For $n=1$, (11.24) is (11.9). For $n\ge2$, (11.9) is not defined: $Q_P(H(P)-E_b)^{-1}Q_P$ does not exist on the rest of $\mathcal E_b$.
- **Members.** Degenerate levels exist in $\mathcal W$.
  - For a diagonal on-site binding, by (11.10) a root of $F_-$ can be placed on a root of $F_+$ by the choice of $w_{12}$, $w_{21}$. All six operators commute with $\Xi$ of Step 6, so the vectors with a definite $\Xi$ are branch vectors of every parameter.
  - For the on-site binding $W_0=-G(E_0,P_0)^{-1}$ on the span of $\lvert11\rangle$, $\lvert22\rangle$, with $G$ of Step 6 and $E_0$ outside the bands, $\mathbb 1_4+GW_0$ vanishes on that span, so $E_0$ is a twofold level of $H(P_0)$. For one such member the six matrices do not commute; this statement rests on the code (`code/v3/check_bound_pair.py`, set X).

### 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. The flat value of Step 1a (equal data, $P\in\{0,\pi\}$, $E=0$) lies in the bands and is not among these solutions.

### 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 $\epsilon^2$.

The forms of $\rho$ and (11.13) are for $j(P)\neq0$. $j(P)=0$ occurs only for $a_1=a_2$ at $P=\pm\frac\pi2$. Then $J=0$, $\alpha$ is not defined and not needed, the chain has no hopping, and the eigenvector of the changed site is $\psi_P(r)=\delta_{r,0}\lvert11\rangle$ up to $O(\epsilon)$, with $E_b=M_1+M_2+a+w+O(\epsilon^4)$. These are the limits $\rho\to0$, $\mathcal N_P\to1$ of (11.13) and the value of (11.12) at $j=0$.

- **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 $\epsilon^6$. First-order perturbation theory in the number $w_{22}$ alone gives $\partial E_b/\partial w_{22}=\lvert\langle22\vert\psi_P(0)\rangle\rvert^2$; the branch $\lvert22\rangle$ needs two flips, so its amplitude is $O(\epsilon^2)$ and this derivative $O(\epsilon^4)$, and $w_{22}=O(\epsilon^2)$. (In $F_+$: the term $w_{11}w_{22}g_\times^2$ is $O(\epsilon^4)$ against a slope $w_{11}\partial_EG_{11}$ of order $\epsilon^{-2}$.) The $O(\epsilon^4)$ remainder of (11.12) has other sources. $w_{12},w_{21}$ 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 $\mathsf R$, taken as ordered limits (stated below the table), with relative corrections $O(u^2)$ (wide) and $O(u^{-2})$ (tight):

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

**Order of the limits.** Every wide-pair and tight-pair form of this package is an ordered limit: the table (11.15), the statements after (11.16), (11.17), (11.20) and (11.26), and those of the Result.
1. First $\epsilon\to0$ at fixed hatted data, hence at a fixed $u\in(0,\infty)$. This gives (11.14): each quantity there is $\epsilon^2$ times a function of the hatted data, up to $O(\epsilon^4)$. The constant of this $O(\epsilon^4)$ depends on the hatted data; Steps 7 and 8 do not bound it for $u\to0$ or $u\to\infty$.
2. Then $u\to0$ (wide) or $u\to\infty$ (tight) in the functions (11.14), with the other hatted data fixed. As statements on the functions (11.14) the entries of (11.15) are exact up to the relative corrections named.

So a wide or tight form is the leading term of the coefficient of $\epsilon^2$ in the quantity (of $\epsilon^{-2}$ for $\mathfrak m_b$, of $\epsilon^{0}$ in (11.17)). It is not a statement at a fixed $\epsilon$. The value $u=0$ is not a member of these limits: without a binding there is no bound level, and (11.18) holds instead.

**Joint range (sufficient conditions, for the level only).** Let $\tau:=\max_i\tau_i$, $M:=\min_iM_i$ and $\bar w:=\max\lvert w_{s_1s_2}\rvert$. The part of $H_{\rm eff}$ that Step 7 neglects is of third order (two factors $A_i$ with one factor $U$, or with the shift of the denominators $2M_i$ by the level) and of fourth order (four factors $A_i$). Its norm is therefore at most $\varrho$ below, with numbers $c$, $\varepsilon_0$ that do not depend on the data. Let

$$
\tau+\bar w\le\varepsilon_0M\quad(\text{gap to the other branch groups}),\qquad \sqrt{w^2+a^2}-a>2\varrho\quad(\text{isolation}),\qquad \varrho:=c\,\frac{\tau^2}{M}\Bigl(\frac{\tau^2}{M^2}+\frac{\bar w}{M}\Bigr).
\tag{11.30}
$$

For every $P$ the level of (11.11) is simple and at the distance $\sqrt{w^2+j^2}-j\ge\sqrt{w^2+a^2}-a$ from the rest of the spectrum of (11.11). Under (11.30) the neglected part is smaller than half this distance, so $H(P)$ has exactly one eigenvalue within $\varrho$ of (11.12), and it is simple.
- *Wide* ($u\le1$, the other hatted data in a fixed bounded set): $\varrho=O(\epsilon^4)$, and (11.30) reads $u^2\ge K\epsilon^2$ with a constant $K$ of that set. Then $E_{\rm bind}=-\frac{w^2}{2a}\bigl[1+O(u^2)+O(\epsilon^2/u^2)\bigr]$ for $w<0$; for $w>0$ the same bracket multiplies the term $\frac{w^2}{2a}$ of $2a+\frac{w^2}{2a}$.
- *Tight* ($u\ge1$): $\bar w=O(\epsilon^2u)$ and $\varrho=O(\epsilon^2u\,a)$, the isolation condition holds for small $\epsilon$, and (11.30) reads $u\le\epsilon^{-2}/K$, that is $\lvert w_{11}\rvert\ll M$. Then $E_{\rm bind}=w+a+O(a/u)+O(\epsilon^2u\,a)$: the first two terms of the tight form hold for $1\ll u\ll\epsilon^{-2}$, and its third term $\frac{a^2}{2w}$ exceeds the remainder only for $u\ll\epsilon^{-1}$.
- (11.30) is sufficient. It is not necessary: for $w_{11}<0$ at $P=0$ the lower band edge is exactly $M_1+M_2$, and in the code the relative deviation of $E_{\rm bind}$ from (11.14) stays below $4\times10^{-3}$ for $u$ from $2$ down to $\frac1{128}$ at $\max_i\tau_i/M_i=0.2$, where $\epsilon^2/u^2$ reaches $655$. This statement rests on the code (Part 3). The scale $u\sim\epsilon$ is real elsewhere: for $w_{11}>0$, and for $w_{11}<0$ at $P=\frac\pi2$, the remainder tends to a constant of order $\epsilon^4$ and exceeds the isolation (same runs).
- For $E_b''(0)$, the expectation values, $\mathfrak m_b$ and $\Gamma_i$ no joint limit is derived: the first sentence of this step differentiates the remainder at fixed hatted data.

### 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 $\epsilon^{-2}$.

Order of the limits (Step 8): (11.16) is the coefficient of $\epsilon^{-2}$ at a fixed $u$; "for every $u$" means every fixed $u\in(0,\infty)$, and the wide and tight values of the difference are the limits $u\to0$ and $u\to\infty$ of this coefficient. At a fixed $u$ the difference has a relative remainder $O(\epsilon^2)$, whose constant Step 8 does not bound for $u\to0$, where the difference itself vanishes as $u^2$, or for $u\to\infty$. No joint limit is derived.

**(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 ordered limit of Step 8 (first $\epsilon\to0$ at fixed $u$, then $u\to0$ with $w_{11}<0$), in which the leading difference tends to zero, 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 $M_i\to m_i+d^{(i)}_k$.

Class of $g$: normalized, with its support in a closed interval $I_g$ inside an open interval $\mathcal P_0\ni0$ on which the bound level is simple and lies outside the bands for all $M_i$ in the range $[m_i,m_i+d^{(i)}_{N_i-1}]$ of the levels, with $\int P\,\lvert g\rvert^2=0$ and $\delta P^2:=\int P^2\lvert g\rvert^2$. The reading of the clock label uses no position label, so (11.19) below holds for every such $g$, and the constants of its $\delta P^2$ term are maxima over $I_g$ of second derivatives in $P$. Its term $O(\kappa)$ is claimed for $g\in\mathcal G(\mathcal P_0)$ of Step 4.

Let $D_i:=d^{(i)}_{N_i-1}$ be the spread of the levels of the clock, $\bar d_i:=\frac1{N_i}\sum_kd^{(i)}_k=\frac{\pi(N_i-1)}{N_i\lambda_{0,i}}=\frac{D_i}2$ their mean and $\bar M_i:=m_i+\bar d_i$ the mean rest cost. Let the bound level at $P=0$ be simple and at a distance of at least $g_0$ from the rest of the spectrum for all $M_i\in[m_i,m_i+D_i]$, and let $\delta_i:=D_i/g_0$. By analytic perturbation theory in $M_i$ ($\partial_{M_i}H=\sigma^{(i)}_3$ has norm one), $E_b$ and $\psi_P$ are analytic in $M_i$ on this range, with $\lVert\partial_{M_i}\psi_P\rVert\le1/g_0$, $\lvert\partial^2_{M_i}E_b\rvert\le2/g_0$ and $\partial^n_{M_i}E_b=O(g_0^{1-n})$. With (11.5), around the mean level:

$$
E^{[k]}_b(P)=\Bigl[E_b(P)+\bigl(d^{(i)}_k-\bar d_i\bigr)\langle\sigma^{(i)}_3\rangle_P+\tfrac12\bigl(d^{(i)}_k-\bar d_i\bigr)^2\,\partial^2_{M_i}E_b(P)\Bigr]_{M_i=\bar M_i}+O\bigl(D_i^3/g_0^2\bigr).
$$

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 up to $O(\delta P^2)$, by (11.29) for every $g$ of the class above and by (11.8) for $g\in\mathcal G(\mathcal P_0)$, 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 $N_i^{-2}\sum_{k,k'}e^{2\pi i(k-k')a/N_i}\langle\Psi_{k'}(\lambda)\vert\Psi_k(\lambda)\rangle$. Four contributions enter the overlaps.

- **Linear term.** It gives the phases $e^{-i(d_k-d_{k'})\Gamma_i\lambda}$ with $\Gamma_i:=\langle\sigma^{(i)}_3\rangle_0$ at $M_i=\bar M_i$. With $d^{(i)}_k=\frac{2\pi k}{N_i\lambda_{0,i}}$ this is $P_{N_i}$ of (5.1) exactly.
- **Quadratic term.** It is a phase $\beta_k$ with $\beta_k=\beta_{N_i-1-k}$ and $\lvert\beta_k\rvert\le\frac18D_i^2\,\lvert\partial^2_{M_i}E_b\rvert\,\lambda\le\frac\pi2\,\delta_i\,\lambda/\lambda_{0,i}$. A phase that is even about the middle of the levels changes the statistics only at second order, because $\sum_k\beta_ke^{ik\phi}$ and $\sum_ke^{ik\phi}$ have the same phase $e^{i(N_i-1)\phi/2}$. This is why the level is expanded around $\bar M_i$.
- **Cubic term.** It is odd, enters at first order and is $O(\delta_i^2\,\lambda/\lambda_{0,i})$.
- **Eigenvector of the start.** The start has on every level the eigenvector of the level $0$, which differs from that of the level $k$ by at most $d^{(i)}_k/g_0$ in norm. This changes the overlaps by $O(\delta_i^2)$ for every $\lambda$.

In the form of (5.21):

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

Expanding $\Gamma_i$ around $M_i=m_i$, with $Q_0$ of (11.9) at $P=0$:

$$
\Gamma_i=\Gamma^{(0)}_i+\bar d_i\,\partial^2_{M_i}E_b(0)+O(\delta_i^2),\qquad \Gamma^{(0)}_i:=\langle\sigma^{(i)}_3\rangle_0\Big\vert_{M_i=m_i},\qquad \partial^2_{M_i}E_b(0)=\partial_{M_i}\langle\sigma^{(i)}_3\rangle_0=-2\,\bigl\langle\sigma^{(i)}_3\psi_0\big\vert Q_0\bigl(H(0)-E_b\bigr)^{-1}Q_0\big\vert\sigma^{(i)}_3\psi_0\bigr\rangle .
\tag{11.25}
$$

- **Rate and ratio.** 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 $\Gamma_i\le1$: the expectation value $\langle\sigma^{(i)}_3\rangle_0$ of item 2(a), taken at the mean rest cost $\bar M_i$ of the part $i$ and on the level $k_j$ of the other part. (11.25) is the rate to first order in the level spacing. Applied to one free body, with $\omega_0(p_0)$ at the rest cost $\mu$ in place of $E_b$, it gives $\gamma\bigl[1+\frac{(N-1)\epsilon}2(1-\gamma^2)\bigr]$ of (5.21), where $\frac{(N-1)\epsilon}2=\bar d/m$.
- **Two orders.** The level to first order in $d^{(i)}_k$ gives only $\Gamma^{(0)}_i$, the rate to zeroth order: the advance $\Gamma_i\lambda/\lambda_{0,i}$ is then wrong by $\bar d_i\,\partial^2_{M_i}E_b\,\lambda/\lambda_{0,i}$, which is of first order in $\delta_i$. The rate to first order in the level spacing needs the level to second order in $d^{(i)}_k$: a sum over the rest of the spectrum, as in (11.9).
- **Range of $\lambda$.** $\mathcal R_i$ is small for $\delta_i\ll1$ on $0\le\lambda\ll\lambda_{0,i}/\delta_i$, with $\delta P^2\lambda/\lambda_{0,i}\ll1$ and $\kappa\to0$ at fixed $\lambda$. On $\lambda_{0,i}\lesssim\lambda\ll\lambda_{0,i}/\delta_i$ the first-order term of (11.25) changes the statistics by more than $\mathcal R_i$. The condition $\delta_i\ll1$ is sufficient, not necessary (see (b)).

Class: every diagonal on-site binding and every bound level at $P=0$ that is simple for all $M_i$ in the range of the levels. Status: the rate to first order in the level spacing; the statistics up to the remainder $\mathcal R_i$, which is of second order.

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

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

To first order in the level spacing, with $a_i$, $a$, $u$, $\Lambda$ taken at $M_i=m_i$:

$$
\Gamma_i=1-\frac{\tau_i^2}{4m_i^2}\Bigl[\Lambda-\frac{\bar d_i}{m_i}\Bigl(2\Lambda+\varsigma\,\frac{a_i}{a}\,\frac{u^2}{(1+u^2)^{3/2}}\Bigr)\Bigr]+O\Bigl(\frac{a_i\,\bar d_i^{\,2}}{m_i^3}\Bigr)+O(\epsilon^4).
\tag{11.26}
$$

This is (11.25) with $\partial^2_{M_i}E_b(0)=\frac{a_i}{M_i^2}\bigl[2\Lambda+\varsigma\frac{a_i}{a}\frac{u^2}{(1+u^2)^{3/2}}\bigr]$, from (11.12) and $\partial_{M_i}a=-a_i/M_i$. In $\mathsf R$ the bounds through $g_0=O(\epsilon^2)$ are not sharp. By (11.12), (11.13) and the analyticity used in Step 8, $E_b-M_1-M_2$ and $\psi_0$ depend on $M_i$ on the scale $M_i$: $\partial^n_{M_i}E_b=O(a_i/M_i^{\,n})$ for $n\ge2$ and $\lVert\partial_{M_i}\psi_0\rVert=O(1/M_i)$. So $\mathcal R_i$ of (11.19) holds with $\delta_i$ replaced by $D_i/M_i$, also for $D_i\gg g_0$. (11.26) and these bounds are for fixed hatted data, hence a fixed $u\in(0,\infty)$; their constants are not bounded here for $u\to0$ or $u\to\infty$.

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). The third relation of (11.20) compares the two ratios at one pair $(M_1,M_2)$. The two readings are two starts, with the pairs $(\bar M_1,M^{(2)}_{k_2})$ and $(M^{(1)}_{k_1},\bar M_2)$, which changes the relation by relative terms of first order in $d^{(i)}_k/M_i$. 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 $\Gamma_i=1$. The wide and the tight form are ordered limits (Step 8): the coefficient $\frac{\tau_i^2}{4M_i^2}\Lambda(u)$ of (11.20) is obtained first, for $\epsilon\to0$ at fixed $u$, and then $\Lambda(u)$ is expanded for $u\to0$ or $u\to\infty$; no joint limit is derived.

## Result

Classes, regimes and statuses. Every result refers to $C_{\rm bod}$ of (11.21), under the assumption $\langle\chi\vert X\vert\chi\rangle=0$. Items 1 and 2 hold for every $W\in\mathcal W$, all data, every bound level and every $P\in\mathcal P$ at which the level lies outside the bands; they are exact. By (11.27) this is every $P\in\mathcal P$ except a flat value (equal data, $P\in\{0,\pi\}$, $E_b=0$), where only (11.6) is derived. For a simple level they are (11.5)–(11.9); for a degenerate level they hold in the form of Step 5a, (11.23) and (11.24). 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, at fixed hatted data. Their wide-pair and tight-pair forms are ordered limits: first $\epsilon\to0$ at fixed $u$, then $u\to0$ or $u\to\infty$ in the leading coefficient. A joint range of $\epsilon$ and $u$ is derived for the level $E_b(P)$ only, under the sufficient conditions (11.30). Item 5(a) holds for every diagonal on-site binding and all data, for a bound level at $P=0$ that is simple over the range of the levels of the clock: the rate to first order in the level spacing, the statistics up to a remainder of second order.

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 $W$. A band is a single point if and only if the data are equal and $P\in\{0,\pi\}$ (11.27). A bound level lies outside the bands, with finite multiplicity, at every $P$ of its interval except possibly such a point with the value $0$; near it (11.28) holds, and a continuous member of $\mathcal W$ with such a value exists (Step 1a). The replacements are (11.3); the range condition fixes $U$, and every single-object term can still be listed in a body term or in a record term (within the stated forms: only constants). The operator to which the results refer is $C_{\rm bod}$ (11.21). The seven answers are (11.4).
   - Computed from the listed terms, only $W(r)$ is well defined.
   - Computed from $C_{\rm bod}$: $W(r)$ yes; $M_i$ yes as half the gap, no as a cost of $b_i$; $E_b(P)$ yes as a total cost (1.8) only; $E_b(P)-(M_1+M_2)$, $E_b(P)-E_b(0)$, $E_b''(0)$ and $\langle\sigma^{(i)}_3\rangle_P$ yes.
   - The expression $E_b(P)-(M_1+M_2)$. Formed from the listed pair contract and the listed coefficients: not well defined. Formed from $C_{\rm bod}$ (the eigenvalue of its fibre and the half gaps): well defined under every replacement (11.3), and for the $C$ of the question equal to the distance $E_{\rm bind}(P)$ from the band edge (11.22). A constant $c$ added to $C$ is not a replacement (11.3); it changes this expression by $c$ and leaves $E_{\rm bind}(P)$ of (11.22) unchanged.
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). In a state built from the bound level on an interval $\mathcal P_0$ where it is simple and outside the bands, the two bodies have the same mean velocity: for every normalized $g$ supported in $\mathcal P_0$ the two velocity operators have the expectation value $\int\lvert g\rvert^2E_b'$ (11.29); for $g$ of the class $\mathcal G(\mathcal P_0)$ the mean position labels exist and both change at this rate, which is $E_b'(P_0)+O(\delta P^2)$ (11.8). $E_b''(P)$ is (11.9). For a degenerate level: (11.6), $\langle V_1\rangle=\langle V_2\rangle$ and the mean rule of (11.23) hold for every eigenvector; (11.5) and (11.7) hold for the branch vectors of the parameter that is varied (11.23); (11.24) replaces (11.9); a state built from one level, with $g$ of the classes named, has equal mean velocities of the two bodies.
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 $O(\epsilon)$; at $j(P)=0$ it is $\delta_{r,0}\lvert11\rangle$), from the effective contract (11.11); only $w_{11}$ enters ($w_{22}$ at order $\epsilon^6$). The five leading terms are (11.14), the limits (11.15), as ordered limits. For the level itself (11.30) gives a sufficient joint range: $u^2\ge K\epsilon^2$ for a wide pair, $u\le\epsilon^{-2}/K$ for a tight pair.
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 $O(\epsilon^2)$, at every fixed $u$. 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 $\Gamma_i=\langle\sigma^{(i)}_3\rangle_0$ at the mean rest cost $\bar M_i$, for every normalized $g$ of mean $0$ with support in a closed interval inside $\mathcal P_0$ (the term $O(\kappa)$ for $g\in\mathcal G(\mathcal P_0)$). To first order in the level spacing the ratio is (11.25). The remainder is of second order in $\delta_i$ and small for $\lambda\ll\lambda_{0,i}/\delta_i$. In $\mathsf R$, $\Gamma_i$ is (11.20) at $M_i=\bar M_i$, and (11.26) to first order in the level spacing, both at fixed $u$: 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 every simple level of the class and (11.23), (11.24) for every degenerate one, at every $P$ where the level lies outside the bands, so no member violates them there. The member of Step 1a is the control for the exception: at its flat value $E_b'(0)$ does not exist. (11.9) fails for a degenerate level, and (11.19) with $\Gamma^{(0)}_i$ in place of $\Gamma_i$ fails at first order (sets X and the clock runs under "Code"). (11.8) fails as a statement on mean position labels for a normalized $g$ with a sharp cutoff, while (11.29) holds (Step 4). An operator read from the listed terms fails as the operator of the two bodies in a listing that differs by a single-object term (Step 2; the last check of Part 2 under "Code"). 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"). At a fixed $\epsilon$ the wide form of $E_{\rm bind}$ for $w_{11}>0$ fails for $u$ of the order of $\epsilon$, and the tight forms fail for $\lvert w_{11}\rvert$ of the order of $M$ (Part 3 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. $D_i$, $\bar d_i$, $g_0$ are costs, $\delta_i$ and $\bar d_i\,\partial^2_{M_i}E_b$ in (11.25) are numbers. 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 $\langle\sigma^{(i)}_3\rangle_0\to1$, $\langle A_i\rangle_0\to0$, $\langle U\rangle_0\to0$, $\Gamma_i\to1$ (an ordered limit in the sense of Step 8: the limit $u\to0$ of the leading-order forms). 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 derived for a bound level that stays simple over the range of the levels of the clock. The condition $D_i\ll g_0$ is sufficient; in $\mathsf R$, where $g_0$ is of order $\epsilon^2$, it is replaced by $D_i\ll M_i$ (Step 10b). Item 5 for a level that is degenerate at some $M_i$ in that range is not derived.
- The results refer to $C_{\rm bod}$ under the assumption $\langle\chi\vert X\vert\chi\rangle=0$. Without it they hold at leading order in $\kappa$ only; the uniform gradient $\kappa\langle X\rangle(\hat x_1+\hat x_2)$ is a position-dependent term and is not treated (out of scope).
- $E_b(P)$ is well defined only as a total cost of the given $C$. Whether the zero of the total cost has any content (a constant in $C$ changes no reading) is not a question of this package.
- Flat values (Step 1a). At a flat value only (11.6) is derived; the other relations of item 2 are not. Whether a bound level can have a flat value and be differentiable there is open; by (11.28) its derivative would be at least $\Omega-M$ in absolute value.
- No joint limit in $\epsilon$ and $u$ is derived for $E_b''(0)$, the expectation values, $\mathfrak m_b$ and $\Gamma_i$. (11.30) is sufficient and, by the code, not necessary; the numbers $c$, $\varepsilon_0$ and $K$ are not computed.
- The class $\mathcal G(\mathcal P_0)$ is sufficient for the mean position labels. The largest class of $g$ for which they exist is not determined.

## Code

`code/v3/check_bound_pair.py` (role: check; outputs `code/v3/output/check_bound_pair.txt`, a log, and `code/v3/output/check_bound_pair.json`, all numbers). Parts 1 and 2 are the script of v2, unchanged; the first 230 lines of the log are identical to `code/v2/output/check_bound_pair.txt`. Part 3 checks what is new in v3. The whole script runs in about 7 minutes (404 s in the run that produced the outputs). The script 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, except the statement of Step 5a and the statement of Step 8 that are marked there.

**Part 1 (the check of v1).**

- **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 Part 1 of the output files; in Part 2 the keys `Gamma_...` are the ratios $\Gamma_i$).
- **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.

**Part 2 (new in v2; its labels I1–I5 are the issues of external round 1).**

- **I3, clock of a part: (11.19), (11.25), (11.26).** For each of the $N_i$ levels $M_i=m_i+d^{(i)}_k$ the matrix $H(0)$ is diagonalized, the eigenvector of the level $0$ is evolved on every level, and the exact statistics $q_i(a;\lambda)$ are formed (total label $P=0$ sharp, no records). They are compared with $P_{N_i}(a;\Gamma\lambda/\lambda_{0,i})$ at $\lambda/\lambda_{0,i}=0.25,0.5,\dots,2N_i$; the deviation is the largest $\lvert q_i-P_{N_i}\rvert$ over $a$ and $\lambda$. $\Gamma_i$ is $\langle\sigma^{(i)}_3\rangle_0$ at $\bar M_i$, $\Gamma^{(0)}_i$ the same at $m_i$ (the rate of v1), and (11.25) uses the sum over the rest of the spectrum.

| set, part, $N_i$ | $D_i$ | $D_i/g_0$ | $\Gamma^{(0)}_i$ | $\Gamma_i$ | (11.25) | deviation with $\Gamma_i$ | deviation with $\Gamma^{(0)}_i$ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| E, 1, 4 | 0.06 | 0.21 | 0.856470 | 0.862190 | 0.862334 | $9.5\times10^{-4}$ | $7.1\times10^{-2}$ |
| E, 1, 4 | 0.03 | 0.10 | 0.856470 | 0.859366 | 0.859402 | $2.5\times10^{-4}$ | $3.6\times10^{-2}$ |
| E, 1, 4 | 0.015 | 0.05 | 0.856470 | 0.857927 | 0.857936 | $6.1\times10^{-5}$ | $1.8\times10^{-2}$ |
| B, 1, 4 | 0.10 | 27.9 | 0.9977124 | 0.9978094 | 0.9978119 | $4.7\times10^{-4}$ | $1.4\times10^{-3}$ |
| B, 1, 4 | 0.05 | 13.9 | 0.9977124 | 0.9977615 | 0.9977621 | $1.3\times10^{-4}$ | $5.9\times10^{-4}$ |
| B, 1, 4 | 0.025 | 7.0 | 0.9977124 | 0.9977371 | 0.9977373 | $3.3\times10^{-5}$ | $3.0\times10^{-4}$ |
| B, 2, 5 | 0.10 | 27.9 | 0.9997133 | 0.9997276 | 0.9997282 | $7.3\times10^{-6}$ | $2.2\times10^{-4}$ |
| B, 2, 5 | 0.05 | 13.9 | 0.9997133 | 0.9997206 | 0.9997208 | $1.9\times10^{-6}$ | $1.1\times10^{-4}$ |
| B, 2, 5 | 0.025 | 7.0 | 0.9997133 | 0.9997170 | 0.9997170 | $4.9\times10^{-7}$ | $5.7\times10^{-5}$ |

  Each halving of the spread at fixed $\lambda/\lambda_{0,i}$ divides the deviation with $\Gamma_i$ by $3.6$ to $4.1$ (second order) and the deviation with $\Gamma^{(0)}_i$ by $1.9$ to $2.4$ (first order). A least-squares fit of the rate gives $0.862146$ (E, $D_i=0.06$) and $0.9977620$ (B, part 1, $D_i=0.05$). In set B the spread is far above $g_0=0.0036$ and the remainder is small all the same, as stated in Step 10(b) with $D_i/M_i$. In $\mathsf R$ (set B): $\partial^2_{M_i}E_b(0)$ is $1.990\times10^{-3}$ against $1.996\times10^{-3}$ from (11.26) (part 1) and $2.976\times10^{-4}$ against $2.952\times10^{-4}$ (part 2); (11.20) at $\bar M_1$ gives $0.9978101$ against $0.9978094$ ($D_1=0.10$).
- **I4, order of $w_{22}$** (set B scaled as above, $\epsilon'=2,1,\frac12,\frac14$): $E_b$ with $w_{22}$ minus $E_b$ with $w_{22}=0$ is $2.9\times10^{-7}$, $4.4\times10^{-9}$, $6.9\times10^{-11}$, $1.1\times10^{-12}$. Each halving divides it by $64.5$, $64.1$, $64.2$ (order $\epsilon^6$) and the weight $\lvert\langle22\vert\psi_0(0)\rangle\rvert^2$ by $16.1$, $16.0$, $16.0$ (order $\epsilon^4$); the difference agrees with $w_{22}\lvert\langle22\vert\psi_0(0)\rangle\rvert^2$ to four digits or better for $\epsilon'\ge\frac12$, and to two digits at $\epsilon'=\frac14$, where it is near the rounding error.
- **I5, $j(P)=0$** (set A, $a_1=a_2$, at $P=\frac\pi2$, $\epsilon'=1,\frac12,\frac14$): $E_b-(M_1+M_2+a+w_{11})$ is $6.2\times10^{-7}$, $3.9\times10^{-8}$, $2.4\times10^{-9}$ (ratios $16.0$); one minus the weight on $(r=0,\lvert11\rangle)$ is $2.5\times10^{-3}$, $6.2\times10^{-4}$, $1.6\times10^{-4}$ (ratios $4.0$, the $O(\epsilon)$ part of the eigenvector); the weight of $\lvert11\rangle$ at $r\neq0$ is $3\times10^{-7}$ at $\epsilon'=1$.
- **I2, twofold levels: (11.23), (11.24)** (set X: $M=(1,0.7)$, $\tau=(0.5,0.35)$, $E_0=1$, the on-site binding $W_0=-G(E_0,P_0)^{-1}$ on the span of $\lvert11\rangle$, $\lvert22\rangle$, which is not diagonal, with $w_{12}=0.15$, $w_{21}=-0.10$; at $P_0=0.3$ and at $P_0=0$). The two eigenvalues at $E_0$ differ by less than $4\times10^{-15}$ and are $0.54$ away from the next one.
  - (11.23): the eigenvalues of $\Pi_b(\partial_\xi H)\Pi_b$ against the slopes of the two branches (five-point differences, the branches followed by their overlap with the branch vectors): below $10^{-11}$ for $M_i$, $\tau_i$, $\theta$ and $3\times10^{-10}$ for $P$.
  - (11.24) against the second differences of the branches: below $5\times10^{-10}$. At $P_0=0$ both first derivatives in $P$ vanish, the branch vectors come from the second-order matrix, and $E''=-0.1063$ and $+0.1929$.
  - (11.9) with $Q_P$: at $P_0=0.3$ the term of the other vector of the level is $6.5\times10^{-4}$ divided by $2\times10^{-15}$.
  - Every eigenvector (five random vectors of the level): the residual of (11.6) is $2\times10^{-15}$, $\langle V_1\rangle-\langle V_2\rangle$ is $10^{-16}$, and $\lVert\Pi_b(V_1-V_2)\Pi_b\rVert$ is $2\times10^{-16}$.
  - The six matrices do not commute: at $P_0=0.3$ the first branch vector of $M_1$ has the squared overlap $0.28$ with that of $\tau_1$ and $0.51$ with that of $P$. In the branch vectors of $M_1$, $\langle U\rangle$ is $+2.0135$ and $-0.7226$, against the slopes $+2.0170$ and $-0.7261$ of the branches of $\theta$.
- **I1, the operator (11.21)** (a small system given in the places: two bodies with a branch label on an open chain of four positions, a medium with three places, a random $X$ and $\chi$, $\lambda=3$). Listing B lists $h\,\sigma^{(1)}_3$, a constant and a part of the motion term of $b_2$ in the record terms. $C_{\rm bod}$ of the two listings agrees to $10^{-15}$, while the listed pair term of B differs from it by $2.6$ in norm. The view of the two bodies against the evolution by $C_{\rm bod}$: $4.1\times10^{-2}$, $1.05\times10^{-2}$, $2.6\times10^{-3}$ for $\kappa=0.04,0.02,0.01$ (ratios $3.9$, $4.0$: second order). With a medium that is not centred, $C_{\rm bod}$ minus the listed $C_{\rm pair}$ is the gradient of (11.21) to $10^{-15}$, and the view against the evolution by the listed $C_{\rm pair}$ is $0.153$, $0.074$, $0.037$ (ratios $2.05$, $2.01$: first order).

**Part 3 (new in v3; I3 and I2 of external round 2).** For the wide pair the level is taken on the infinite chain, as the root of $F_+$ of (11.10) with the integrals over $q$ as means over $2^{15}$ values and the level measured from $M_1+M_2$ without cancellation; a pair with $u\le\frac1{16}$ does not fit into $L=120$. For $u=1$ and $u=\frac12$, both signs of $w_{11}$, the root agrees with the dense diagonalization at $L=120$ to $10^{-14}$. Data: set B with $w_{11}=\mp u\,a$ and the other three $w_{s_1s_2}$ of set B. Here $\epsilon$ stands for $\max_i\tau_i/M_i=0.2$, and every run is repeated at $0.1$ ($\tau_i\to\tau_i/2$, $w_{12},w_{21},w_{22}\to$ a quarter).

- **I3, wide pair.** "Remainder" is the level minus (11.12); "isolation" is the distance of the level from the exact band edge. $\epsilon=0.2$:

| $u$ | $\epsilon^2/u^2$ | remainder over isolation, $w_{11}<0$, $P=0$ | the same, $w_{11}<0$, $P=\frac\pi2$ | the same, $w_{11}>0$, $P=0$ | rel. deviation of the two sides of the second equality of (11.16) |
| --- | --- | --- | --- | --- | --- |
| 2 | 0.01 | $3.9\times10^{-3}$ | $3.7\times10^{-3}$ | 0.018 | $-4.4\times10^{-3}$ |
| 1 | 0.04 | $2.6\times10^{-3}$ | $2.1\times10^{-3}$ | 0.045 | $-6.1\times10^{-4}$ |
| 1/2 | 0.16 | $1.5\times10^{-3}$ | $3.9\times10^{-4}$ | 0.15 | $+5.4\times10^{-4}$ |
| 1/4 | 0.64 | $7.3\times10^{-4}$ | $7.2\times10^{-3}$ | 0.55 | $+6.8\times10^{-4}$ |
| 1/8 | 2.6 | $3.6\times10^{-4}$ | 0.034 | 2.2 | $+5.8\times10^{-4}$ |
| 1/16 | 10 | $1.8\times10^{-4}$ | 0.14 | 8.6 | $+4.8\times10^{-4}$ |
| 1/32 | 41 | $8.8\times10^{-5}$ | 0.58 | 34 | $+4.1\times10^{-4}$ |
| 1/64 | 164 | $4.4\times10^{-5}$ | 2.3 | 137 | $+3.7\times10^{-4}$ |
| 1/128 | 655 | $2.2\times10^{-5}$ | 9.3 | 546 | $+3.4\times10^{-4}$ |

  - $w_{11}<0$, $P=0$: the remainder stays far below the isolation for every $u$. The third column is also the relative deviation of $E_{\rm bind}$ from (11.14); at $\epsilon=0.1$ it is $4.0$ times smaller in every row. So (11.30) is not necessary. The two sides of the second equality of (11.16) (the numerical $E_{\rm bind}$, and $E_b''(0)$ from five-point differences of the roots) differ by less than $7\times10^{-4}$ for $u\le\frac12$, and by $3.9$ to $4.6$ times less at $\epsilon=0.1$.
  - $w_{11}<0$, $P=\frac\pi2$: the remainder tends to $-5.26\times10^{-6}$ ($-3.29\times10^{-7}$ at $\epsilon=0.1$; ratio $16.0$, order $\epsilon^4$). It exceeds the isolation for $u\le\frac1{64}$ (at $\epsilon=0.1$: for $u=\frac1{128}$), and there the value of (11.12) lies above the exact lower band edge.
  - $w_{11}>0$, $P=0$: the remainder tends to $-2.01\times10^{-4}$ ($-1.28\times10^{-5}$ at $\epsilon=0.1$; ratio $15.8$). It exceeds the isolation for $u\le\frac18$ (at $\epsilon=0.1$: for $u\le\frac1{16}$), and there $E_{\rm bind}-2a$ is negative, against $+\frac{w^2}{2a}$ of (11.15). The distance of the level from the exact upper edge agrees with $\sqrt{w^2+a^2}-a$ to $0.5$ per cent ($0.13$ per cent at $\epsilon=0.1$) for every $u$.
- **I3, tight pair** ($w_{11}=-u\,a$, $u=2,4,\dots,128$, dense diagonalization at $L=120$, $P=0$). The relative deviation of $E_{\rm bind}-w_{11}$ from (11.14) is $0.26$ to $0.56$ times $\lvert w_{11}\rvert/M$ for $\lvert w_{11}\rvert/M\le0.39$, at both $\epsilon$ (for example $4.0\times10^{-2}$ and $4.5\times10^{-2}$ at $\lvert w_{11}\rvert/M=0.097$). That of $E_b''(0)$ is $0.19$ to $0.41$ times $\lvert w_{11}\rvert/M$, and that of $1-\langle\sigma^{(1)}_3\rangle_0$ is $0.4$ to $1.4$ times $\lvert w_{11}\rvert/M$. At $\lvert w_{11}\rvert/M=1.55$ ($\epsilon=0.2$, $u=128$) the forms fail: $E_b''(0)$ has the opposite sign. The remainders grow in proportion to $\lvert w_{11}\rvert/M$, that is as $\epsilon^2u$, like $\varrho$ of (11.30).
- **I2, the member with a flat value** ($M_1=M_2=\tau_1=\tau_2=1$, $W_0=w\,\lvert v\rangle\langle v\rvert$ with $w=1$; $\Omega-M=0.414214$). The weight of $v$ on the kernel of $h_0(q)$ is below $10^{-31}$ for all $q$. For $P=0.8,0.4,0.2,0.1,0.05$ the equation $1+w\,g_v(E,P)=0$ has exactly one root above the mixed bands and none below them: $E_b-\Delta(P)=3.2\times10^{-3}$, $4.1\times10^{-4}$, $5.2\times10^{-5}$, $6.5\times10^{-6}$, $8.1\times10^{-7}$ (between $6.3$ and $6.5\times10^{-3}\lvert P\rvert^3$), and $E_b/\lvert P\rvert=0.3754$, $0.4043$, $0.4117$, $0.4136$, $0.4141$. The root at $P=-0.2$ equals that at $P=0.2$, and a finer grid ($2^{17}$ values of $q$) changes no digit shown. A dense diagonalization at $L=240$ gives the same eigenvalue to $6\times10^{-16}$ ($P=0.8$) and to $6\times10^{-11}$ ($P=0.4$, with the weight $2\times10^{-4}$ at $\lvert r\rvert>120$), as the only eigenvalue between $\Delta(P)$ and $1$. At $P=0$ it gives $962=2(2L+1)$ eigenvalues with $\lvert E\rvert\le7\times10^{-15}$, all states of the mixed bands; the nearest other eigenvalue is $-1.81616$, which is the root of $1+w\,g_v(E,0)=0$.

No check failed. The member of Step 1a and the bound (11.30) were derived before Part 3 was run, and the run agrees with them. The sentence of Step 8 that (11.30) is not necessary was written after the run and rests on it.

## Methods used

- Fibre decomposition of a shift-invariant two-body contract (total plane-wave label)
- Weyl's theorem on the essential spectrum; discrete spectrum; degree of a trigonometric polynomial (flat bands)
- Hellmann–Feynman theorem; second-order perturbation theory; Euler's relation for homogeneous functions
- Rellich's theorem; degenerate perturbation theory to second order (reduced resolvent)
- Partial expectation value of the contract in the start of the medium; unique split into single-object terms and a finite-range pair term
- Resolvent (Green function) equation for a rank-four on-site change; monotonicity of a resolvent matrix element in a gap
- Degenerate perturbation theory (Schrieffer–Wolff) and the one-defect chain; stability of an isolated simple eigenvalue under a change of small norm
- Commutator identity for the relative coordinate in a bound state
- Packets of fibre eigenvectors: position labels as derivatives in the total label; Taylor's formula with remainder
- Taylor expansion of the level around the mean clock level; cancellation of an even phase in the reading statistics
- Numerical diagonalization, finite differences, root of the bound-level equation on the infinite chain (check)

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