Version 1 · earlier version; the current one is v2 · External review, round 1: major errors

# Bound level of the pair at the next order in the regime R

- **Subproject:** 05-ilang-free-fall
- **Package:** 14-pair-level-next-order
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-18

## Setup and assumptions

**Inputs** (11-bound-pair@v3, as quoted in question.md): the fibre contract (11.1), $H(P)=A_1+A_2+M_1\sigma^{(1)}_3+M_2\sigma^{(2)}_3+U$ on $\psi(r)\in\mathbb C^2\otimes\mathbb C^2$; the identities (11.5); the reduction of Step 7, $H_{\rm eff}=M_1+M_2+\Pi U\Pi+\Pi\bigl(\frac{A_1^2}{2M_1}+\frac{A_2^2}{2M_2}\bigr)\Pi+O(\epsilon^4)$, and its form (11.11); the level (11.12), the eigenvector (11.13), the values (11.14), the relation (11.16), and the statements of Step 7 (odd orders of $E_b$ vanish; $w_{22}$ enters $E_b$ at order $\epsilon^6$, $w_{12},w_{21}$ never). Notation of question.md: $a_i,a,j(P),\alpha(P),J(P),w=w_{11},\varsigma,u,\Lambda,\mathfrak m_i,\mathfrak m_b,\omega_i$, $p_1+p_2=P$.

**Notation of this package.**
- $E_0:=M_1+M_2$; $\Pi$ projects on the branch $\lvert11\rangle$ (all $r$); $\Pi_0:=\lvert0\rangle\langle0\rvert$ on $r$.
- $T_1:=\frac{i\tau_1}{2}(e^{i\eta P}S^\dagger-e^{-i\eta P}S)$, $T_2:=\frac{i\tau_2}{2}(e^{i(1-\eta)P}S-e^{-i(1-\eta)P}S^\dagger)$, so $A_i=\sigma^{(i)}_1T_i$. The $T_i$ are self-adjoint, commute (functions of $S$), $\lVert T_i\rVert\le\tau_i$; on $e^{iqr}$ they act as $-\tau_i\sin p_i$, $p_1=q+\eta P$, $p_2=(1-\eta)P-q$.
- $e_1:=e^{2i\eta P}$, $e_2:=e^{-2i(1-\eta)P}$, $D_i:=e_iS^{\dagger2}$ (unitary, $(S^{\dagger2}\psi)(r)=\psi(r+2)$); then $T_i^2=\frac{\tau_i^2}{4}(2-D_i-D_i^\dagger)$.
- $\beta_i:=a_i/M_i=\tau_i^2/(4M_i^2)$, $\beta:=\beta_1+\beta_2$, $\nu:=a_1a_2/a^2\in(0,\frac14]$, $\bar s:=\sqrt{1+u^2}$, $s(P):=\sqrt{w^2+j(P)^2}$ (so $s(0)=a\bar s$), $\rho(P)$ as in (11.13).
- $\gamma_s:=\frac{a_1^2}{M_1}+\frac{a_2^2}{M_2}$, $\gamma_c:=\frac{a_2^2}{M_1}+\frac{a_1^2}{M_2}$, $\gamma_m:=a_1a_2\bigl(\frac1{M_1}+\frac1{M_2}\bigr)$; note $a\beta=\gamma_s+\gamma_m$.
- Orders in $\mathsf R$: $a_i,w,\beta_i=O(\epsilon^2)$; $\gamma_s,\gamma_c,\gamma_m=O(\epsilon^4)$; $u,\bar s,\nu$ fixed.

**Assumptions.**
- (S1) The setting, the binding, the regime $\mathsf R$ and the level are those of question.md (M8). The eigenvector $\psi_P$ of the level is a distinguished state (M9, A11). Equal parameters of the two types are never assumed (M7).
- (S2) Isolation: $\lVert A+U\rVert\le\epsilon(\hat\tau_1+\hat\tau_2)+\epsilon^2\max\lvert\hat w_{s_1s_2}\rvert$ is small against $2\min(M_1,M_2)$, the distance from $E_0$ to the other branch values of $H_0$. The Schrieffer–Wolff (SW) series then converges and is analytic in $\epsilon$, $P$ and $M_i$.
- (S3) Hatted data fixed, hence $u\in(0,\infty)$ fixed; the constants in $O(\epsilon^k)$ depend on them and are not claimed uniform in $u$.
- **Class:** every diagonal on-site binding with $\hat w_{11}\neq0$, both signs, all hatted data. (14.1)–(14.10) are derived for every member and (14.12) for every member with $\hat w_{11}<0$, so the class contains no control for them; the control is item 3, (14.11).

## Derivation

### Step 1. A conserved parity

Each $A_i$ shifts $r$ by $\pm1$ and flips the branch of body $i$; $H_0$ and $U$ are diagonal in $r$ and in the branches. Hence
$$
[H(P),\Theta]=0,\qquad \Theta:=(-1)^{\hat r}\,\sigma^{(1)}_3\sigma^{(2)}_3 .
\tag{14.1}
$$
On the coefficient of $\lvert11\rangle$, $\Theta=(-1)^r$. In the sector $\Theta=+1$ the binding acts at $r=0$ only on $\lvert11\rangle$ (by $w_{11}$) and $\lvert22\rangle$ (by $w_{22}$); $\lvert12\rangle,\lvert21\rangle$ at $r=0$ lie in $\Theta=-1$. The vector (11.13) lives on even $r$; the level is simple and $\Theta$ is conserved, so the exact $\psi_P$ lies in $\Theta=+1$, and $w_{12},w_{21}$ enter $E_b$ at no order.

### Step 2. Effective contract through order $\epsilon^4$ (item 1)

Write $H(P)=H_0+V$, $H_0=M_1\sigma^{(1)}_3+M_2\sigma^{(2)}_3$, $V=A+U$, $A=A_1+A_2=O(\epsilon)$, $U=O(\epsilon^2)$, and $R:=(1-\Pi)(E_0-H_0)^{-1}(1-\Pi)$; $R$ multiplies the branches $\lvert21\rangle,\lvert12\rangle,\lvert22\rangle$ by $\frac1{2M_1},\frac1{2M_2},\frac1{2(M_1+M_2)}$. Two facts select the terms: (i) $\Pi A\Pi=0$, $\Pi U(1-\Pi)=0$, $\Pi U\Pi=w\Pi_0$; (ii) a product between $\Pi$ and $\Pi$ with an odd number of factors $A$ vanishes (each $A_i$ flips one branch).

The canonical SW operator of the isolated group at $E_0$ is, to fourth order in $V$ (standard): $E_0+\Pi V\Pi+\Pi VRV\Pi+\Pi VRVRV\Pi-\frac12\{\Pi VR^2V\Pi,\Pi V\Pi\}+\Pi VRVRVRV\Pi-\frac12\{\Pi VR^2V\Pi,\Pi VRV\Pi\}+Y+O(\lVert V\rVert^5)$, where $Y$ collects the fourth-order terms containing the factor $\Pi V\Pi$. By (i)–(ii), $\Pi VRV\Pi=\Pi ARA\Pi=O(\epsilon^2)$, and a chain of three $V$ between projectors reduces to terms $A\,U\,A$ of order $\epsilon^4$. Every term of $Y$ is $\Pi V\Pi=O(\epsilon^2)$ times either a chain of three $V$ or further factors $\Pi V\Pi$ and a chain of two $V$, hence $Y=O(\epsilon^6)$. In the other fourth-order terms only the parts with four $A$ are of order $\epsilon^4$, and the order-$\epsilon^5$ part of the fifth-order terms has five factors $A$ and vanishes. With $K:=\Pi AR^2A\Pi$:
$$
H_{\rm eff}=E_0+w\Pi_0+\Pi ARA\Pi+\Pi ARURA\Pi+\Pi ARARARA\Pi-\tfrac12\bigl\{K,\;w\Pi_0+\Pi ARA\Pi\bigr\}+O(\epsilon^6)
\tag{14.2}
$$
in norm, uniformly in $P$ (since $\lVert T_i\rVert\le\tau_i$), by a unitary that differs from $\mathbb 1$ by $O(\epsilon)$.

Path sums on the coefficient of $\lvert11\rangle$:
- $\Pi ARA\Pi=\frac{T_1^2}{2M_1}+\frac{T_2^2}{2M_2}$ and $K=\frac{T_1^2}{4M_1^2}+\frac{T_2^2}{4M_2^2}$ (one flip and back).
- $\Pi ARURA\Pi=\frac{w_{21}}{4M_1^2}T_1\Pi_0T_1+\frac{w_{12}}{4M_2^2}T_2\Pi_0T_2$: $U$ acts on $\lvert21\rangle$, $\lvert12\rangle$ as $w_{21}\Pi_0$, $w_{12}\Pi_0$, and the way back flips the same branch.
- $\Pi ARARARA\Pi$: the middle state is $\lvert22\rangle$; the paths $11{\to}21{\to}22{\to}21{\to}11$, $11{\to}21{\to}22{\to}12{\to}11$, $11{\to}12{\to}22{\to}21{\to}11$, $11{\to}12{\to}22{\to}12{\to}11$ give $\frac{T_1^2T_2^2}{8(M_1+M_2)}\bigl(\frac1{M_1}+\frac1{M_2}\bigr)^2=\frac{(M_1+M_2)T_1^2T_2^2}{8M_1^2M_2^2}$.
- $-\frac12\{K,\Pi ARA\Pi\}=-\frac{T_1^4}{8M_1^3}-\frac{T_2^4}{8M_2^3}-\frac{(M_1+M_2)T_1^2T_2^2}{8M_1^2M_2^2}$.

The mixed terms cancel:
$$
H_{\rm eff}=E_0+\sum_{i=1,2}\Bigl(\frac{T_i^2}{2M_i}-\frac{T_i^4}{8M_i^3}\Bigr)+w\Pi_0-\frac w2\{K,\Pi_0\}+\frac{w_{21}}{4M_1^2}T_1\Pi_0T_1+\frac{w_{12}}{4M_2^2}T_2\Pi_0T_2+O(\epsilon^6).
\tag{14.3}
$$
On $e^{iqr}$ the sum is $\omega_1(p_1)+\omega_2(p_2)-E_0$ through order $\tau^4$. With $T_i^2=\frac{\tau_i^2}4(2-D_i-D_i^\dagger)$, $\frac{\tau_i^4}{128M_i^3}=\frac{a_i\beta_i}8$, $T_1\lvert0\rangle=\frac{i\tau_1}2(e^{i\eta P}\lvert{-1}\rangle-e^{-i\eta P}\lvert1\rangle)$ and $T_2\lvert0\rangle=\frac{i\tau_2}2(e^{i(1-\eta)P}\lvert1\rangle-e^{-i(1-\eta)P}\lvert{-1}\rangle)$:
$$
\begin{aligned}
(H_{\rm eff}\psi)(r)={}&\Bigl(E_0+\textstyle\sum_i a_i\bigl(1-\frac34\beta_i\bigr)\Bigr)\psi(r)-\frac12\bigl[J_2\,\psi(r+2)+J_2^*\,\psi(r-2)\bigr]-\frac18\bigl[J_4\,\psi(r+4)+J_4^*\,\psi(r-4)\bigr]\\
&+\delta_{r,0}\Bigl[w\bigl(1-\tfrac\beta2\bigr)\psi(0)+\tfrac w8\bigl(J_\beta\,\psi(2)+J_\beta^*\,\psi(-2)\bigr)\Bigr]+\tfrac w8\bigl(\delta_{r,-2}J_\beta+\delta_{r,2}J_\beta^*\bigr)\psi(0)\\
&+\tfrac\Omega4\bigl(\delta_{r,1}+\delta_{r,-1}\bigr)\psi(r)-\tfrac14\bigl(\delta_{r,-1}L\,\psi(1)+\delta_{r,1}L^*\psi(-1)\bigr)+O(\epsilon^6),\\
J_2={}&\textstyle\sum_ia_i(1-\beta_i)e_i,\quad J_4=\sum_ia_i\beta_ie_i^2,\quad J_\beta=\sum_i\beta_ie_i,\quad \Omega=\beta_1w_{21}+\beta_2w_{12},\quad L=\beta_1w_{21}e_1+\beta_2w_{12}e_2 .
\end{aligned}
\tag{14.4}
$$
Ranges: the first line couples $r$ to $r,r\pm2,r\pm4$ for all $r$; the $w_{11}$ terms act at $r=0$ and between $0$ and $\pm2$; the $w_{12},w_{21}$ terms act at $r=\pm1$ and between $-1$ and $1$. Even and odd $r$ stay uncoupled, by (14.1). Through order $\epsilon^2$, (14.4) is (11.11) ($J_2=J+O(\epsilon^4)$). At order $\epsilon^4$ enter $w_{11}$ (also at order $\epsilon^2$), $w_{12}$ and $w_{21}$ (odd $r$ only); $w_{22}$ does not, since $\lvert22\rangle$ needs two flips each way. The code (Part B) confirms (14.4) on a ring.

### Step 3. The level for every $P$ (item 2a)

On even $r$, (14.4) is $E_0+h_2+h_4+O(\epsilon^6)$, with $h_2$ the operator (11.11) minus $E_0$ and $h_4:=-\sum_iT_i^4/(8M_i^3)-\frac w2\{K,\Pi_0\}=O(\epsilon^4)$. With $r=2n$, the vector (11.13), $\psi_P(2n)=\mathcal N_Pe^{-i\alpha n}\rho^{\lvert n\rvert}$, is an exact normalized eigenvector of $E_0+h_2$ with eigenvalue $E_0+a+\varsigma s$: substitution gives $-\frac j2(\rho+\rho^{-1})=\varsigma s$ for $n\ne0$ and $w-j\rho=\varsigma s$ at $n=0$. Its distance from the continuous spectrum $[a-j,a+j]$ of $h_2$ is $s-j=w^2/(s+j)$, of order $\epsilon^2$ at fixed $u$. Perturbation theory for an isolated simple eigenvalue, analytic by (S2), gives $E_b(P)=E_0+a+\varsigma s+\langle\psi_P\vert h_4\vert\psi_P\rangle+O(\epsilon^6)$, with a remainder analytic in $P$ and $M_i$, so that its derivatives in $P$ and $M_i$ are $O(\epsilon^6)$ too.

Moments of (11.13). With $\mathcal N_P^2=\lvert w\rvert/s$, $\rho^2=\frac{s-\lvert w\rvert}{s+\lvert w\rvert}$, and the sum split into $n\ge0$, $n\le-m$ and the $m-1$ terms between, for $m\ge0$:
$$
\langle\psi_P\vert S^{\dagger2m}\vert\psi_P\rangle=e^{-im\alpha}c_m,\qquad c_m=\mathcal N_P^2\sum_{n\in\mathbb Z}\rho^{\lvert n\rvert+\lvert n+m\rvert}=\rho^m\Bigl(1+m\frac{\lvert w\rvert}{s}\Bigr),
\tag{14.5}
$$
so $c_1=-\varsigma j/s$ and $c_2=\frac{j^2(s+2\lvert w\rvert)}{s(s+\lvert w\rvert)^2}$. Define $\theta_i$ by $je^{i\theta_1}=a_1+a_2e^{2iP}$, $je^{i\theta_2}=a_2+a_1e^{-2iP}$; since $e^{-i\alpha}=J^*/j$, $e_ie^{-i\alpha}=e^{i\theta_i}$ and $\langle D_i^m\rangle=e^{im\theta_i}c_m$. Put $\kappa_i:=j\cos\theta_i$, $\lambda_i:=j^2\cos2\theta_i$: $\kappa_1=a_1+a_2\cos2P$, $\lambda_1=a_1^2+2a_1a_2\cos2P+a_2^2\cos4P$, and $\kappa_2,\lambda_2$ by $a_1\leftrightarrow a_2$.

Now $\frac{T_i^4}{8M_i^3}=\frac{a_i\beta_i}8(6-4D_i-4D_i^\dagger+D_i^2+D_i^{\dagger2})$ has expectation $\frac{a_i\beta_i}8(6-8c_1\cos\theta_i+2c_2\cos2\theta_i)$. With $K=\frac\beta2-\frac14\sum_i\beta_i(D_i+D_i^\dagger)$, $-\frac w2\langle\{K,\Pi_0\}\rangle=-w\,{\rm Re}\bigl[\psi_P(0)^*(K\psi_P)(0)\bigr]=-\frac{\varsigma w^2}{2s}(\beta-\rho g)$, $g:=\beta_1\cos\theta_1+\beta_2\cos\theta_2$. With $\varsigma\rho=-j/(s+\lvert w\rvert)$:
$$
E_b(P)=E_0+a+\varsigma s-\sum_{i=1,2}\frac{a_i^2}{8M_i}\Bigl[6+\frac{8\varsigma\kappa_i}{s}+\frac{2(s+2\lvert w\rvert)\lambda_i}{s(s+\lvert w\rvert)^2}\Bigr]-\frac{w^2}{2s}\Bigl[\varsigma\beta+\frac{\beta_1\kappa_1+\beta_2\kappa_2}{s+\lvert w\rvert}\Bigr]+O(\epsilon^6).
\tag{14.6}
$$
It is even in $P$ and independent of $\eta$.

### Step 4. Binding cost and rest terms (item 2b)

At $P=0$: $j=a$, $\kappa_i=a$, $\lambda_i=a^2$, $s=a\bar s$, and $\frac{\bar s+2u}{\bar s(\bar s+u)^2}=1-2u^2+\frac{2u^3}{\bar s}$ (use $(\bar s+u)^{-1}=\bar s-u$). Then (14.6) gives
$$
E_{\rm bind}=a(1+\varsigma\bar s)+\mathcal E_4+O(\epsilon^6),\qquad \mathcal E_4=-\frac{\gamma_s}2\,\phi_A-\frac{\gamma_s+\gamma_m}2\,\phi_B,\qquad \phi_A=2-u^2+\frac{2\varsigma+u^3}{\bar s},\quad \phi_B=\frac{u^2(\varsigma+\bar s-u)}{\bar s}.
\tag{14.7}
$$
By (11.5), $\langle\sigma^{(i)}_3\rangle_0=1+\partial E_{\rm bind}/\partial M_i$ at fixed $\tau_i$ and $w_{s_1s_2}$. With $\partial_{M_i}a=-\beta_i$, $\partial_{M_i}u=u\beta_i/a$, $\partial_{M_i}\gamma_s=-3\beta_i^2$ and $\partial_{M_i}(a\beta)=-\beta_i\beta-2a\beta_i/M_i$, the leading term gives $-\beta_i\Lambda$ (as in (11.14)), and
$$
\langle\sigma^{(i)}_3\rangle_0=1-\beta_i\Lambda+\beta_i\,\Xi_i+O(\epsilon^6),\qquad \Xi_i=\frac32\beta_i\phi_A-\frac{\gamma_s}{2a}\,u\phi_A'+\frac12\Bigl(\beta+\frac{2a}{M_i}\Bigr)\phi_B-\frac\beta2\,u\phi_B',
\tag{14.8}
$$
with $u\phi_A'=-2u^2+\frac{3u^3}{\bar s}-\frac{u^2(2\varsigma+u^3)}{\bar s^3}$ and $u\phi_B'=2u^2-\frac{3u^3}{\bar s}+\frac{\varsigma u^2(2+u^2)+u^5}{\bar s^3}$.

### Step 5. Curvature and curvature inertia (items 2b, 2c)

The order-$\epsilon^4$ part of (14.6) is $-\sum_i\frac{a_i\beta_i}8(6-8c_1\cos\theta_i+2c_2\cos2\theta_i)-\frac{\varsigma w^2}2\bigl(\frac\beta s-\frac\rho s\,g\bigr)$. All blocks are even in $P$; $c_1,c_2,s,\rho$ depend on $P$ through $j^2=a^2-4a_1a_2\sin^2P$, and $\sin\theta_1=a_2\sin2P/j$, $\sin\theta_2=-a_1\sin2P/j$ give $\theta_1'=2a_2/a$, $\theta_2'=-2a_1/a$ (primes: $\partial_P$ at $P=0$). Differentiating through $j^2$ ($\partial_{j^2}s=\frac1{2s}$, $\partial_{j^2}\ln\rho=\frac{\lvert w\rvert}{2j^2s}$) and using $(\bar s-u)^2(u+2\bar s)=2\bar s^3-3u-2u^3$:
$c_1''=\frac{4\varsigma\nu u^2}{\bar s^3}$, $c_2''=-\frac{8\nu u^2(2\bar s^3-3u-2u^3)}{\bar s^3}$, $a(1/s)''=\frac{4\nu}{\bar s^3}$, $a(\rho/s)''=\frac{4\varsigma\nu(\bar s-u)(u\bar s-1)}{\bar s^3}$, $g''=-\frac{4\nu(\gamma_c+\gamma_m)}a$, at $c_1=-\frac\varsigma{\bar s}$, $c_2=1-2u^2+\frac{2u^3}{\bar s}$, $a\rho/s=-\frac{\varsigma(\bar s-u)}{\bar s}$, $g=\beta$.
The second derivative of the order-$\epsilon^4$ part is $-\sum_i\frac{a_i\beta_i}8\bigl[-8c_1''+2c_2''+8\theta_i'^2(c_1-c_2)\bigr]-\frac{\varsigma w^2}2\bigl[\beta(1/s)''-\beta(\rho/s)''-(\rho/s)g''\bigr]$. With $\sum_ia_i\beta_i\theta_i'^2=4\nu\gamma_m$ and $(\bar s-u)(u\bar s-1)=2u+u^3-\bar s^3$:
$$
\begin{aligned}
E_b''(0)={}&-\frac{4\varsigma\nu a}{\bar s}+\nu\,\mathcal C+O(\epsilon^6),\\
\mathcal C={}&\frac{2u^2}{\bar s^3}\bigl[\gamma_s(\varsigma+\bar s^3-u-u^3)-\gamma_m(\varsigma+\bar s^3-2u-u^3)\bigr]+4\gamma_m\Bigl(1+\frac\varsigma{\bar s}-2u^2+\frac{2u^3}{\bar s}\Bigr)+\frac{2u^2(\bar s-u)}{\bar s}(\gamma_c+\gamma_m).
\end{aligned}
\tag{14.9}
$$
The first term is $E_b''(0)$ of (11.14). Since $\mathfrak m_1+\mathfrak m_2=a/(4a_1a_2)$, inverting (14.9) gives
$$
\mathfrak m_b=-\varsigma(\mathfrak m_1+\mathfrak m_2)\sqrt{1+u^2}\,\Bigl[1+\frac{\varsigma\bar s}{4a}\,\mathcal C+O(\epsilon^4)\Bigr].
\tag{14.10}
$$

### Step 6. Inertia against the binding cost (item 3)

With $4\mathfrak m_1\mathfrak m_2=1/(4a_1a_2)$ and the identity $\mathfrak m_1+\mathfrak m_2-\frac{a(1+\varsigma\bar s)}{4a_1a_2}=-\varsigma\bar s(\mathfrak m_1+\mathfrak m_2)$ (the leading order of (11.16)), (14.7) and (14.10) give
$$
\mathfrak m_b-\bigl(\mathfrak m_1+\mathfrak m_2-4\mathfrak m_1\mathfrak m_2E_{\rm bind}\bigr)=-\frac{(1+u^2)\,\mathcal C-4\,\mathcal E_4}{16\,a_1a_2}+\mathfrak m_b\,O(\epsilon^4),
\tag{14.11}
$$
whose first term, divided by $\mathfrak m_b$, is $\frac{\varsigma[(1+u^2)\mathcal C-4\mathcal E_4]}{4a\bar s}+O(\epsilon^4)$, of order $\epsilon^2$. It does not vanish identically. For $w_{11}<0$ and $u\to0$, (14.12) gives $(1+u^2)\mathcal C-4\mathcal E_4=2(\gamma_c-2\gamma_m)u^2+O(u^3)$, nonzero at small enough $u$ unless $\gamma_c=2\gamma_m$ (equal data of the two bodies: $\gamma_c-2\gamma_m=-2a_1^2/M_1$). For the data sets D1, D2, D3 of the code, the predicted relative differences are $-7.0\times10^{-4}\,\epsilon^2$, $1.07\,\epsilon^2$, $2.40\times10^{-3}\,\epsilon^2$; the numerical differences agree with them to $O(\epsilon^4)$ (log, Part A, "item 3").

### Step 7. Wide pair (item 4)

For $\varsigma=-1$ and $u\to0$ in the coefficients ($\bar s=1+\frac{u^2}2+O(u^4)$): $\phi_A=u^3-\frac34u^4$, $u\phi_A'=3u^3-3u^4$, $\phi_B=-u^3+\frac12u^4$, $u\phi_B'=-3u^3+2u^4$, each up to $O(u^5)$. Inserting into (14.7)–(14.10):
$$
\begin{aligned}
&\mathcal E_4=\tfrac12\gamma_m\,u^3+O(u^4),\qquad \nu\,\mathcal C=2\nu(\gamma_c-2\gamma_m)\,u^2+2\nu(5\gamma_m-\gamma_s-\gamma_c)\,u^3+O(u^4),\\
&\beta_1\Xi_1=\beta_1\Bigl\{\frac{a_2}{2a}\Bigl[\frac{a_1-2a_2}{M_1}+\frac{2a_1-a_2}{M_2}\Bigr]u^3+\Bigl[\frac{a_1^2-7a_1a_2+4a_2^2}{8aM_1}+\frac{3a_2(a_2-a_1)}{4aM_2}\Bigr]u^4\Bigr\}+O(u^5),\\
&\frac{\varsigma\bar s}{4a}\,\mathcal C=-\frac{\gamma_c-2\gamma_m}{2a}\,u^2-\frac{5\gamma_m-\gamma_s-\gamma_c}{2a}\,u^3+O(u^4),
\end{aligned}
\tag{14.12}
$$
and $\beta_2\Xi_2$ by exchanging the labels 1 and 2. In each line the leading term is the first displayed one if its coefficient is nonzero, otherwise the second if its coefficient is nonzero. $\gamma_m>0$, so the leading term of $\mathcal E_4$ never vanishes. For equal data the $u^3$ term of $\beta_i\Xi_i$ vanishes and $\beta_i\Xi_i=-\frac{\beta_i^2}8u^4+O(u^5)$. Code, Part C: at $u=10^{-3}$ the ratios of the exact coefficients of (14.7)–(14.10) to these leading terms lie between 0.9970 and 1.0040, with deviations proportional to $u$.

## Result

- **(14.1)** [theorem; every diagonal on-site binding, every $P$, exact]: $\Theta=(-1)^{\hat r}\sigma^{(1)}_3\sigma^{(2)}_3$ commutes with $H(P)$. In the class, the level lies in $\Theta=+1$, so $w_{12},w_{21}$ enter $E_b$ at no order.
- **(14.3)–(14.4)** [formula; every member of the class, every $P$, under (S2)]: the operator on $\psi(r)$ that replaces (11.11), unitarily equivalent to the part of $H(P)$ near $M_1+M_2$ through order $\epsilon^4$, with remainder $O(\epsilon^6)$ in norm uniformly in $P$. The terms and ranges are those of (14.4). At this order enter $w_{11}$ (orders $\epsilon^2$, $\epsilon^4$), and $w_{12},w_{21}$ (order $\epsilon^4$, on odd $r$ only); $w_{22}$ does not enter.
- **(14.6)** [formula; class, every $P$, fixed hatted data, both signs]: $E_b(P)$ through order $\epsilon^4$, remainder $O(\epsilon^6)$ with constants depending on the hatted data; even in $P$ and independent of $\eta$.
- **(14.7)** [formula; class, fixed hatted data]: $E_{\rm bind}=a(1+\varsigma\bar s)+\mathcal E_4+O(\epsilon^6)$.
- **(14.8)** [formula; class, fixed hatted data]: $\langle\sigma^{(i)}_3\rangle_0=1-\beta_i\Lambda+\beta_i\Xi_i+O(\epsilon^6)$, $i=1,2$, from (11.5).
- **(14.9)** [formula; class, fixed hatted data]: $E_b''(0)=-4\varsigma\nu a/\bar s+\nu\,\mathcal C+O(\epsilon^6)$.
- **(14.10)** [formula; class, fixed hatted data]: $\mathfrak m_b=-\varsigma(\mathfrak m_1+\mathfrak m_2)\sqrt{1+u^2}\,[1+\frac{\varsigma\bar s}{4a}\mathcal C+O(\epsilon^4)]$, the relative correction being of order $\epsilon^2$.
- **(14.11)** [negative; class, fixed hatted data; the control of item 3]: with $E_{\rm bind}$ of (14.7), the relation (11.16) holds at relative order $\epsilon^2$ only on the data with $(1+u^2)\mathcal C=4\mathcal E_4$. Otherwise it fails, with leading difference $-[(1+u^2)\mathcal C-4\mathcal E_4]/(16a_1a_2)$, of relative order $\epsilon^2$. The difference does not vanish identically: it is nonzero for $w_{11}<0$ at small enough $u$ when $\gamma_c\neq2\gamma_m$ (e.g. equal data), and for the data sets D1–D3.
- **(14.12)** [formula; $w_{11}<0$, ordered limit: $\epsilon\to0$ at fixed $u$, then $u\to0$ in the coefficients]: $\mathcal E_4\simeq\frac12\gamma_mu^3$. For $\nu\mathcal C$ and the relative correction of $\mathfrak m_b$ the leading term is of order $u^2$, with coefficients $2\nu(\gamma_c-2\gamma_m)$ and $-(\gamma_c-2\gamma_m)/(2a)$. Where these vanish it is the $u^3$ term of (14.12), if that term's coefficient is nonzero. For $\beta_i\Xi_i$ the leading term is of order $u^3$. Where its coefficient vanishes and the $u^4$ coefficient does not (e.g. equal data), it is the $u^4$ term. All coefficients are as displayed in (14.12).

## Consistency checks

1. **Dimensions and scaling.** $a_i,w,s,\kappa_i,\gamma_{s,c,m}$ are costs; $\lambda_i$ is a squared cost; $\beta_i,u,\nu,c_m$ are dimensionless. Every term of (14.6), (14.7) and (14.9) is a cost, and (14.8) and the bracket of (14.10) are dimensionless. Under $(M_i,\tau_i,w_{s_1s_2})\to\ell(M_i,\tau_i,w_{s_1s_2})$, $H(P)\to\ell H(P)$; (14.6), (14.7) and (14.9) scale as $\ell$, (14.8) is invariant, and (14.10) scales as $\ell^{-1}$, as they must. Passed.
2. **No binding, ordered limit $u\to0$, $w_{11}<0$.** In (14.6), $s\to j$, the first bracket tends to $6-8\cos\theta_i+2\cos2\theta_i=4(1-\cos\theta_i)^2$, and the $w^2$ term vanishes. Since $a_1\sin\theta_1+a_2\sin\theta_2=0$ and $\theta_1-\theta_2=2P$, $(p_1^*,p_2^*)=(\theta_1/2,-\theta_2/2)$ minimizes the leading band. The order-$\epsilon^4$ part therefore tends to $-\sum_i\frac{\tau_i^4\sin^4p_i^*}{8M_i^3}$, and $E_b(P)\to\min_{p_1+p_2=P}[\omega_1+\omega_2]$ through order $\epsilon^4$: by stationarity, the shift of the minimizer does not enter at this order. At $P=0$ every order-$\epsilon^4$ coefficient tends to 0 (14.12). Passed.
3. **Numerical diagonalization** (Code, log Parts A–C). For D1–D3, the deviations from (11.14) scale as $\epsilon^4$ (fitted slopes 3.94–4.27), those from (14.6)–(14.9) as $\epsilon^6$ (slopes 5.94–6.02; one slope of 5.37 at a deviation of $6\times10^{-12}$, the finite-difference noise level). The eigenvalue deviations of the effective operator (14.4) on a ring scale as $\epsilon^6$ (slopes 5.97, 5.99), and without the $w_{12},w_{21}$ terms they approach $\epsilon^4$ (slopes 4.85, 4.26). Passed.

## Open issues

- The remainders are not shown to be uniform in $u$ (the gap $s-j$ vanishes as $u\to0$); (14.12) is derived only as the ordered limit.
- Where both displayed coefficients of a line of (14.12) vanish, the leading term in $u$ is not given.
- The data on which $(1+u^2)\mathcal C=4\mathcal E_4$, where (11.16) survives at relative order $\epsilon^2$, are not characterized.
- On odd $r$ (sector $\Theta=-1$), the $w_{12},w_{21}$ terms of (14.4) may produce a further level near the band; other levels are out of scope.

## Code

- `code/v1/check_pair_level.py` (role: check). Output: `code/v1/output/check_pair_level.log`, which contains every number cited here.
  - **Part A.** Diagonalizes $H(P)$ of (11.1) in the sector $\Theta=+1$ on the open range $r\in[-L,L]$, with $L=2\lceil40/\ln(1/\lvert\rho(0)\rvert)\rceil$ (so $\lvert\psi_P(\pm L)\rvert\lesssim e^{-40}$): $L=112,80,408$ for D1, D2, D3. It takes $\eta=\frac12$ and $\epsilon\in\{0.2,0.1,0.05\}$. The data sets (hatted; all $w_{s_1s_2}$ nonzero) are:
    - D1: $\hat\tau=(1,1.6)$, $M=(1,2.5)$, $\hat w_{11}=-0.4$, $u=0.79$;
    - D2: $\hat\tau=(1.3,0.8)$, $M=(1.5,0.7)$, $\hat w_{11}=+0.6$, $u=1.18$;
    - D3: $\hat\tau=(1.2,0.9)$, $M=(0.8,1.3)$, $\hat w_{11}=-0.12$, $u=0.198$.

    The level is the lowest ($w_{11}<0$) or highest ($w_{11}>0$) eigenvalue of the cluster above $E_0-\min M_i$. $E_b''(0)$ comes from a 7-point finite-difference stencil in $P$ with step $h=0.05$ (using $E_b(-P)=E_b(P)$), with a control stencil of step $0.1$. Convergence was checked by repeating every run with $1.5L$, which changes all quantities by at most $6\times10^{-12}$. The two stencils differ by at most $2\times10^{-9}$, the truncation error of the step-$0.1$ stencil, which is $2^6$ times larger than that of the step-$0.05$ one. Part A compares $E_{\rm bind}$, $E_b''(0)$, $\langle\sigma^{(i)}_3\rangle_0$ and $E_b(0.3)$ with (11.14) and with (14.6)–(14.9), gives the slopes in $\epsilon$, and compares item 3 with (14.11) and $\mathfrak m_b$ with (14.10).
  - **Part B.** Compares the cluster eigenvalues of $H(P)$ on a ring of 40 sites ($P=0.37$, $\eta=0.3$, D1) with (14.4) built from its explicit coefficients.
  - **Part C.** Evaluates the ratios of the order-$\epsilon^4$ coefficients of (14.7)–(14.10) to the leading terms of (14.12) at $u=10^{-2},10^{-3}$, for the data of D3 and for equal data.

## Methods used

- Canonical Schrieffer–Wolff reduction to fourth order
- Conserved parity and sector reduction
- Perturbation theory of an isolated simple eigenvalue; Hellmann–Feynman identities (11.5)
- Lattice contact bound state, geometric sums
- Taylor expansion in $P$, asymptotic expansion in $u$
- Exact diagonalization, finite-difference stencils, convergence in system size

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