Version 4 · current

# Inertia of a pair bound through the field

- **Subproject:** 07-ilang-weight
- **Package:** 06-inertia-of-pair
- **Version:** v4
- **Mode:** external regeneration
- **Date:** 2026-10-21

## Changes from previous version

- Setup: new B0 (no cancellation of the relative hopping, $\nu_t$ not small, with its hierarchy) for all items; B3 replaced by the hierarchy $\eta_4\ll1$ of (6.19), with sufficient conditions in $\epsilon_4$, $\nu_t$ and the relative-level gap $\Delta_{\rm rel}$; (N) made a class condition of item 4; the gap of (6.16) renamed $\Delta_{\rm rel}$. Reason: note (i), (ii) of 2026-10-21; I1, I2, I3.
- Step 2: the sign argument of (6.3) rests on B0, and (6.3) is stated under R, B0 and $\chi L^3\gg1$. Reason: I2.
- Step 3: the curvature error from the omitted orders is bounded by (6.19), through the gauge frame, $\nu_t$ and $\Delta_{\rm rel}$. Reason: I1.
- Step 4: "outside B2" no longer includes the near-cancellation of $\mu_1+\mu_2$ (excluded by B0). Reason: note (i).
- Step 5: the zero family is checked against B0 and B3; with the R1 estimate of $\epsilon_4$ it misses the gap part of B3, so it is no longer claimed inside B; its gaps use the margins of Step 7. Reason: I1, I4.
- Step 6: (6.14) stated in relative form under (N); the paragraph on $g_1+g_2=0$ and the corresponding parts of (6.13), (6.14) removed. Reason: note (ii), I3.
- Step 7: the R1 gaps are distances between sets of held-pair eigenvalues; the $R_i$-based formulas are leading-order estimates with margin $w$; the $t_f$ power laws hold at leading order, with the isolated-body quantities at that accuracy; the $t_f$-dependence of B3 added. Reason: I4, I5, I1.
- Code: `check_inertia.py` adds the omitted terms $\Delta_P$ (exact effective contract on $\mathcal G$ minus its order-$\tau^2$ part), the ratios of B3 and $\eta_4$, and the R1 estimate with its margin; `check_weight.py` drops set D. Reason: I1, I3, I4.

## Response to verification

- **I1:** Accepted and fixed in the Setup (B0, B3) and in Step 3, (6.19). In the gauge frame of Step 3 the order-$\tau^2$ part has no first $P$-derivative, and the error of the curvature has three terms: the second $P$-derivative of the omitted terms, the shift of $f_0$ (reduced resolvent, gap $\Delta_{\rm rel}$), and a resolvent term in their first derivative. The gauge phases have derivatives up to $2/\nu_t$, and $\lvert m_0\rvert=\nu_t/(8\min_i\lvert\bar\kappa_i\rvert)$. So $\eta_4=O(\epsilon_4/(\beta_0\nu_t\min_i\lvert\bar\kappa_i\rvert))+O(\epsilon_4/(\beta_0\Delta_{\rm rel}))$, where $\epsilon_4$ contains each body's own order-$\tau^4$ dispersion. The near-cancellation of your example ($\nu_t\to0$) is excluded by note (i) through B0. Conclusions are restricted accordingly: the zero family of Step 5 is no longer claimed inside B.
- **I2:** Accepted and fixed in the Class and Step 2. (6.3) is stated under B0, which keeps every bond $t(\rho;0)$ at the sign of $\bar t_0$ and excludes $\bar t_0=0$.
- **I3:** Accepted and settled by note (ii). Item 4 is now asked only under (N); the material on $g_1+g_2=0$ is removed from Step 6 and the Result, except one sentence in Open issues. Set D is dropped from `check_weight.py`; it stays in `check_inertia.py`, since items 1–3 include it.
- **I4:** Accepted and fixed in Step 7 and Step 5. $\delta_1$ and $\delta$ are distances between sets of held-pair eigenvalues. The $R_i$-based formulas are leading-order estimates with margin $w=2\max_r\lvert M(r)\rvert+n(\lvert g_1\rvert+\lvert g_2\rvert)$, which contains the width of $\mathcal G$. The code finds $\lvert\delta_1-{\rm estimate}\rvert\le w$ in all 12 cases.
- **I5:** Accepted and fixed in Step 7 and (6.15). The power laws hold at leading order in $\varepsilon$, with $\mu_i$, $\bar\kappa_i$, $\bar t_0$, $\nu_t$, $c_i$ at that accuracy. Their relative $O(\varepsilon^2)$ corrections and the $t_f$-dependence of these corrections are not tracked.

## Setup and assumptions

**Setting (assumptions, M8, M9).** As in question.md: $N_f\ge1$ field objects ($c=+1$) on a ring of even $L\ge6$ places with own term $C_1=-t_f\sum_x(\lvert x\rangle\langle x+1\rvert+{\rm h.c.})$, $\Phi_x$ the number at $x$, $n=N_f/L$; two Model D bodies $b_1,b_2$ of different types, clock levels $k_i$ with half-gaps $M_i$, branches $\sigma_i$, coupled by (a) or (c) with constants $\gamma_i$, $g_i:=\sigma_i\gamma_iM_i$, no pair term between them, records at $\kappa\to0$; the field in its lowest state adapted to the bodies; regime R (R1, R2) as stated there. Order counting: $\varepsilon:=\max_i\lvert\gamma_i\rvert$ ($g_i=O(\varepsilon)$). Speeds of (2.14): $c_i=\tau_i$ (a), $c_i=\tau_i(1+\gamma_in)$ (c); $s_c:=+1$ (a), $-1$ (c).

**Inputs** (verbatim in question.md). From 07-ilang-weight/02-two-free-bodies: the joint translation $\mathcal T=e^{-i\hat P}$, $[\mathcal T,C]=0$ (2.1); $E(x_1,x_2)=\sigma_1M_1+\sigma_2M_2+N_f\varepsilon_0(x_1,x_2)$ (2.3); the band contract $H_{\rm eff}=W+K_1+K_2+O(\tau^4)$, $W={\rm diag}\,E(j,l)$ (2.10), $K_i$ moving $b_i$ by $0,\pm2$; $E''_{k_1,\sigma_1}(0)=-8\bar\kappa$ (2.11), hence $\mu_i=-1/(8\bar\kappa_i)$; $\mu_i=\sigma_iR_i/c_i^2+O(\gamma_i^2)$ with $R_i=M_i(1+\gamma_in)+O(\gamma_i^3)$ (2.14); $\delta\kappa(j,l)/\bar\kappa=\mp\gamma_1\overline{\delta\rho}_l(j)+r_1$, $\gamma_1\delta\rho_l(y)=\sigma_1M^{(2)}(y,l)/M_1+O(\gamma_1g_2^2)$, $r_1=O(\varepsilon^3)$ (2.16), upper sign (a). From 06-ilang-source/04-profile: $M=N_f[\varepsilon_0(g_1,g_2)-\varepsilon_0(g_1,0)-\varepsilon_0(0,g_2)+\varepsilon_0(0,0)]$ (4.2). From 06-ilang-source/07-dressed-fall: $a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}W'_{k,\sigma}(j)+O(g_s^2)+O(\tau^4\gamma g_s)$ (7.7). From 07-ilang-weight/04-weight-of-bond: $\ell(r)=\frac{L^2-1}{12L}-\frac{r(L-r)}{2L}$ and $E(r)-E(r')=\frac{ng_1g_2}{t_fL}[r(L-r)-r'(L-r')]+O(n\lvert g_1g_2\rvert\ell_0\eta/t_f)$ (4.3); $W_i(x)=ng_i\phi(x)+O(\varepsilon^2g_s)$ with $\phi(x)=-\frac{2g_s}{t_f}\ell(x-x_s)$ (4.5); $W_{\rm pair}=W_1(x_1)+W_2(x_2)+W_b$ with $W_b$ (4.7).

**Notation.** $\bar t_0:=\bar\kappa_1+\bar\kappa_2=-\tfrac18(\mu_1^{-1}+\mu_2^{-1})$; $m_0:=\mu_1+\mu_2$; $S:=c_1^{-2}+c_2^{-2}$; $F:=n\lvert g_1g_2\rvert/t_f$; binding ratio $\chi:=F/\lvert\bar t_0\rvert$; signs $s_e:={\rm sgn}(g_1g_2)$, $s_\mu:={\rm sgn}(\mu_1^{-1}+\mu_2^{-1})=-{\rm sgn}\,\bar t_0$, $s_\beta:=s_es_\mu$; $c_{\min}$, $c_{\max}$ the smaller and the larger of $c_1,c_2$; $E_i(P)$ the band level of $b_i$ alone with the field, $E_\varnothing$ the lowest eigenvalue of the field alone. Non-cancellation ratio of the relative hopping $\nu_t:=\lvert\mu_1^{-1}+\mu_2^{-1}\rvert/\max_i\lvert\mu_i^{-1}\rvert=\lvert\bar t_0\rvert/\max_i\lvert\bar\kappa_i\rvert$, so that $\lvert m_0\rvert=\nu_t\max_i\lvert\mu_i\rvert=\nu_t/(8\min_i\lvert\bar\kappa_i\rvert)$. $\Delta_{\rm rel}$: the distance of the selected level from the other levels of $h_0$ (relative-level gap). $\epsilon_4$: a bound on the norms, at $P=0$, of the terms of the sector band contract beyond order $\tau^2$ and of their first two $P$-derivatives (body 1 as representative). R1 gives $\epsilon_4=O(\tau'^4/(\delta_1^2\delta))$. $\epsilon_4$ contains each body's own order-$\tau^4$ dispersion, of amplitude $O(\tau_i'^4/R_i^3)$ ($-\sigma_i\tau_i^4\sin^4p/(8M_i^3)$ for $\gamma_i=0$).

**Binding regime B (assumption).**
- B0 (no cancellation of the relative hopping; note (i) of 2026-10-21): $\nu_t$ is not small, with the hierarchy $\max_{\rho,i}\lvert\kappa_i(\rho)-\bar\kappa_i\rvert\ll\lvert\bar t_0\rvert$ (at fixed $n,t_f,L$: $\nu_t\gg\varepsilon^2$) and $\epsilon_4\ll\nu_t\min_i\lvert\bar\kappa_i\rvert$.
  - The first condition keeps every relative bond $t(\rho;0)=\bar t_0+\sum_i(\kappa_i(\rho)-\bar\kappa_i)$ at the sign of $\bar t_0$.
  - The second keeps the omitted orders, in particular the bodies' own order-$\tau^4$ dispersions, small against the retained relative hopping $\lvert\bar t_0\rvert\ge\nu_t\min_i\lvert\bar\kappa_i\rvert$.
  - For $\sigma_1=\sigma_2$, $\nu_t\ge1$. B0 excludes $\bar t_0=0$, where $\chi$ and (6.3) are not defined, and, since $\lvert\mu_1+\mu_2\rvert=\nu_t\max_i\lvert\mu_i\rvert$, the near-cancellation of $\mu_1+\mu_2$.
- B1: the selected level (Step 2) decays on a scale $\xi_0\ll L$; this needs $\chi L^3\gg1$, with $\xi_0\simeq(4/\chi)^{1/3}$ for $\chi\lesssim1$ and $\xi_0\simeq1$ for $\chi\gtrsim1$.
- B2 (no cancellation of the pair inertia by the mark): $\bar x:=\max_\rho\lvert m(\rho)-m_0\rvert/\lvert m_0\rvert\ll1$, with $m(\rho)$ of (6.5). By (6.7) it holds if $S\max_\rho\lvert\bar M^{(2)}(\rho)\rvert\ll\lvert\mu_1+\mu_2\rvert$ and $\varepsilon^3\ll\nu_t$; for $\sigma_1=\sigma_2$, when $\lvert M^{(2)}(0)\rvert\ll\min_iR_i$.
- B3 (hierarchy of the omitted orders; note (i)): $\eta_4\ll1$, with $\eta_4$ of (6.19). Sufficient: $\epsilon_4\ll\beta_0\nu_t\min_i\lvert\bar\kappa_i\rvert$ and $\epsilon_4\ll\beta_0\Delta_{\rm rel}$ (Step 3).
- B4 (ring hierarchy): the ring remainder $\rho_L$ of (6.16) is small against every retained relative term: $\lvert\rho_L\rvert\ll1-\beta_0$ for the leading statements, and $\lvert\rho_L\rvert\ll\lvert M^{(2)}(0)\rvert/R_i$ where the mark terms are kept.

**Condition (N)** (item 4 only; note (ii) of 2026-10-21): the first-order weights do not cancel, $\nu_g:=\lvert g_1+g_2\rvert/\max_i\lvert g_i\rvert$ is not small, with the hierarchy that $n\lvert g_1+g_2\rvert\lvert\phi'\rvert$ is large against the second-order weight differences $O(\varepsilon^2g_s)$; at fixed $n,t_f,L$ this reads $\nu_g\gg\varepsilon$.

**Class.** Item 1: couplings (a), (c), $N_f\ge1$, even $L\ge6$, all branch combinations, both signs of $\gamma_1\gamma_2$, under R, B0 and $\chi L^3\gg1$. Items 2–5: the descriptions of this class in which the selected level is bound ($s_\beta=+1$, (6.3)), under R and B (B0–B4); item 4 in addition under (N). Results are at order $\tau_i^2$ in the band contract (the omitted orders are bounded by (6.19)) and, unless stated otherwise, at leading order in $\varepsilon$ at fixed $\chi$.

## Derivation

### Step 1. The band contract in a sector of $P$

By (2.1) the group $\mathcal G$ and (2.10) are invariant under $\mathcal T$; take every held field state $\Omega_{jl}$ real (lowest state of a real symmetric contract) and $\Omega_{j+1,l+1}$ the translate of $\Omega_{jl}$. Then $E(j,l)=E(l-j)$, and with $\rho:=l-j$ define $\kappa_1(\rho):=\langle e_{j+1,l}\vert K_1\vert e_{j-1,l}\rangle$ (the $\kappa(j,l)$ of (2.16)), $\kappa_2(\rho):=\langle e_{j,l+1}\vert K_2\vert e_{j,l-1}\rangle$, and $d_i(r)$ the diagonal of $K_i$. They are real: two motion terms give $(\mp i\sigma_1)(\mp i\sigma_1)$ or $\sigma_1^2$, and field overlaps and denominators are real. They are even in $\rho$: the reflection $x\to-x$ of all places times $\sigma_3$ on both bodies maps $A\to A$, $B\to B$ and (a), (c) into themselves. By (2.16) and its $1\leftrightarrow2$ form, $\kappa_i(\rho)=\bar\kappa_i(1+O(\varepsilon^2))$; the same energy-denominator argument gives $d_i(r)=\bar d_i(1+O(\varepsilon^2))$. A $\mathcal T$-eigenstate with $\mathcal T\psi=e^{-iP}\psi$ is $\psi(j,l)=e^{iPj}g(l-j)$, and $H_{\rm eff}$ acts on $g$ as
$$(h_Pg)(r)=U(r)g(r)+t(r+1;P)\,g(r+2)+t(r-1;P)^*g(r-2),\qquad t(\rho;P):=\kappa_1(\rho)e^{-2iP}+\kappa_2(\rho),\qquad U:=E+d_1+d_2. \tag{6.1}$$
Each $K_i$ moves its body by an even number of places, so $r$ even and $r$ odd form two separate rings of $L/2$ sites. (6.1) is defined for every real $P$: it is the sector contract with body 1 as representative, continued in $P$ (the continuation used throughout and in the code). $E_b(P)$ is an eigenvalue of $h_P$.

### Step 2. Bound levels and the selected level (item 1)

By (2.3) and (4.2), $E(r)=E_1^{(0)}+E_2^{(0)}-E_\varnothing+M(r)$, $E_i^{(0)}$ the $\tau=0$ level of $b_i$ alone with the field. Second-order perturbation of $\varepsilon_0$ (one-object ground state $L^{-1/2}\sum_x\lvert x\rangle$, excitation energies $2t_f(1-\cos q)$) gives the mixed term $-2g_1g_2\ell(r)/(t_fL)$, consistent with (4.3):
$$M(r)=M^{(2)}(r)+O\big(n\lvert g_1g_2\rvert\ell_0\eta/t_f\big),\quad M^{(2)}(r)=-\frac{2ng_1g_2}{t_f}\ell(r),\quad U(r)-U(0)=V(r)+O\big(n\lvert g_1g_2\rvert\ell_0\eta/t_f\big)+O(\varepsilon^2\lvert\bar\kappa\rvert),\quad V(r):=\frac{ng_1g_2}{t_fL}\,r(L-r). \tag{6.2}$$
$V$ is even and monotone in $\lvert r\rvert$ on $[0,L/2]$. Near $r=0$ it rises like $F\lvert r\rvert$ for $g_1g_2>0$ (a well) and falls like $-F\lvert r\rvert$ for $g_1g_2<0$ (a hill).

*Criterion of localization:* the real eigenvector $f$ of $h_0$ decays away from $r=0$ on a scale $\xi\ll L$, so that its weight near $r=L/2$ is exponentially small in $L/\xi$ (B1).

The uniform relative band is $2\bar t_0\cos2k$. Its edge at $k=0$, where both bodies are near $p_i=0$ (near rest), is the bottom for $s_\mu=+1$ (positive relative inertia) and the top for $s_\mu=-1$. *Selected level* (item 1, note of 2026-10-21): the extreme level at that edge, the lowest if $s_\mu=+1$ and the highest if $s_\mu=-1$.

On a chain the sign of the hopping is a gauge ($f(r)\to(-1)^{r/2}f(r)$). So a well binds levels below the band and a hill above it, for either sign of $\bar t_0$ (standard for one-dimensional chains), and the level near the opposite extremum $r=L/2$ fails the criterion. The selected level is therefore bound iff a well sits at the bottom with $s_\mu=+1$ or a hill at the top with $s_\mu=-1$, i.e. iff $s_e=s_\mu$. Then every bond $t(\rho;0)$ of each sublattice ring has the sign of $\bar t_0$ (B0). $-s_\mu h_0$ is irreducible on the ring with nonnegative off-diagonal elements, so Perron–Frobenius gives an extreme eigenvector $f_0>0$ on the whole ring, for every even $L$, without gauge or frustration. With $g_1g_2=\sigma_1\sigma_2\gamma_1\gamma_2M_1M_2$:
$$\text{bound levels: the lowest if }\sigma_1\sigma_2\gamma_1\gamma_2>0,\ \text{the highest if }<0;\qquad\text{selected level bound}\iff s_\beta:=s_es_\mu=+1,\ \text{and then }f_0>0. \tag{6.3}$$
Examples:
- $\sigma_i=+1$ ($\mu_i>0$): the selected level is the lowest and is bound for $\gamma_1\gamma_2>0$.
- $\sigma_i=-1$ ($\mu_i<0$): it is the highest and is bound for $\gamma_1\gamma_2<0$.
- $\sigma_1=-\sigma_2$: $s_\mu$ is the sign of the $\mu_i$ of smaller modulus.

For $s_\beta=-1$ the bound levels lie at the other edge, with an envelope that alternates in sign near contact (bodies near $p_1=-p_2=\pm\pi/2$). They, and the excited bound levels (with nodes), are named here only: items 2–5 concern the selected level.

With uniform hopping and the ring flux dropped (B1), the gauge leaves the magnitude $\lvert\bar t(P)\rvert=(\bar\kappa_1^2+\bar\kappa_2^2+2\bar\kappa_1\bar\kappa_2\cos2P)^{1/2}$. With $E_i(0)=E_i^{(0)}+\bar d_i+2\bar\kappa_i$ the selected level is
$$E_b(P)=E_1(0)+E_2(0)-E_\varnothing+M(0)-2\bar t_0+s_e\Big[-2\lvert\bar t(P)\rvert+a_0\big(4\lvert\bar t(P)\rvert F^2\big)^{1/3}\Big],\qquad s_e:={\rm sgn}(g_1g_2), \tag{6.4}$$
for $s_\beta=+1$ and $F\ll\lvert\bar t(P)\rvert$ (weak binding), up to relative corrections $O(\chi^{2/3})$ and $O(\xi_0/L)$. Reason: the continuum limit of the chain, $-4\tau_hf''+F\lvert r\rvert f=(\epsilon+2\tau_h)f$ with $\tau_h=\lvert\bar t(P)\rvert$, has its extreme level at $a_0$ in units of $(4\tau_hF^2)^{1/3}$, where $a_0=1.0188$ is the first zero of ${\rm Ai}'$ (even ground state). This holds for both classes ($r$ even, $r$ odd), which differ at $O(\chi^{2/3})$. Outside weak binding $E_b(P)$ is an eigenvalue of (6.1).

### Step 3. Curvature of the bound level (item 2)

Let $f_0$ be the real normalized eigenvector of $h_0$ for $E_b(0)$ of the selected level ($f_0>0$ by (6.3)), and
$$b_0(\rho):=f_0(\rho-1)f_0(\rho+1),\qquad\beta_0:=\sum_\rho b_0(\rho),\qquad m(\rho):=-\frac1{8\kappa_1(\rho)}-\frac1{8\kappa_2(\rho)}. \tag{6.5}$$
$b_0>0$. By Cauchy–Schwarz $\beta_0\le1$, with equality only if $f_0(r+2)=f_0(r)$ for all $r$, which B1 excludes; hence $0<\beta_0<1$.
1. A gauge $g(r)\to e^{i\theta(r)}g(r)$ removes the phases of all bonds of the ring of (6.1) except their sum $\tilde\Phi(P):=\sum_\rho\arg t(\rho;P)-\sum_\rho\arg t(\rho;0)$, which is put on one bond. The bond magnitudes become $s_\rho(P):={\rm sgn}\,t(\rho;0)\,\lvert t(\rho;P)\rvert$, real. So $E_b(P)=\epsilon_0(\{s_\rho(P)\},\tilde\Phi(P))$.
2. $\lvert t(\rho;P)\rvert^2=t(\rho;0)^2-4\kappa_1\kappa_2\sin^2P$ is even in $P$, so $s_\rho'(0)=0$ and $s_\rho''(0)=-4\kappa_1(\rho)\kappa_2(\rho)/t(\rho;0)=1/(2m(\rho))$ (B0: $t(\rho;0)\neq0$). $\tilde\Phi$ is odd in $P$, with $\tilde\Phi'(0)=\sum_\rho{\rm Im}\,\partial_P\ln t=-2\sum_\rho\kappa_1/(\kappa_1+\kappa_2)$. $\epsilon_0$ is even in $\tilde\Phi$ (complex conjugation).
3. Hellmann–Feynman: $\partial\epsilon_0/\partial s_\rho=2b_0(\rho)$. Hence, exactly for the band contract,
$$E_b''(0)=\sum_\rho\frac{b_0(\rho)}{m(\rho)}+\tilde\Phi'(0)^2\,\partial_{\tilde\Phi}^2\epsilon_0,\qquad\tilde\Phi'(0)\simeq-\frac{L\mu_2}{\mu_1+\mu_2}. \tag{6.6}$$

The same derivation holds for any bound level with its own $b_\nu$. The second term (*ring term*) is the only place where the continuation enters: body 2 as representative replaces $\mu_2$ by $\mu_1$ in $\tilde\Phi'(0)$.

Write $E''_{\rm line}:=\sum_\rho b_0/m$ and $E_b''(0)=E''_{\rm line}(1+\rho_L)$. Since $b_0>0$ on every bond, $E''_{\rm line}$ has the sign of $m$ whenever $m(\rho)$ has one sign along the ring, in particular under B2 (the sign of $m_0$). So $E''_{\rm line}$ can vanish only if $m(\rho)$ changes sign along the ring, which B2 excludes.

$\partial^2_{\tilde\Phi}\epsilon_0$ does not depend on which bond carries the flux; put it on the bond $(r_*,r_*+2)$ farthest from contact, with amplitude $s_*$. Use second-order perturbation in $\tilde\Phi$: the first-order term is $-2s_*b_0(r_*+1)$, and the matrix elements are $\lvert s_*\rvert\,\lvert f_{\nu'}(r_*)f_0(r_*+2)-f_{\nu'}(r_*+2)f_0(r_*)\rvert$. Cauchy–Schwarz and $\sum_{\nu'}f_{\nu'}(r)^2=1$ then give
$$\lvert\rho_L\rvert\le\frac{\tilde\Phi'(0)^2}{\lvert E''_{\rm line}\rvert}\Big[2\lvert s_*b_0(r_*+1)\rvert+\frac{4s_*^2}{\Delta_{\rm rel}}\big(f_0(r_*)^2+f_0(r_*+2)^2\big)\Big], \tag{6.16}$$
with $\Delta_{\rm rel}$ the distance of $\epsilon_0$ from the other levels of $h_0$. By B1 the right side is exponentially small in $L/\xi_0$ (times $L^2$). Both continuations obey (6.16), so $E_b''(0)$ is independent of the continuation to relative accuracy $\lvert\rho_L\rvert$, which B4 makes smaller than every retained term. If $\kappa_i(\rho)=\bar\kappa_i$, then $m=\mu_1+\mu_2$ and $1/E''_{\rm line}=(\mu_1+\mu_2)/\beta_0$.

*Omitted orders.* Let $\Delta_P$ be the terms of the sector band contract beyond order $\tau^2$ (dropped by R2): the band level of the full sector contract is the selected eigenvalue of $h_P+\Delta_P$; write $E''_{\rm full}$ for its curvature at $P=0$, against $E_b''(0)$ of (6.6) for $h_P$.
1. On the line the gauge of item 1 is a diagonal unitary $G_P$ whose phase difference across the bond $\rho$ is $\arg t(\rho;P)-\arg t(\rho;0)$, odd in $P$, with $P$-derivative $2\lvert\kappa_1/t\rvert\le2\nu_t^{-1}(1+O(\varepsilon^2))$ at $P=0$. In this frame $\tilde h_P:=G_P^\dagger h_PG_P$ has the real bonds $s_\rho(P)$. So $\tilde h'_0=0$, and $\lVert\tilde h''_0\rVert\le\max_\rho\lvert m(\rho)\rvert^{-1}$ (bonds $1/(2m)$), while $\lvert E''_{\rm line}\rvert\ge\beta_0\min_\rho\lvert m(\rho)\rvert^{-1}$ ($b_0>0$).
2. Second-order perturbation theory in $P$ for the isolated level of $\tilde H:=\tilde h+\tilde\Delta$ gives $E''_{\rm full}=\langle\psi\vert\tilde H''_0\vert\psi\rangle-2\langle\psi\vert\tilde\Delta'_0\mathcal R\tilde\Delta'_0\vert\psi\rangle$, with $\mathcal R$ the reduced resolvent, $\lVert\mathcal R\rVert\le1/\Delta_{\rm rel}$ at leading order. Against $E''_{\rm line}=\langle f_0\vert\tilde h''_0\vert f_0\rangle$ there are three terms: $\langle\tilde\Delta''_0\rangle$; the shift of $f_0$ by at most $\lVert Q_0\tilde\Delta_0f_0\rVert/\Delta_{\rm rel}$ ($Q_0:=1-\vert f_0\rangle\langle f_0\vert$) in $\langle\tilde h''_0\rangle$; and the resolvent term. Hence, to leading order in $\lVert\Delta\rVert/\Delta_{\rm rel}$ and up to a factor $1+O(\bar x)$,
$$\frac{\lvert E''_{\rm full}-E_b''(0)\rvert}{\lvert E''_{\rm line}\rvert}\le\eta_4,\qquad\eta_4:=\frac{\lvert m_0\rvert}{\beta_0}\Big(\lVert\tilde\Delta''_0\rVert+\frac{2\lVert\tilde\Delta'_0f_0\rVert^2}{\Delta_{\rm rel}}\Big)+\frac{2\lVert Q_0\tilde\Delta_0f_0\rVert}{\beta_0\Delta_{\rm rel}}. \tag{6.19}$$
3. $\Delta_P$ moves body 1 by at most four places, and the gauge adds commutators with phase derivatives of at most $2\nu_t^{-1}$ per bond. So $\lVert\tilde\Delta^{(k)}_0\rVert=O(\epsilon_4\nu_t^{-k})$ for $k\le2$, and with $\lvert m_0\rvert=\nu_t/(8\min_i\lvert\bar\kappa_i\rvert)$,
$$\eta_4=O\Big(\frac{\epsilon_4}{\beta_0\nu_t\min_i\lvert\bar\kappa_i\rvert}\Big)+O\Big(\frac{\epsilon_4}{\beta_0\Delta_{\rm rel}}\Big),$$
where the middle term of (6.19) is bounded by the product of the two. This gives the sufficient form of B3.
4. As $\nu_t\to0$ the first term grows as $1/\nu_t$: the retained relative dispersion vanishes, while the bodies' own order-$\tau^4$ dispersions in $\epsilon_4$ do not. B0 and B3 exclude this.
5. On the ring, the flux part of $\tilde\Delta$ adds a term bounded as in (6.16).

In the code sets the measured $\epsilon_4$ satisfies both sufficient conditions of B3, with ratios at most $0.02$. The R1 estimate $\tau'^4/(\delta_1^2\delta)$ exceeds it there by factors 7–180 (`code/v4/output/inertia_check.txt`).

### Step 4. Local inertia and the difference from $\mu_1+\mu_2$ (item 2)

$m(\rho)=m_1(\rho)+m_2(\rho)$ with $m_i(\rho):=-1/(8\kappa_i(\rho))$. By (2.11), $m_i$ is the inertia $b_i$ would have with the amplitude it has at distance $\rho$ from the other body. By (2.16), $m_1=\mu_1(1\pm\sigma_1\bar M^{(2)}(\rho)/M_1+O(\varepsilon^3))$, and by (2.14) $\mu_1\sigma_1/M_1=(R_1/M_1)/c_1^2\,(1+O(\gamma_1^2))=c_1^{-2}(1+O(\varepsilon))$; the same holds for $b_2$. With $\bar M^{(2)}(\rho):=\tfrac14[M^{(2)}(\rho-1)+2M^{(2)}(\rho)+M^{(2)}(\rho+1)]$:
$$m(\rho)=\mu_1+\mu_2+s_c\,\bar M^{(2)}(\rho)\Big(\frac1{c_1^2}+\frac1{c_2^2}\Big)+O\big(\varepsilon^3(\lvert\mu_1\rvert+\lvert\mu_2\rvert)\big). \tag{6.7}$$
Each body's local inertia contains the second-order mutual term divided by the square of its own speed: once per body, with the sign $s_c$.

*Inverse expansion.* Write $m=m_0+\delta m$ and $A_0:=\sum_\rho b_0\,\delta m$. Expanding $1/E''_{\rm line}=[\sum_\rho b_0/(m_0+\delta m)]^{-1}$ in $\delta m$ requires $\lvert A_0\rvert\ll\lvert m_0\rvert\beta_0$. Since $b_0>0$, $\lvert A_0\rvert\le\bar x\lvert m_0\rvert\beta_0$, so B2 implies this condition, and $1/E''_{\rm line}=m_0/\beta_0+A_0/\beta_0^2+O(\bar x^2\lvert m_0\rvert/\beta_0)$. With (6.7) and $E_b''(0)=E''_{\rm line}(1+\rho_L)$:
$$D_0:=\frac1{E_b''(0)}-\mu_1-\mu_2=\underbrace{(\mu_1+\mu_2)\frac{1-\beta_0}{\beta_0}}_{D^{\rm lat}_0}+\underbrace{\frac{s_c}{\beta_0^2}\Big(\frac1{c_1^2}+\frac1{c_2^2}\Big)\sum_\rho b_0(\rho)\bar M^{(2)}(\rho)}_{D^{\rm mark}_0}-\frac{\rho_L}{E''_{\rm line}}+O\Big(\frac{\varepsilon^3(\lvert\mu_1\rvert+\lvert\mu_2\rvert)}{\beta_0},\frac{\bar x^2\lvert m_0\rvert}{\beta_0},\frac{\rho_L^2}{E''_{\rm line}}\Big). \tag{6.8}$$
$\beta_0$ and $b_0$ are those of the full band chain. The terms of (6.8):
- $D^{\rm lat}_0$ is of order zero in $\varepsilon$ at fixed $\chi$ and nonzero, since $0<\beta_0<1$. It has the sign of $\mu_1+\mu_2$. It comes from $K_1+K_2$ (the $P$-dependence of the relative hopping $\lvert t(\rho;P)\rvert$, i.e. the form $\cos2p$ of each body's band (2.11)), together with $W$, which localizes $f_0$.
- $D^{\rm mark}_0$ comes from the dependence of $K_i$ on the other body's mark (2.16). It obeys $\lvert D^{\rm mark}_0\rvert\le\bar x\lvert m_0\rvert/\beta_0$ up to the $O(\varepsilon^3)$ remainder, and is of order $\varepsilon^2$ at fixed $n,t_f,L$.
- The ring part is smaller than both by B4.

*Outside B2* (a mark comparable with $\lvert m_0\rvert/S$; the near-cancellation of $\mu_1+\mu_2$ is excluded by B0), (6.8) is not used. There $1/E_b''(0)=[(1+\rho_L)\sum_\rho b_0(\rho)/m(\rho)]^{-1}$ by (6.6), with $m(\rho)$ of (6.5) unexpanded.

*Dependence on the level* (the selected level of the two classes). In weak binding, Hellmann–Feynman for the uniform chain, $\partial\epsilon_0/\partial\bar t_0=2\beta_0$, applied to the bracket of (6.4), gives $\beta_0\simeq1-\frac{a_0}6(4\chi^2)^{1/3}\simeq1-0.270\chi^{2/3}$ in both classes. In tight binding $\beta_0\simeq1/\chi$ (even class) and $\beta_0\to\frac12$ (odd class; check 3). So $D^{\rm lat}_0$ grows from $\simeq0.270\chi^{2/3}m_0$ to $\simeq\chi m_0$ (even) or $m_0$ (odd).

*Control:* none in the class. In the wider class of unbound pairs on the line, $\chi\to0$ gives $\beta_0\to1$ and $D_0\to0$.

### Step 5. Inertia against cost (item 3)

As in (4.2), the cost of the bound pair is $\Delta E_0:=E_b(0)-E_1(0)-E_2(0)+E_\varnothing$. These are eigenvalues of the total contracts of four descriptions (pair, $b_1$ alone, $b_2$ alone, field alone), so no split of contract terms enters (M6). With $E_b(0)=\langle f_0\vert h_0\vert f_0\rangle=\sum_rf_0^2U+2\sum_\rho b_0\,t(\rho;0)$ and Step 1,
$$\Delta E_0=\langle M\rangle_0+K_0+O(\varepsilon^2\lvert\bar\kappa\rvert),\qquad K_0:=2\bar t_0(\beta_0-1),\qquad\langle M\rangle_0=\sum_rf_0(r)^2M(r)=M(0)+\langle V\rangle_0+O\big(n\lvert g_1g_2\rvert\ell_0\eta/t_f\big). \tag{6.9}$$
$K_0$ is the kinetic cost of the relative motion, of sign $s_\mu$. Since $\mu_1+\mu_2=-\bar t_0/(8\bar\kappa_1\bar\kappa_2)$ and $\mu_1\mu_2=1/(64\bar\kappa_1\bar\kappa_2)$, $D^{\rm lat}_0=4\mu_1\mu_2K_0/\beta_0$. Splitting $\bar M^{(2)}=M^{(2)}(0)+\bar V$ in (6.8) (B2):
$$D_0=\frac{4\mu_1\mu_2}{\beta_0}K_0+\frac{s_cS}{\beta_0}M(0)+\frac{s_cS}{\beta_0^2}\sum_\rho b_0\bar V-\frac{\rho_L}{E''_{\rm line}},\qquad\frac{\Delta E_0}{c^2}=\frac{K_0+M(0)+\langle V\rangle_0}{c^2}. \tag{6.10}$$
*No universal identity.* The coefficients of the two leading terms differ:
- of $K_0$: $4\mu_1\mu_2/\beta_0$ against $c^{-2}$, a ratio of size $R_1R_2/c^2\gg1$ by R1;
- of the contact value $M(0)=-\frac{ng_1g_2}{t_f}\frac{L^2-1}{6L}$: $s_cS/\beta_0$ against $c^{-2}$; at equal speeds, $2s_c/\beta_0$ against 1.

So (6.10) gives no identity $D_0=\Delta E_0/c^2$ with one speed $c$ fixed by the contracts. That the relation indeed fails on the class, for every $c\ge c_{\min}$, follows from case (i) of (6.17); whether it holds in individual descriptions is decided there.

*Individual equalities.* For a speed $c$ let
$$\Delta_0(c):=D_0-\frac{\Delta E_0}{c^2}=C_KK_0+C_MM(0)+\mathcal V_0-\frac{\rho_L}{E''_{\rm line}},\quad C_K:=\frac{4\mu_1\mu_2}{\beta_0}-\frac1{c^2},\quad C_M:=\frac{s_cS}{\beta_0}-\frac1{c^2},\quad\mathcal V_0:=\frac{s_cS}{\beta_0^2}\sum_\rho b_0\bar V-\frac{\langle V\rangle_0}{c^2}.$$
Let $c\ge c_{\min}$; this includes $c_1$, $c_2$ and equal speeds. The signs of the terms:
- $S/\beta_0>c^{-2}$, so ${\rm sgn}\,C_M=s_c$ and $\lvert C_M\rvert\ge c_{\max}^{-2}$.
- $4\lvert\mu_1\mu_2\rvert c^2\ge4R_1R_2/c_{\max}^2\gg1$ (R1: $c_i\simeq\tau_i\ll R_i$), so ${\rm sgn}\,C_K=\sigma_1\sigma_2$.
- ${\rm sgn}\,K_0=s_\mu$ and ${\rm sgn}\,M(0)=-s_e=-s_\mu$.

With $\lvert\bar V(\rho)\rvert\le F(\lvert\rho\rvert+\frac12)$ and $b_0(\rho)\le\frac12[f_0(\rho-1)^2+f_0(\rho+1)^2]$, $\lvert\mathcal V_0\rvert\le4F(\langle\lvert r\rvert\rangle_0+\frac32)/(c_{\min}^2\beta_0^2)$, which is below $\lvert C_MM(0)\rvert$ if
$$\text{(S)}\qquad\langle\lvert r\rvert\rangle_0+\tfrac32<\frac{\beta_0^2\,(L-L^{-1})}{24}\,\frac{c_{\min}^2}{c_{\max}^2}.$$
(S) may be replaced by $\lvert\mathcal V_0\rvert<\lvert C_KK_0\rvert$. This holds in weak binding, where $\langle V\rangle_0\simeq2\lvert K_0\rvert$ (virial theorem of the linear potential) gives $\lvert\mathcal V_0/C_KK_0\rvert=O(c_{\max}^2/R_1R_2)$. Hence, at leading order and under B2,
$$\begin{aligned}&\text{(i) }s_c=-\sigma_1\sigma_2:\ \Delta_0(c)\neq0,\ \text{of sign }\sigma_1\sigma_2s_\mu,\ \text{for every }c\ge c_{\min},\ \text{under (S) and B4};\\&\text{(ii) }s_c=\sigma_1\sigma_2:\ C_KK_0\ \text{and}\ C_MM(0)\ \text{have opposite signs, and }\Delta_0(c)=0\ \text{where they cancel (codimension one); at equal speeds}\\&\qquad\text{in weak binding near }L^*_0\simeq2\cdot4^{1/3}a_0\,\big\lvert C_K/C_M\big\rvert_{\beta_0=1}\,\chi^{-1/3}\ \text{at fixed }n,\chi.\end{aligned} \tag{6.17}$$
$L^*_0$ solves $\lvert C_KK_0\rvert=\lvert C_M\rvert FL/6$ with $\lvert K_0\rvert=\lvert\bar t_0\rvert\frac{a_0}34^{1/3}\chi^{2/3}$ (Step 4), up to relative $O(\chi^{2/3})$ and $O(c^2/R_1R_2)$. For $\sigma_1=\sigma_2$ (code sets A, B, D), case (i) is coupling (c) and case (ii) coupling (a); for $\sigma_1=-\sigma_2$ the reverse.

*A zero of the leading-order expressions.* Take case (ii) with:
- $\sigma_1=\sigma_2=-1$ (then $\delta_1\ge2R_i-w$ for every $t_f$, Step 7), coupling (a), $M_i=M$, $\tau_i=\tau$, $\gamma_1=-\gamma_2=\varepsilon$. Then $s_e=s_\mu=-1$, $s_\beta=+1$, the selected level is the highest, and B0 ($\nu_t=2$) and B2 hold since $\sigma_1=\sigma_2$.
- $n=\varepsilon^{-3/4}$, $\tau/M=\varepsilon^{1/2}$, and $t_f=TM^3/(\tau^2\chi^{2/3})$ with fixed $T\gg4.3$; then $\chi^{1/3}=4\varepsilon^{5/4}/T$.
- $L$ near $L^*_0\simeq13M^2/(\tau^2\chi^{1/3})$.

As $\varepsilon\to0$:
- spectral hierarchy (H): the width of $\mathcal G$ in a sector, $W_{\mathcal G}\simeq\lvert V(L/2)\rvert=FL/4\simeq0.81M\chi^{2/3}$, stays below the first excitation of the held field, $\omega_1\simeq4\pi^2t_f/L^2\simeq0.24T\tau^2/M$: $W_{\mathcal G}/\omega_1\simeq55\varepsilon^{3/2}/T^3\to0$. So, with the margins of Step 7, the field-excited levels of the $\mathcal G$ branches lie at least $\omega_1-W_{\mathcal G}-2\varepsilon M/L\simeq\omega_1$ from $\mathcal G$, the flipped levels at least $2R-w\simeq2M$ ($w\simeq2\lvert M(0)\rvert+2n\varepsilon M\to0$ against $M$), and $\delta\simeq\omega_1$. (H) needs $\tau/M\gg7.4\,\varepsilon^{5/4}T^{-3/2}$.
- R1: $\tau\ll\delta_1\simeq2M$ and $\tau^2/(\delta_1\delta)\simeq2.1/T\ll1$;
- B3 with the R1 estimate $\epsilon_4\simeq\tau^4/(\delta_1^2\omega_1)\simeq1.07\tau^2/(TM)$: the kinetic part $\epsilon_4/(\beta_0\nu_t\min_i\lvert\bar\kappa_i\rvert)\simeq4.3/T$ is small. The gap part $\epsilon_4/(\beta_0\Delta_{\rm rel})\simeq0.13\,T\varepsilon^{-5/2}$ diverges, with $\Delta_{\rm rel}\simeq2.09\lvert\bar t_0\rvert\chi^{2/3}$ the Airy spacing $(a_1-a_0)(4\lvert\bar t_0\rvert F^2)^{1/3}$, $a_1=2.338$ the first zero of ${\rm Ai}$;
- perturbative control: $n\varepsilon=\varepsilon^{1/4}\to0$ and $\eta\simeq1.1\varepsilon\chi^{1/3}/T\to0$;
- B1, B4 and (S): $L^*_0/\xi_0\simeq8M^2/\tau^2=8/\varepsilon\gg1$;
- B2 and the mark expansion: $\bar x\simeq\lvert M(0)\rvert/M\simeq0.54\chi^{2/3}\to0$;
- remainders against the cancelling terms, $\lvert C_KK_0\rvert\simeq0.54M\chi^{2/3}/\tau^2$. Those of (6.8), as stated, are smaller by $0.23T^2\varepsilon^{1/2}$ and $O(\chi^{2/3})$. That of (6.9) is smaller by $(\tau/M)^2T^2/(69\varepsilon^{1/2})=\varepsilon^{1/2}T^2/69$, or by $O(\varepsilon)$ if counted by the actual size $\lvert\bar\kappa\rvert\chi^{2/3}$ of the mark. Together with the lower bound from (H), this sets the window $7.4\,\varepsilon^{5/4}T^{-3/2}\ll\tau/M\ll8\varepsilon^{1/4}/T$.

So the leading-order $\Delta_0$ of (6.8)–(6.10) changes sign across $L^*_0$, and tuning $\varepsilon$ at fixed $L$ gives a zero of the leading-order expressions: $D_0=\Delta E_0/c^2$ with $c=\tau$.

*Status:* established along the family are R, (H), B0, B1, B2, B4, the kinetic part of B3 and the zero of the leading-order $\Delta_0$ (code: line chain). Not established:
- the gap part of B3: with the R1 estimate of $\epsilon_4$ it fails, and the actual omitted terms are not evaluated along the family;
- the uniformity of the remainders of the quoted inputs (2.14), (2.16), (4.3), and hence of (6.7)–(6.9): they are stated at fixed $n,L,t_f$, while here $n,L,t_f\to\infty$ together.

So the family is not shown to lie in the class, and whether the class or the full contract contains a description with $D_0=\Delta E_0/c^2$ is **unverified**.

*Not a universal speed.* In case (ii) the zero is a codimension-one condition. At fixed $c_i$ the fitted ratio $c_0^2:=\Delta E_0/D_0$, where positive, varies with $L$ and $\chi$ and equals $c^2$ only at $L^*_0$: it is a property of the description, not a speed fixed by the contracts. In case (i), under (S) and B4, the relation holds for no $c\ge c_{\min}$.

### Step 6. Weight of the bound level and fall (item 4)

A held source $g_s\Phi_{x_s}$ changes the $\tau=0$ levels of $\mathcal G$ by the held-pair weight $W_{\rm pair}(j,l)$ at first order in $g_s$. It changes the $K_i$ at relative order $\gamma_ig_s/t_f$ (through the field states, as in (2.16)), which enters the fall at $O(\tau^4\gamma g_s)$, as in (7.7). Let the source be at a distance $d_s\gg\xi_0$ from the pair, and work on the line (B1, B4). The weight of the bound level is
$$W_0(X):=\sum_rf_0(r)^2\,W_{\rm pair}\big(X-\tfrac r2,X+\tfrac r2\big). \tag{6.11}$$
$W_b$ vanishes for $g_1=0$ and for $g_2=0$, hence is $O(g_1g_2g_s)$; by (4.5), $W_{\rm pair}=ng_1\phi(x_1)+ng_2\phi(x_2)+O(\varepsilon^2g_s)$. $\phi$ is quadratic on each arc between images of $x_s$ ($\phi''=-2g_s/(t_fL)$), and $\sum_rf_0^2\,r=0$ ($f_0^2$ is even by the reflection of Step 1). Hence, for pair positions not reaching $x_s$,
$$W_0(X+1)-W_0(X-1)=n(g_1+g_2)\,[\phi(X+1)-\phi(X-1)]+O(\varepsilon^2g_s)+O(e^{-d_s/\xi_0}): \tag{6.12}$$
at leading order the weight difference of the bound level is that of its members.

*Bound band with $P$-dependent profile.* Written symmetrically, the sector eigenvector of (6.1) is $\psi_P(j,l)=L^{-1/2}e^{iP(j+l)/2}f_{0,P}(l-j)$, with $f_{0,P}$ the eigenvector of the chain with bonds $\kappa_1(\rho)e^{-iP}+\kappa_2(\rho)e^{iP}=e^{iP}t(\rho;P)$; this chain is unitarily equivalent to $h_P$. The gauge of Step 3 gives $f_{0,P}=e^{i\theta_P}F_{0,P}$ with $F_{0,P}$ real and $F_{0,P}=f_0+O(P^2)$ ($s_\rho$ is even in $P$). From $\arg(\kappa_1e^{-iP}+\kappa_2e^{iP})=-P\frac{\kappa_1-\kappa_2}{\kappa_1+\kappa_2}+O(P^3)$ and $\kappa_i=-1/(8m_i)$: $\theta_P=P\theta'+O(P^3)$, with $\theta'(r+2)-\theta'(r)=[m_2(\rho)-m_1(\rho)]/m(\rho)$ ($\rho=r+1$) and $\theta'$ odd. Hence $\psi_P=L^{-1/2}e^{iPY}F_{0,P}[1+O(P^3)]$ with the centre
$$Y(j,l):=\frac{j+l}2+\theta'(l-j)=\frac{\mu_1j+\mu_2l}{\mu_1+\mu_2}+O\big(\varepsilon^2\lvert l-j\rvert\big): \tag{6.18}$$
$P$ is conjugate to the inertia-weighted centre, not to the midpoint.

The source matrix elements on the band are $\langle P',0\vert W\vert P,0\rangle=L^{-1}\sum_{j,l}e^{i(P-P')Y}F_{0,P'}F_{0,P}W_{\rm pair}[1+O(P^3,P'^3)]$, not $f_0^2$ at all momenta. For a start of width $w$ in $Y$ ($\lvert P\rvert\lesssim1/w$), $F_{0,P'}F_{0,P}=f_0^2[1+O(w^{-2})]$. So on the band the source acts as $\tilde W_0(\hat Y)$ up to relative $O(w^{-2})$. Here $\tilde W_0(Y):=\sum_rf_0^2W_{\rm pair}(Y-\frac r2-\theta'(r),Y+\frac r2-\theta'(r))$, with $W_{\rm pair}$ continued between places by its form on the arc. At first order in $g_i$, $\tilde W_0-W_0$ is constant on the arc: $f_0^2$ is even, $\theta'$ odd and $\phi''$ constant. So the weight differences of the band are those of (6.11), (6.12), up to $O(\varepsilon^2g_s)$.

*Acceleration from rest.* Take the band contract with source, $H=H_{\rm eff}+W_{\rm pair}$, and a start $\psi(j,l)=\varphi(Y)f_0(l-j)$, with a real envelope $\varphi$ of width $w$ that is even about a point $x_0$ with $2x_0\in\mathbb Z$. Let $a_0(0):=\frac{{\rm d}^2}{{\rm d}\lambda^2}\langle\hat Y\rangle\vert_{\lambda=0}=-\langle[H,[H,\hat Y]]\rangle$.
- This needs no band projection and contains the interband terms.
- It is exactly linear in $W_{\rm pair}$, since $[W,[W,\hat Y]]=0$.
- Its $g_s$-independent part vanishes on the line: the reflection about $x_0$ (Step 1) leaves $H_{\rm eff}$ invariant and maps $\hat Y-x_0\to-(\hat Y-x_0)$ and $\psi\to\pm\psi$.

The linear part is $-\sum_{a,b}\psi_a\psi_b(W_a-W_b)H_{ab}(Y_b-Y_a)$. A hop of $b_i$ by $\pm2$ across the bond $\rho$ has $H_{ab}=\kappa_i(\rho)$ and, by (6.18), $Y_b-Y_a=\pm2m_i(\rho)/m(\rho)$, so $\kappa_i(Y_b-Y_a)=\mp1/(4m(\rho))$ for both bodies. On each bond $\psi_a\psi_b=\varphi^2b_0(\rho)[1+O(w^{-2})]$. Hence
$$a_0(0)=-\sum_X\varphi_X^2\sum_\rho\frac{b_0(\rho)}{m(\rho)}\,(\partial_1+\partial_2)W_{\rm pair}\big\vert_{X,\rho}\big[1+O(w^{-2})\big]=-E''_{\rm line}\sum_X\varphi_X^2\,W'_0(X)\big[1+O(w^{-2})\big]+O(\varepsilon^2g_sE''_{\rm line})+O(g_s^2)+O(\tau^4\gamma g_s), \tag{6.13}$$
with $\partial_i$ the centred difference in $x_i$ and $W'_0(X):=\tfrac12[W_0(X+1)-W_0(X-1)]$; the $O(g_s^2)$ is zero within the band contract.

The second form of (6.13) uses $(\partial_1+\partial_2)W_{\rm pair}=n(g_1+g_2)\phi'(X)+(\text{odd in }\rho)+O(\varepsilon^2g_s)$, whose odd part cancels against the even $b_0/m$, and (6.12). It is informative where $n\lvert g_1+g_2\rvert\lvert\phi'\rvert$ exceeds its additive remainders, i.e. under (N), which is part of the class of item 4. For one body the same computation gives the leading form of (7.7). So (7.7) carries over to the bound level, for the centre $\hat Y$, with $E''\to E''_{\rm line}$ and $W'\to W'_0$. The midpoint $(\hat x_1+\hat x_2)/2$ moves by $\pm1$ per hop and gets instead $-\frac{\beta_0}2\sum_X\varphi_X^2(W_1'/\mu_1+W_2'/\mu_2)$ at $\lambda=0$ (uniform hopping), the instantaneous response of the relative motion.

*Fall factor.* $\Pi_0:=E''_{\rm line}W'_0$, and for $b_i$ alone $\Pi_i:=E_i''(0)W_i'=ng_i\phi'/\mu_i+O(\varepsilon^2g_s/\lvert\mu_i\rvert)$ by (4.5), with $ng_i\phi'/\mu_i=n\gamma_ic_i^2\phi'(1+O(\varepsilon))$ by (2.14). By (6.12) and (4.5), $W'_0=W'_1+W'_2+O(\varepsilon^2g_s)=\mu_1\Pi_1+\mu_2\Pi_2+O(\varepsilon^2g_s)$, with $\mu_1\Pi_1+\mu_2\Pi_2=n(g_1+g_2)\phi'+O(\varepsilon^2g_s)$; by B2, $E''_{\rm line}=(\beta_0/m_0)(1+O(\bar x))$. Under (N) the additive $O(\varepsilon^2g_s)$ is a relative $O(\varepsilon/\nu_g)$ (at fixed $n,t_f,L$), hence
$$\Pi_0=\beta_0\,\frac{\mu_1\Pi_1+\mu_2\Pi_2}{\mu_1+\mu_2}\Big[1+O\Big(\frac{\varepsilon}{\nu_g}\Big)+O(\bar x)\Big]. \tag{6.14}$$

Under the named condition $\gamma_1c_1^2=\gamma_2c_2^2$ (M7), $\Pi_2=\Pi_1(1+O(\varepsilon))$ and $n(g_1+g_2)\phi'=(\mu_1+\mu_2)\Pi_1(1+O(\varepsilon))$. Then $g_i\propto\mu_i(1+O(\varepsilon))$, so $\nu_g=\nu_t+O(\varepsilon)$ and (N) follows from B0 ($\nu_t$ not small, $\varepsilon\ll1$), and
$$\Pi_0-\Pi_1=-(1-\beta_0)\Pi_1+O(\varepsilon\Pi_1/\nu_t)+O(\bar x\Pi_1).$$
The single bodies have equal fall factors only under that named condition. Even then the bound pair differs at leading order (order zero in $\varepsilon$ at fixed $\chi$) by the factor $\beta_0\in(0,1)$: it falls slower than its parts.

*Control:* none in the class. Unbound pairs on the line ($\chi\to0$) give $\beta_0\to1$, and equality under the named condition.

### Step 7. Dependence on $t_f$ (item 5)

At fixed $n,\gamma_i,M_i,\tau_i$ and at leading order in $\varepsilon$, the accuracy of this step: the isolated-body quantities are taken at the accuracy of (2.14), $\mu_i=\sigma_iR_i/c_i^2$ with $R_i=M_i(1+\gamma_in)$, and $\bar\kappa_i=-1/(8\mu_i)$, $\bar t_0$, $\nu_t$, $c_i$. At that accuracy they do not depend on $t_f$. Their corrections (relative $O(\gamma_i^2)$ in (2.14), $O(\gamma_i^3)$ in $R_i$) do depend on it and are not tracked, so every power law below holds up to relative $O(\varepsilon^2)$. $M^{(2)}$, $F$ and $\eta$ scale as $1/t_f$; $\phi\propto g_s/t_f$.

*R1 from the held spectra.* $\delta_1$ and $\delta$ are distances between finite sets of held ($\tau=0$) eigenvalues of the sector. The first set is the group $\{E(j,l)\}$ of $\mathcal G$, of width $W_{\mathcal G}=\max_rM(r)-\min_rM(r)$. It is compared with the held-pair levels reached by one motion term ($b_i$ flipped and moved by one place, all field states), resp. with all other held levels. The code computes them so.

At leading order they follow from $R_i$. When $b_i$ is flipped, by (4.2) the field ground energies differ by $2\sigma_iR_i$ plus the change of the mutual term $M$; the excited levels lie higher by the excitation energies of the held field. Hence, for every pair of connected configurations and with $\omega^{(i)}_n$ now the excitation energies of the free field, at leading order
$$\big\lvert E(j,l)-E^{(i)}_n(j',l')-(2\sigma_iR_i-\omega^{(i)}_n)\big\rvert\le w:=2\max_r\lvert M(r)\rvert+n(\lvert g_1\rvert+\lvert g_2\rvert).$$
- The first term of $w$ bounds the change of the mutual term, which reverses sign with $g_i$ at second order; it contains $W_{\mathcal G}$.
- The second bounds the first-order shifts of the excitation energies by the marks: at most $(\lvert g_1\rvert+\lvert g_2\rvert)/L$ per excited object (degenerate perturbation of $\pm q$).

Here $\omega\in\{2t_f\sum_kn_k(1-\cos\frac{2\pi k}L):\sum_kn_k\le N_f\}$; for $N_f=1$, $\omega\in\{4t_f\sin^2(\pi k/L)\}$. The $R_i$-based statements are leading-order estimates with this margin:
- For $\sigma_i=-1$ all distances are $\ge2R_i-w$: there is no resonance condition on $t_f$.
- For $\sigma_i=+1$, R1 excludes the $t_f$ for which some $\omega$ lies within $\delta_1+w$ of $2R_i$. For $N_f=1$ these are the intervals $[(2R_i-\delta_1-w),(2R_i+\delta_1+w)]/(4\sin^2(\pi k/L))$, $k=1,\dots,L/2$.
- Sufficient conditions are $4N_ft_f\le2R_i-\delta_1-w$ (all excitations below) or $4t_f\sin^2(\pi/L)\ge2R_i+\delta_1+w$ (all above).

In the code ($N_f=1$, 12 cases, $t_f=1$ and $40$) the actual $\delta_1$ lies within $w$ of the estimate $\min_{i,n}\lvert2\sigma_iR_i-\omega_n\rvert$, at up to $0.98\,w$.

$\delta$ is the distance of $\mathcal G$ from all other held levels of the sector. These are:
- the field-excited levels of the $\mathcal G$ branches, at least $4t_f\sin^2(\pi/L)-W_{\mathcal G}-(\lvert g_1\rvert+\lvert g_2\rvert)/L$ away;
- the doubly flipped levels, at $\lvert2\sigma_1R_1+2\sigma_2R_2-\omega\rvert$ up to a margin of the same kind.

So $\delta$ is not automatically the field gap. With the weak-binding $\beta_0$ of Step 4:
$$\chi=\frac{n\lvert g_1g_2\rvert}{t_f\lvert\bar t_0\rvert}\propto\frac1{t_f},\quad1-\beta_0\simeq0.270\,\chi^{2/3}\propto t_f^{-2/3},\quad\frac{\tau'^2}{\delta_1\delta}\ll1\ll\chi L^3,\quad\min_n\big\lvert2R_i-\omega_n\big\rvert\ge\delta_1+w\ \ (\sigma_i=+1). \tag{6.15}$$
The bounds on $t_f$:
- The second inequality bounds $t_f$ from below where $\delta$ is the field gap: $t_f\gg\tau'^2/(4\delta_1\sin^2(\pi/L))$.
- $\chi L^3\gg1$ (B1) bounds it from above. B4 bounds it further from above, since $\xi_0\propto t_f^{1/3}$ while the mark terms fall as $1/t_f$.
- B3 with the R1 estimate of $\epsilon_4$, in weak binding ($\Delta_{\rm rel}\propto\chi^{2/3}\propto t_f^{-2/3}$): where $\delta$ is the field gap, $\epsilon_4\propto1/t_f$, and both parts bound $t_f$ from below; where $\delta$ does not depend on $t_f$, the gap part bounds it from above.
- In B2 the mark part of $\bar x$ falls as $1/t_f$; $\nu_t$ (B0) does not depend on $t_f$.

Hence:
- (6.3) does not depend on $t_f$.
- $D^{\rm lat}_0$ and the relative deviation $1-\beta_0$ of $\Pi_0$ from the inertia-weighted $\Pi_i$ fall as $t_f^{-2/3}$; $D^{\rm mark}_0$ and $M(0)$ fall as $1/t_f$; the zeros of the leading-order $\Delta_0$ in (6.17)(ii) move as $L^*_0\propto t_f^{1/3}$.
- No $t_f$ in the window gives $\beta_0=1$, so (6.10) and (6.14) persist.

A faster field weakens the binding and brings the pair's inertia and fall factor towards $\mu_1+\mu_2$ and the inertia-weighted $\Pi_i$. The limit is reached only where the pair stops being bound on the ring.

## Result

- **(6.1)** Class above, order $\tau^2$: in the sector $P$ (body 1 as representative, continued to real $P$) the band contract is the relative chain $h_P$ with bond amplitudes $t(\rho;P)=\kappa_1(\rho)e^{-2iP}+\kappa_2(\rho)$ and separate rings for $r$ even and odd.
- **(6.2)** Leading order in $\varepsilon$: $M^{(2)}(r)=-2ng_1g_2\ell(r)/t_f$ and the relative potential $V(r)=\frac{ng_1g_2}{t_fL}r(L-r)$, a well at contact for $g_1g_2>0$ and a hill for $g_1g_2<0$.
- **(6.3)** Under R, B0 and $\chi L^3\gg1$, for every branch combination and both signs of $\gamma_1\gamma_2$, bound levels exist: the lowest relative levels if $\sigma_1\sigma_2\gamma_1\gamma_2>0$, the highest if $<0$. The selected level is the extreme level at the $k=0$ edge of the relative band (bodies near rest): the lowest if $s_\mu=+1$, the highest if $s_\mu=-1$. It is bound iff $s_\beta={\rm sgn}[g_1g_2(\mu_1^{-1}+\mu_2^{-1})]=+1$, and then $f_0>0$ on the whole relative ring. Bound levels with $s_\beta=-1$ (alternating near contact) and excited bound levels are named only.
- **(6.4)** Selected level, $s_\beta=+1$, leading order in $\varepsilon$, uniform hopping, weak binding $F\ll\lvert\bar t(P)\rvert$, up to $O(\chi^{2/3})$ and $O(\xi_0/L)$: $E_b(P)=E_1(0)+E_2(0)-E_\varnothing+M(0)-2\bar t_0+s_e[-2\lvert\bar t(P)\rvert+a_0(4\lvert\bar t(P)\rvert F^2)^{1/3}]$, with $a_0=1.0188$ the first zero of ${\rm Ai}'$.
- **(6.5)** Definitions for the selected level: bond weights $b_0>0$, bond overlap $0<\beta_0<1$, local pair inertia $m(\rho)$.
- **(6.6)** Exact for the band contract (order $\tau^2$): $E_b''(0)=E''_{\rm line}(1+\rho_L)$ with $E''_{\rm line}=\sum_\rho b_0(\rho)/m(\rho)$ and the ring remainder $\rho_L$, the only part that depends on the continuation. Since $b_0>0$, $E''_{\rm line}$ can vanish only if $m(\rho)$ changes sign along the ring, which B2 excludes.
- **(6.7)** Order $\varepsilon^2$, error $O(\varepsilon^3(\lvert\mu_1\rvert+\lvert\mu_2\rvert))$: $m(\rho)=\mu_1+\mu_2+s_c\bar M^{(2)}(\rho)(c_1^{-2}+c_2^{-2})$. The mutual term enters each body's inertia once, over its own speed squared, with sign $+$ for (a) and $-$ for (c).
- **(6.8)** Selected level, under B2, which implies the inverse-expansion condition $\lvert A_0\rvert\le\bar x\lvert\mu_1+\mu_2\rvert\beta_0\ll\lvert\mu_1+\mu_2\rvert\beta_0$. Through order $\varepsilon^2$, to first order in $\rho_L$, with remainders $O(\varepsilon^3(\lvert\mu_1\rvert+\lvert\mu_2\rvert)/\beta_0,\bar x^2\lvert\mu_1+\mu_2\rvert/\beta_0)$: $1/E_b''(0)-\mu_1-\mu_2=D^{\rm lat}_0+D^{\rm mark}_0-\rho_L/E''_{\rm line}$.
  - The leading part $D^{\rm lat}_0=(\mu_1+\mu_2)(1-\beta_0)/\beta_0$ comes from the $P$-dependence of $K_1+K_2$ with the binding by $W$. It is nonzero, with the sign of $\mu_1+\mu_2$.
  - $D^{\rm mark}_0$ comes from the $\rho$-dependence of $K_i$, with $\lvert D^{\rm mark}_0\rvert\le\bar x\lvert\mu_1+\mu_2\rvert/\beta_0$.
  - $\beta_0\simeq1-0.270\chi^{2/3}$ in weak binding (both classes); in tight binding $\beta_0\simeq1/\chi$ (even class) and $\beta_0\to\frac12$ (odd class).
  - Outside B2 (a mark comparable with $\lvert\mu_1+\mu_2\rvert/(c_1^{-2}+c_2^{-2})$) only (6.6), unexpanded, is derived.
- **(6.9)** Selected level, order $\tau^2$, through $\varepsilon^2$: the cost of the bound pair is $\Delta E_0=\langle M\rangle_0+K_0$, with the kinetic cost $K_0=2\bar t_0(\beta_0-1)$ of sign $s_\mu$.
- **(6.10)** Negative, selected level, leading order, both couplings, under R and B: $1/E_b''(0)-\mu_1-\mu_2=\Delta E_0/c^2$ is not an identity on the class for any speed $c\ge\min(c_1,c_2)$, in particular $c_1$, $c_2$ and equal speeds. The coefficients of $K_0$ ($4\mu_1\mu_2/\beta_0$ against $c^{-2}$) and of $M(0)$ ($s_c(c_1^{-2}+c_2^{-2})/\beta_0$ against $c^{-2}$) differ, and the relation fails in every description of case (i) of (6.17).
- **(6.11)** Definition: the weight of the bound level is the $f_0^2$-average of the held-pair weight, at first order in $g_s$.
- **(6.12)** First order in $g_i$ and $g_s$, source far from the pair: the weight difference of the bound level is that of its members, $n(g_1+g_2)\Delta\phi$, with additive error $O(\varepsilon^2g_s)$.
- **(6.13)** (7.7) carries over to the bound level for the centre $\hat Y$ of (6.18), with $E''_{\rm line}$ and $W'_0$. This holds on the line, for a start $\varphi(Y)f_0$ with a real envelope of width $w$ even about a point, at first order in $g_s$, up to a relative $O(w^{-2})$ and additive $O(\varepsilon^2g_sE''_{\rm line})$, $O(\tau^4\gamma g_s)$, which are small under (N). The midpoint of the pair has in general a different acceleration from rest.
- **(6.14)** Negative, selected level, both couplings, under R, B and (N).
  - $\Pi_0=\beta_0(\mu_1\Pi_1+\mu_2\Pi_2)/(\mu_1+\mu_2)$ up to relative $O(\varepsilon/\nu_g)+O(\bar x)$.
  - Under the named condition $\gamma_1c_1^2=\gamma_2c_2^2$, where (N) follows from B0: $\Pi_0-\Pi_1=-(1-\beta_0)\Pi_1+O(\varepsilon\Pi_1/\nu_t)+O(\bar x\Pi_1)$, so the bound level falls slower than its parts at leading order.
  - No control in the class.
- **(6.15)** Item 5, selected level, at fixed $n,\gamma_i,M_i,\tau_i$, at leading order in $\varepsilon$, with the isolated-body quantities $\mu_i$, $\bar\kappa_i$, $\bar t_0$, $\nu_t$, $c_i$ at that accuracy, where they do not depend on $t_f$ (their relative $O(\varepsilon^2)$ corrections are not tracked):
  - $\chi\propto1/t_f$; in weak binding $1-\beta_0\simeq0.270\chi^{2/3}$, so the deviations of items 2 and 4 fall as $t_f^{-2/3}$, and the mark terms and the mark part of $\bar x$ (B2) as $1/t_f$.
  - The resonance condition of R1 restricts $t_f$ only for $\sigma_i=+1$. As a leading-order estimate of the actual held-pair gaps, with margin $w=2\max_r\lvert M(r)\rvert+n(\lvert g_1\rvert+\lvert g_2\rvert)$, it excludes the intervals where an excitation of the free field lies within $\delta_1+w$ of $2R_i$. Sufficient conditions are $4N_ft_f\le2R_i-\delta_1-w$ or $4t_f\sin^2(\pi/L)\ge2R_i+\delta_1+w$.
  - $\tau'^2/(\delta_1\delta)\ll1\ll\chi L^3$ bound $t_f$ from below (where $\delta$ is the field gap) and from above. With the R1 estimate of $\epsilon_4$, B3 bounds $t_f$ from below where $\delta$ is the field gap.
  - No $t_f$ in the window removes (6.10) or (6.14).
- **(6.16)** Bound on the ring remainder at the bond farthest from contact: $\lvert\rho_L\rvert\le\tilde\Phi'(0)^2[2\lvert s_*b_0\rvert+4s_*^2(f_0(r_*)^2+f_0(r_*+2)^2)/\Delta_{\rm rel}]/\lvert E''_{\rm line}\rvert$, exponentially small in $L/\xi_0$ (times $L^2$). $E_b''(0)$ is independent of the continuation to relative accuracy $\lvert\rho_L\rvert$.
- **(6.17)** Selected level, leading order, under B2, every speed $c\ge\min(c_1,c_2)$ (including $c_1$, $c_2$ and equal speeds).
  - (i) If $s_c=-\sigma_1\sigma_2$, then $1/E_b''(0)-\mu_1-\mu_2\neq\Delta E_0/c^2$. This needs (S), or $\lvert\mathcal V_0\rvert<\lvert C_KK_0\rvert$ (e.g. weak binding), and B4.
  - (ii) If $s_c=\sigma_1\sigma_2$, the two leading terms have opposite signs and equality holds where they cancel, a codimension-one condition. At equal speeds in weak binding this is near $L^*_0\simeq2\cdot4^{1/3}a_0\lvert C_K/C_M\rvert\chi^{-1/3}$.
  - For the leading-order expressions such a zero occurs along a family ($\sigma_i=-1$, coupling (a), Step 5) that satisfies R with the spectral hierarchy (H), B0, B1, B2, B4 and the kinetic part of B3. The gap part of B3 and the uniformity of the quoted remainders are not established there. So whether the class, or the full contract, contains a description with equality is unverified. A fitted ratio $\Delta E_0/D_0$ varies with $L$ and $\chi$ and is not a speed fixed by the contracts.
- **(6.18)** On the line, the plane-wave label $P$ of the bound band is conjugate to the centre $Y=\frac{j+l}2+\theta'(l-j)=\frac{\mu_1j+\mu_2l}{\mu_1+\mu_2}+O(\varepsilon^2\lvert l-j\rvert)$. With the $P$-dependent profile the projected source is $\tilde W_0(\hat Y)$, whose differences equal those of $W_0$ up to $O(\varepsilon^2g_s)$ and relative $O(w^{-2})$.
- **(6.19)** Selected level, under R, B0, B1, B2, to leading order in the omitted terms, on the line (on the ring plus a term bounded as in (6.16)). The curvature $E''_{\rm full}$ of the bound level of the full sector contract differs from $E_b''(0)$ of (6.6) by at most $\eta_4\lvert E''_{\rm line}\rvert$.
  - $\eta_4=\frac{\lvert m_0\rvert}{\beta_0}(\lVert\tilde\Delta''_0\rVert+2\lVert\tilde\Delta'_0f_0\rVert^2/\Delta_{\rm rel})+2\lVert Q_0\tilde\Delta_0f_0\rVert/(\beta_0\Delta_{\rm rel})$, built from the terms beyond order $\tau^2$ in the gauge frame of Step 3.
  - $\eta_4=O(\epsilon_4/(\beta_0\nu_t\min_i\lvert\bar\kappa_i\rvert))+O(\epsilon_4/(\beta_0\Delta_{\rm rel}))$, where $\epsilon_4$ contains each body's own order-$\tau^4$ dispersion. The first term grows as $1/\nu_t$ when the relative hopping cancels.
  - B3 is $\eta_4\ll1$.

## Consistency checks

1. *Dimensions.* $P$ and the positions are dimensionless, so $E_b''$, $E_i''$, $W'_0$, $W_i'$ have the dimension $[C]$, and $\mu_i$, $m$, $1/E_b''$ have $[C]^{-1}$. So do $\bar M^{(2)}/c_i^2$ and $\mu_1\mu_2K_0$ ($c_i^2\sim[C]^2$). $\Pi_0$, $\Pi_i$ and $a_0(0)$ have $[C]^2$, the dimension of a second derivative in $\lambda$. $\beta_0$, $\chi$, $\rho_L$, $\bar x$, $\nu_t$, $\nu_g$, $\eta_4$ are pure numbers, and $\epsilon_4$, $\Delta_{\rm rel}$, $\bar\kappa_i$ have $[C]$, so both terms of $\eta_4$ are ratios of $[C]$. Passed.
2. *Unbound limit $\chi\to0$ on the line* (e.g. $\gamma_2\to0$): $\beta_0\to1$ and $M^{(2)}\to0$, so $1/E_b''(0)\to\mu_1+\mu_2$. This agrees with the curvature of $E_1(p_1)+E_2(p_2)$ at fixed $p_1+p_2=P$ at its extremum $p_i=0$ (bodies at rest), $1/(\mu_1+\mu_2)$. Passed.
3. *Tight binding $\chi\to\infty$.* Even class: $f_0\simeq\delta_{r0}$ and $f_0(\pm2)\simeq\bar t_0/(E_b-U(2))$, so $\beta_0\simeq1/\chi$ and $E_b''(0)\simeq s_e\,8\bar\kappa_1\bar\kappa_2/F$. This agrees with direct second-order perturbation of the contact state, $E_b(P)=U(0)-s_e\lvert\bar t(P)\rvert^2/F$. Odd class: $f_0\simeq(\delta_{r,1}+\delta_{r,-1})/\sqrt2$, $\beta_0\to\frac12$, so $1/E_b''(0)=2(\mu_1+\mu_2)$. This agrees with the two-site level $U(1)-s_e\lvert\bar t(P)\rvert$, whose curvature is $4s_e\bar\kappa_1\bar\kappa_2/\lvert\bar t_0\rvert$. Passed.

## Open issues

- Bound levels with $s_\beta=-1$ (bodies near $p_1=-p_2=\pm\pi/2$) and excited bound levels are out of scope for items 2–5 (note of 2026-10-21). (6.6) holds for them, but their inertia, cost and fall are not derived here.
- A near-cancellation of the relative hopping (small $\nu_t$, possible only for $\sigma_1=-\sigma_2$) is out of scope (note (i)). Outside B2 within B0 only the unexpanded (6.6) is available; the cost comparison and the fall factor there are not derived.
- The code sets do not meet B4 for the mark terms: $\lvert\rho_L\rvert=0.2$–$12\%$, against $\lvert D^{\rm mark}\rvert\approx0.2$–$0.3\%$ of $1/E_b''$. The code therefore checks (6.6) with the ring term included.
- The closed forms (6.2), (6.4), (6.15), (6.17) use $M^{(2)}$; at $\eta\approx0.3$ (code) the exact binding slope is about half of $F$. Statements in terms of $\beta_0$, $b_0$, $M$ are not affected.
- The $O(\varepsilon^3)$ remainder of (6.7) is large at moderate couplings (code: ratio 1.40 for (a), $-0.19$ for (c) at $\eta\approx0.3$), so $D^{\rm mark}$ is reliable only for small $\eta$.
- The second-order corrections of (6.13), (6.14) are not evaluated (the bond function $B$ of (4.7) is not quoted). In the code they are 13–16% in $\Delta W_0/(\Delta W_1+\Delta W_2)$.
- On the ring, (6.13) has a $g_s$-independent part from the relative cut $r=L/2$, where the reflection argument fails. In the code it is up to 0.6% of the first-order part (set A) and drops to $10^{-4}$ of it when the profile is cut there; it is not bounded analytically.
- Item 4 where the first-order weights cancel ($g_1+g_2$ small against $\max_i\lvert g_i\rvert$) is out of scope (note (ii)); there the first-order fall factor of the pair is small or zero, and the next order is not derived.
- For the even class in tight binding, $\beta_0\sim1/\chi$, and the kinetic part of B3 fails when $\chi\gtrsim\nu_t\min_i\lvert\bar\kappa_i\rvert/\epsilon_4$; the order-$\tau^4$ band contract is then needed.
- B3 is stated through $\epsilon_4$, whose R1 estimate $\tau'^4/(\delta_1^2\delta)$ is crude: in the code it exceeds the measured omitted terms by factors 7–180, and it would place set A outside B3 (kinetic ratio up to 2.6). Where only this estimate is available, B3 can fail although the omitted terms are small.
- For $\sigma_i=+1$ and $N_f>1$ the excitation energies near $2R_i$ become dense, and the windows of (6.15) shrink. Whether weakly coupled resonances (matrix elements $O(\gamma^k)$ for $k$ excited objects) can be tolerated is open.
- The zero of (6.17)(ii) is established only for the leading-order expressions, on the line chain. Along the family of Step 5 the gap part of B3 is not established (with the R1 estimate of $\epsilon_4$ it fails), and the remainders of (2.14), (2.16), (4.3) are not shown to be uniform in $n,L,t_f\to\infty$. Whether the class contains a description with $D_0=\Delta E_0/c^2$ is open.
- Out of scope: full grip, tori, the highest field state, bodies of one type, the speed of a front of the field; how typical the structure is (item 5 of the base problem).

## Code

- `code/v4/check_inertia.py` (check). It treats $N_f=1$, $t_f=1$ and three parameter sets, all with $s_\beta=+1$ and $\nu_t=1.50$–$1.98$ (B0):
  - A: $\sigma_i=+1$, $M=(3,4)$, $\tau_1=0.2$, $\gamma=(0.09,0.07)$, $r$ even, lowest level; $\tau_2=\tau_1$ for (a) and $\tau_2=\tau_1(1+\gamma_1/L)/(1+\gamma_2/L)$ for (c), so that $c_1=c_2$ exactly for both couplings.
  - B: $\sigma_i=-1$, $M=(2.5,2)$, $\tau=(0.16,0.2)$, $\gamma=(0.10,-0.14)$, $r$ odd, highest level, unequal speeds.
  - D: $\sigma_i=-1$, $M=(3,3)$, $\tau=0.2$, $\gamma=(0.09,-0.09)$, $r$ even, highest level, speeds equalized as in A. It has $g_1+g_2=0$, so it is in the class of items 1–3 only, and it is not used for item 4.

  Each set runs at $L=12,16$ for couplings (a) and (c). The script builds the exact sector contract (body 1 as representative, continued in $P$) and finds the exact bound level by its weight on $\mathcal G$ ($\ge0.998$). It takes the curvature by second-order perturbation in $P$ (cross-checked by finite differences) and builds the band contract by second-order Schrieffer–Wolff. Results:
  - (6.6) with its ring term agrees with the exact curvature to $\le0.28\%$ in all 12 cases.
  - $f_0>0$ in all cases. $\beta_0=0.69$–$0.79$ (A), $0.77$–$0.78$ (B), $0.51$–$0.57$ (D). $1/E_b''(0)=(1.22$–$1.29)(\mu_1+\mu_2)$ for A, B and $(1.76$–$1.92)(\mu_1+\mu_2)$ for D.
  - B2: $\bar x=0.0011$–$0.0054$ and $\lvert A_0\rvert/(\lvert m_0\rvert\beta_0)=0.0003$–$0.005$. The expanded inverse $m_0/\beta_0+A_0/\beta_0^2$ differs from the unexpanded $1/E''_{\rm line}$ by $\le3\times10^{-6}$ (relative), i.e. $O(\bar x^2)$.
  - $\rho_L$ is 12% and 3% (A, $L=12,16$), 6% and 0.6% (B), 2.6–2.8% and 0.2% (D). The bound (6.16) holds in all cases and exceeds $\lvert\rho_L\rvert$ by factors 5.7–11.
  - $D\,c^2/\Delta E$ is $-500$ to $-575$ (A), $-208$ to $-363$ (B, with $c_1$ or $c_2$) and $-562$ to $-734$ (D): far from 1, since these $L$ lie far below $L^*_0$.
  - The ratio of the numerical $m(\rho)-\mu_1-\mu_2$ to (6.7) tends to 1 as the couplings are scaled down (1.04 for (a), 0.93 for (c) at scale 0.03).
  - R1 gaps from the held-sector spectra, as distances between sets of held-pair eigenvalues, at $t_f=1$: $\delta_1=2.0$ (A), $3.9$ (B), $5.9$ (D); $\delta=0.12$–$0.23$, below the free field gaps $0.268$ ($L=12$) and $0.152$ ($L=16$); the width of $\mathcal G$ is $0.010$–$0.018$, so (H) holds. At $t_f=40$, $\delta_1=2.6$ (A), $4.0$ (B), $6.0$ (D) for $L=12$; for $L=16$, A has a resonance ($\delta_1=0.024$, $4t_f\sin^2(\pi/16)\simeq2R_1$), while B and D ($\sigma_i=-1$) keep $4.0$ and $6.0$. In all 12 cases $\delta_1$ lies within the margin $w$ of Step 7 ($w=0.034$–$0.070$) of the estimate $\min_{i,n}\lvert2\sigma_iR_i-\omega_n\rvert$, at $0.01$–$0.98\,w$.
  - Omitted orders (B3, (6.19)): the script builds the exact effective contract on $\mathcal G$ in the sector $P$ (canonical block diagonalization: band levels with the unitary polar factor of their $\mathcal G$-components) and subtracts the order-$\tau^2$ band contract. This gives $\Delta_P$, whose norm and first two $P$-derivatives at $P=0$ are at most $3.7\times10^{-5}$ (finite differences, body 1 as representative). The R1 estimate $\tau'^4/(\delta_1^2\delta)=(0.2$–$4.6)\times10^{-3}$ is larger by factors 7–180. With the measured $\epsilon_4$:
    - the kinetic ratio $\epsilon_4/(\beta_0\nu_t\min_i\lvert\bar\kappa_i\rvert)$ is $0.014$–$0.020$, and the gap ratio $\epsilon_4/(\beta_0\Delta_{\rm rel})$ is $0.005$–$0.008$;
    - $\eta_4$ of (6.19), evaluated with these (ungauged) norms, is $0.005$–$0.011$. It exceeds the actual relative deviation of the exact curvature from the band contract's, $0.08$–$0.28\%$, by factors 3.2–9.9 in all 12 cases.
  - Sign rule (6.17) on the leading-order line chain ($\chi=0.05$, $\sigma_i=-1$, $s_e=-1$, $4\mu_1\mu_2c^2=64$, selected level): $\Delta_0$ changes sign for (a) at $L=564$, against the estimate 553, and never for (c) in $100\le L\le3000$.

  Outputs: `code/v4/output/inertia_check.txt`, `code/v4/output/inertia_check.json`.
- `code/v4/check_weight.py` (check). Set A, couplings (a) and (c), $L=16,20$, held source $g_s=-0.01$ (and $-0.005$ at $L=16$). It computes the exact eigenvalues of the full contract near the bottom of the bound band (weight on $\mathcal G\ge0.9997$) and compares them with three reductions: (i) the first-order projected contract $E_b(P_k)+\langle P',v_{P'}\vert g_s\Phi_{x_s}\vert P,v_P\rangle$ with the exact $P$-dependent sector eigenvectors; (ii) the frozen description $E_b(\hat P)+W_0(\hat X)$ with (6.11); (iii) the same with the pair inertia set to $\mu_1+\mu_2$. Results:
  - Main level spacing: (i) is within 0.15% and (ii) within 0.17% of the exact value; (iii) misses it by 10–20%.
  - Splittings of the near-degenerate pairs ($E_b(P+\pi)=E_b(P)$): (i) reproduces the first to 0.2–1.2% and the second to 3.5–10%; (ii) misses them by up to a factor 30. So the main spacing tests the product $E_b''W_0''$, while only (i) tests the complete first-order projected contract.
  - The common level offsets ($\approx10^{-5}$) scale as $g_s^2$.
  - $\Delta W_0/(\Delta W_1+\Delta W_2)$ is 0.84–0.87 at these couplings and tends to 1 as they are scaled down (0.997 at scale 0.03), as in (6.12).

  Second part, set A at $L=24$: the script computes $a(0)=-\langle[H,[H,\hat Z]]\rangle$ exactly in the position-space band contract (exact Schrieffer–Wolff amplitudes, held weights), for $\hat Z=\hat Y$ of (6.18) and for the midpoint. The start is $\varphi(Y)f_0$ with a Gaussian envelope of width $w=1,1.5,2$ centred at $L/4$. Results:
  - Set A, with the couplings in the weights scaled by 0.05 (first order in $g_i$): the first-order part of $a_Y$ over the right side of (6.13) is 0.88, 0.95, 0.97, a trend $O(w^{-2})$. At full couplings it is 0.90–0.99. The $g_s$-independent part is up to 0.6% of the first-order part at full couplings; with the profile cut at $\lvert r\rvert<L/3$ it drops to $10^{-4}$.

  Outputs: `code/v4/output/weight_check.txt`, `code/v4/output/weight_check.json`.

## Methods used

- band contract (second-order Schrieffer–Wolff), plane-wave sectors of the joint translation
- gauge reduction of a chain to bond magnitudes and one flux; Hellmann–Feynman theorem
- Perron–Frobenius; Cauchy–Schwarz
- second-order perturbation theory, reduced-resolvent bound on the curvature of an isolated level; Airy functions (continuum limit of a linear potential), virial theorem
- controlled inversion of a weighted harmonic mean; sign analysis of coefficients; scaling families of descriptions
- reflection symmetry; double commutator for the acceleration from rest; band projection with momentum-dependent eigenvectors
- exact diagonalization (dense and shift-invert sparse) and the canonical (polar-factor) effective contract on $\mathcal G$ for the code checks

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