Version 1 · earlier version; the current one is v4 · External review, round 1: major errors

# Inertia of a pair bound through the field

- **Subproject:** 07-ilang-weight
- **Package:** 06-inertia-of-pair
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-21

## 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.** $\bar t_0:=\bar\kappa_1+\bar\kappa_2=-\tfrac18(\mu_1^{-1}+\mu_2^{-1})$; $F:=n\lvert g_1g_2\rvert/t_f$; binding ratio $\chi:=F/\lvert\bar t_0\rvert$; $E_i(P)$ the band level of $b_i$ alone with the field, $E_\varnothing$ the lowest eigenvalue of the field alone.

**Binding regime B (assumption).** B1: the relative level $\nu$ considered decays on a scale $\xi_\nu\ll L$ (for the extreme level: $\chi L^3\gg1$; $\xi_0\simeq(4/\chi)^{1/3}$ for $\chi\lesssim1$, $\xi_0\simeq1$ for $\chi\gtrsim1$). B2: $\lvert\bar t_0\rvert\gg\varepsilon^2\max_i\lvert\bar\kappa_i\rvert$ (the relative hopping does not cancel). B3: $\lvert\beta_\nu\rvert\gg\tau'^2/(\delta_1\delta)$ with $\beta_\nu$ of (6.5), so that the order-$\tau^4$ terms dropped by R2 stay small (Step 3).

**Class.** Couplings (a), (c), $N_f\ge1$, even $L\ge6$, all branch combinations, both signs of $\gamma_1\gamma_2$, under R and B. Results are at order $\tau_i^2$ in the band contract 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 $\sigma_1^2$, field overlaps and denominators are real) and 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 (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, monotone in $\lvert r\rvert$ on $[0,L/2]$, rises like $F\lvert r\rvert$ near $r=0$ for $g_1g_2>0$ (well) and falls like $-F\lvert r\rvert$ for $g_1g_2<0$ (hill). *Criterion of localization:* the real eigenvector $f_\nu$ of $h_0$ decays away from $r=0$ on a scale $\xi_\nu\ll L$, so that its weight near $r=L/2$ is exponentially small in $L/\xi_\nu$ (B1). On a chain the sign of the hopping is a gauge ($f(r)\to(-1)^{r/2}f(r)$; on the ring up to terms exponentially small by B1), so the edges of the spectrum of $h_0$ depend on $\lvert\bar t_0\rvert$ and $V$ only; a well binds levels below the uniform band $2\bar t_0\cos2k$, a hill above it (standard for one-dimensional chains), and the level near the opposite extremum $r=L/2$ fails the criterion. The extreme eigenvector of a real chain is of one sign iff the hoppings have the sign that puts that edge at $k=0$ (Perron–Frobenius); $t(\rho;0)$ has the sign of $\bar t_0$ by B2. With $g_1g_2=\sigma_1\sigma_2\gamma_1\gamma_2M_1M_2$:
$$\text{bound levels: the lowest if }\sigma_1\sigma_2\gamma_1\gamma_2>0,\ \text{the highest if }\sigma_1\sigma_2\gamma_1\gamma_2<0;\qquad f_\nu\ \text{of one sign}\iff g_1g_2\,(\mu_1^{-1}+\mu_2^{-1})>0. \tag{6.3}$$
In the second case of (6.3) the bound level is made of bodies near $p_i=0$; otherwise of bodies near $p_i=\pm\pi/2$ ($f_\nu$ alternates). For $\sigma_1=\sigma_2=+1$ it is the lowest level with $\gamma_1\gamma_2>0$; for $\sigma_1=\sigma_2=-1$ the highest level with $\gamma_1\gamma_2<0$, where the relative motion has negative inertia ($\mu_1^{-1}+\mu_2^{-1}<0$). With uniform hopping and the ring flux dropped (B1), the gauge leaves the magnitude $\lvert\bar t(P)\rvert=(\bar\kappa_1^2+\bar\kappa_2^2+2\bar\kappa_1\bar\kappa_2\cos2P)^{1/2}$, and with $E_i(0)=E_i^{(0)}+\bar d_i+2\bar\kappa_i$:
$$E_b(P)=E_1(0)+E_2(0)-E_\varnothing+M(0)-2\bar t_0+s_e\Big[-2\lvert\bar t(P)\rvert+a_\nu\big(4\lvert\bar t(P)\rvert F^2\big)^{1/3}\Big],\qquad s_e:={\rm sgn}(g_1g_2), \tag{6.4}$$
for $F\ll\lvert\bar t(P)\rvert$ (weak binding), up to relative corrections $O(\chi^{2/3})$ and $O(\xi_\nu/L)$: the continuum limit of the chain of step 2, $-4\tau_hf''+F\lvert r\rvert f=(\epsilon+2\tau_h)f$, has the Airy levels $a_0=1.0188$ (first zero of ${\rm Ai}'$), $a_1=2.3381$ (first zero of ${\rm Ai}$) in units $(4\tau_hF^2)^{1/3}$. Outside weak binding $E_b(P)$ is the extreme eigenvalue of (6.1).

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

Let $f_\nu$ be the real normalized eigenvector of $h_0$ for $E_b(0)$ and
$$b_\nu(\rho):=f_\nu(\rho-1)f_\nu(\rho+1),\qquad\beta_\nu:=\sum_\rho b_\nu(\rho),\qquad m(\rho):=-\frac1{8\kappa_1(\rho)}-\frac1{8\kappa_2(\rho)}. \tag{6.5}$$
By Cauchy–Schwarz $\lvert\beta_\nu\rvert\le1$, with equality only if $f_\nu(r+2)=\pm f_\nu(r)$ for all $r$, excluded by B1: $\lvert\beta_\nu\rvert<1$.
1. 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 $\tilde\Phi(P):=\sum_\rho\arg t(\rho;P)-\sum_\rho\arg t(\rho;0)$, put on one bond; the bond magnitudes become $s_\rho(P):={\rm sgn}\,t(\rho;0)\,\lvert t(\rho;P)\rvert$, real. So $E_b(P)=\epsilon_\nu(\{s_\rho(P)\},\tilde\Phi(P))$.
2. $\lvert t(\rho;P)\rvert^2=t(\rho;0)^2-4\kappa_1\kappa_2\sin^2P$ is even in $P$: $s_\rho'(0)=0$, $s_\rho''(0)=-4\kappa_1(\rho)\kappa_2(\rho)/t(\rho;0)$ (B2: $t(\rho;0)\neq0$). $\tilde\Phi$ is odd in $P$ with $\tilde\Phi'(0)=\sum_\rho{\rm Im}\,\partial_P\ln t=-2\sum_\rho\kappa_1/(\kappa_1+\kappa_2)$. $\epsilon_\nu$ is even in $\tilde\Phi$ (complex conjugation).
3. Hellmann–Feynman: $\partial\epsilon_\nu/\partial s_\rho=2b_\nu(\rho)$. Hence, exactly for the band contract,
$$E_b''(0)=\sum_\rho\frac{b_\nu(\rho)}{m(\rho)}+\tilde\Phi'(0)^2\,\partial_{\tilde\Phi}^2\epsilon_\nu,\qquad\tilde\Phi'(0)\simeq-\frac{L\mu_2}{\mu_1+\mu_2}. \tag{6.6}$$
The second term (*ring term*) needs the amplitude of $f_\nu$ around the ring: $\partial^2_{\tilde\Phi}\epsilon_\nu$ is exponentially small in $L/\xi_\nu$, the term vanishes as $L\to\infty$, and it depends on the continuation (body 2 as representative replaces $\mu_2$ by $\mu_1$). The first term is the inverse inertia of the bound level on the line. If $\kappa_i(\rho)=\bar\kappa_i$, $m=\mu_1+\mu_2$ and, on the line, $1/E_b''(0)=(\mu_1+\mu_2)/\beta_\nu$. The order-$\tau^4$ terms dropped by R2 change $E_b''$ by $O(\tau'^4/(\delta_1^2\delta))$, relative $O(\tau'^2/(\beta_\nu\delta_1\delta))$: small by B3.

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

$m(\rho)=m_1(\rho)+m_2(\rho)$, $m_i(\rho):=-1/(8\kappa_i(\rho))$: 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 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 $s_c$. Insert (6.7) into the first term of (6.6), $1/m=(\mu_1+\mu_2)^{-1}(1-\eta)+O(\eta^2)$ with $\eta=O(\varepsilon^2)$, and invert:
$$D_\nu:=\frac1{E_b''(0)}-\mu_1-\mu_2=\underbrace{(\mu_1+\mu_2)\frac{1-\beta_\nu}{\beta_\nu}}_{D^{\rm lat}_\nu}+\underbrace{\frac{s_c}{\beta_\nu^2}\Big(\frac1{c_1^2}+\frac1{c_2^2}\Big)\sum_\rho b_\nu(\rho)\bar M^{(2)}(\rho)}_{D^{\rm mark}_\nu}+O\Big(\frac{\varepsilon^3(\lvert\mu_1\rvert+\lvert\mu_2\rvert)}{\beta_\nu^2}\Big), \tag{6.8}$$
on the line (ring term of (6.6) dropped), $\beta_\nu$ and $b_\nu$ of the full band chain. $D^{\rm lat}_\nu$ is of order zero in $\varepsilon$ at fixed $\chi$ and nonzero for every bound level ($\lvert\beta_\nu\rvert<1$); for $0<\beta_\nu<1$ 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. from the form $\cos2p$ of each body's band (2.11), together with $W$, which localizes $f_\nu$. $D^{\rm mark}_\nu$ is of order $\varepsilon^2$ and comes from the dependence of $K_i$ on the other body's mark (2.16). $D_\nu$ is the leading difference; $D^{\rm lat}_\nu$ dominates when $1-\beta_\nu\gg\lvert M^{(2)}(0)\rvert/R_i$. *Level dependence:* through $\beta_\nu$; in weak binding (Step 7) $1-\beta_\nu\simeq\frac{a_\nu}6(4\chi^2)^{1/3}$, larger for higher relative levels. *Control:* none in the class ($\lvert\beta_\nu\rvert<1$); 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 $\Delta E_\nu:=E_b(0)-E_1(0)-E_2(0)+E_\varnothing$: 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 $E_b(0)=\langle f_\nu\vert h_0\vert f_\nu\rangle=\sum_rf_\nu^2U+2\sum_\rho b_\nu\,t(\rho;0)$ and Step 1,
$$\Delta E_\nu=\langle M\rangle_\nu+K_\nu+O(\varepsilon^2\lvert\bar\kappa\rvert),\qquad K_\nu:=2\bar t_0(\beta_\nu-1),\qquad\langle M\rangle_\nu=\sum_rf_\nu(r)^2M(r)=M(0)+\langle V\rangle_\nu+O\big(n\lvert g_1g_2\rvert\ell_0\eta/t_f\big). \tag{6.9}$$
$K_\nu$ is the kinetic cost of the relative motion. Since $\mu_1+\mu_2=-\bar t_0/(8\bar\kappa_1\bar\kappa_2)$ and $\mu_1\mu_2=1/(64\bar\kappa_1\bar\kappa_2)$, $D^{\rm lat}_\nu=4\mu_1\mu_2K_\nu/\beta_\nu$, and splitting $\bar M^{(2)}=M(0)+\bar V$ in (6.8):
$$D_\nu=\frac{4\mu_1\mu_2}{\beta_\nu}K_\nu+\frac{s_c}{\beta_\nu}\Big(\frac1{c_1^2}+\frac1{c_2^2}\Big)M(0)+\frac{s_c}{\beta_\nu^2}\Big(\frac1{c_1^2}+\frac1{c_2^2}\Big)\sum_\rho b_\nu\bar V,\qquad\frac{\Delta E_\nu}{c^2}=\frac{K_\nu+M(0)+\langle V\rangle_\nu}{c^2}. \tag{6.10}$$
At fixed $n,g_i,t_f,\tau_i,M_i$ the contact value $M(0)=-\frac{ng_1g_2}{t_f}\frac{L^2-1}{6L}$ grows like $L$, while $K_\nu$, $\beta_\nu$, $b_\nu$, $\langle V\rangle_\nu$ change only by $O(\xi_\nu/L)$ (B1). *Equal speeds* $c_1=c_2=c$ (named condition): $D_\nu=\Delta E_\nu/c^2$ for all $L$ would need the coefficients of $M(0)$ to agree, $2s_c/\beta_\nu=1$, impossible since $\lvert\beta_\nu\rvert<1$; the coefficients of $K_\nu$ differ as well, $4\mu_1\mu_2c^2/\beta_\nu=4\sigma_1\sigma_2R_1R_2/(\beta_\nu c^2)$, of size $(R/c)^2\gg1$ by R1. For $\sigma_1=\sigma_2$, $f_\nu$ of one sign and $1-\beta_\nu\gg\lvert M(0)\rvert/R_i$, $D_\nu\simeq D^{\rm lat}_\nu$ has the sign of $\mu_1+\mu_2$ while $\Delta E_\nu\simeq M(0)$ has the opposite sign when $\langle\lvert r\rvert\rangle_\nu\ll L/6$: the pair has more inertia and less cost than its parts ($\sigma_i=+1$). *Unequal speeds:* a single $c$ would need $c^{-2}=4\mu_1\mu_2/\beta_\nu$ and $c^{-2}=s_c(c_1^{-2}+c_2^{-2})/\beta_\nu$, incompatible since the first is larger by $\sim R_1R_2/c^2$; in particular neither $c_1$ nor $c_2$ works. *Control:* none in the class.

### 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 $g_s$, and 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). Project on the bound level (distance of the source $d_s\gg\xi_\nu$, source terms small against the spacing of relative levels): states $\sum_X\varphi_X\sum_rf_\nu(r)e_{X-r/2,X+r/2}$, $X=(j+l)/2$, with center contract $E_b(\hat P)+W_\nu(\hat X)+O(g_s^2)$, where the weight of the bound level is
$$W_\nu(X):=\sum_rf_\nu(r)^2\,W_{\rm pair}\big(X-\tfrac r2,X+\tfrac r2\big). \tag{6.11}$$
$W_b$ 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 $\sum_rf_\nu^2\,r=0$ ($f_\nu^2$ even by the reflection of Step 1). Hence, for pair positions not reaching $x_s$,
$$W_\nu(X+1)-W_\nu(X-1)=n(g_1+g_2)\,[\phi(X+1)-\phi(X-1)]+O(\varepsilon^2g_s)+O(e^{-d_s/\xi_\nu}): \tag{6.12}$$
at leading order the weight difference of the bound level is that of its members, for every level $\nu$. With $E_b(P)=\sum_s\epsilon_se^{isP}$, $[\mathcal T_s,\hat X]=-s\mathcal T_s$ and $\ddot{\langle\hat X\rangle}=-\langle[H,[H,\hat X]]\rangle$, a start with real envelope $\varphi_X$ in the bound band has
$$a_\nu(0)=\sum_ss\,\epsilon_s\sum_X\varphi_{X+s}\varphi_X\,[W_\nu(X+s)-W_\nu(X)]=-E_b''(0)\sum_X\varphi_X^2W'_\nu(X)\,\big[1+O(w^{-2})\big]+O(g_s^2)+O(\tau^4\gamma g_s), \tag{6.13}$$
$W'_\nu(X):=\tfrac12[W_\nu(X+1)-W_\nu(X-1)]$, $w$ the width of $\varphi$ against the range of the $\epsilon_s$. For one body ($\epsilon_{\pm2}=\bar\kappa$ only) the middle form is (7.7); so (7.7) carries over to the bound level with $E''\to E_b''(0)$, $W'\to W'_\nu$, for broad envelopes. The fall factor is $\Pi_\nu:=E_b''(0)W'_\nu$, and for $b_i$ alone $\Pi_i:=E_i''(0)W_i'=ng_i\phi'/\mu_i=n\gamma_ic_i^2\phi'\,(1+O(\varepsilon))$ by (4.5), (2.14). With (6.6)–(6.8) (line) and (6.12):
$$\Pi_\nu=\beta_\nu\,\frac{\mu_1\Pi_1+\mu_2\Pi_2}{\mu_1+\mu_2}\,\big[1+O(\varepsilon)\big];\qquad\text{if }\gamma_1c_1^2=\gamma_2c_2^2:\quad\Pi_\nu-\Pi_1=-(1-\beta_\nu)\,\Pi_1+O(\varepsilon\,\Pi_1). \tag{6.14}$$
The single bodies have equal fall factors only under the named condition $\gamma_1c_1^2=\gamma_2c_2^2$ (M7). Even then the bound pair differs at leading order (order zero in $\varepsilon$ at fixed $\chi$) by the factor $\beta_\nu$: it falls slower for $0<\beta_\nu<1$. The $O(\varepsilon)$ corrections come from the second-order weights ((4.5), $W_b$ of (4.7)) and from $D^{\rm mark}$. *Control:* none in the class; unbound pairs on the line ($\chi\to0$) give $\beta_0\to1$ and equality under the condition.

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

At fixed $n,\gamma_i,M_i,\tau_i$: $R_i$, $\mu_i$, $c_i$, $\bar\kappa_i$, $\bar t_0$ do not depend on $t_f$ at the orders used; $M^{(2)}$, $F$ and $\eta$ scale as $1/t_f$; $\phi\propto g_s/t_f$. The field gap of R1 is $\delta\simeq4\pi^2t_f/L^2$. For $\sigma_i=+1$ the levels reached by flipping $b_i$ include an excited field up to $4N_ft_f$ above its lowest state, so R1 needs $4N_ft_f<2M_i-\delta_1$; for $\sigma_i=-1$, $\delta_1\simeq2M_i$. With the weak-binding $\beta_0=1-\frac{a_0}6(4\chi^2)^{1/3}$ from $\partial\epsilon_0/\partial\tau_h=-2\beta_0$ applied to (6.4):
$$\chi=\frac{n\lvert g_1g_2\rvert}{t_f\lvert\bar t_0\rvert}\propto\frac1{t_f},\quad1-\beta_0\simeq0.270\,\chi^{2/3}\propto t_f^{-2/3},\quad\frac{\tau'^2L^2}{4\pi^2\delta_1}\ll t_f\ll\frac{n\lvert g_1g_2\rvert L^3}{\lvert\bar t_0\rvert},\quad t_f<\frac{2M_i-\delta_1}{4N_f}\ (\sigma_i=+1). \tag{6.15}$$
Hence: (6.3) does not depend on $t_f$, but B1 holds only below the upper bound in (6.15), with $\xi_0\propto t_f^{1/3}$. $D^{\rm lat}_\nu$ and $1-\Pi_\nu/\Pi_1$ fall as $t_f^{-2/3}$, $D^{\rm mark}_\nu$ and $M(0)$ as $1/t_f$. The coefficient mismatches of Step 5 ($2s_c/\beta_\nu$ against 1) and the factor $\beta_\nu$ of (6.14) do not vanish at any $t_f$ inside the window, since $\lvert\beta_\nu\rvert<1$ there. A faster field (larger $t_f$) weakens the binding and brings the pair's inertia and fall towards those of its parts; 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 and B, every branch combination and both signs of $\gamma_1\gamma_2$: bound levels exist; they are the lowest levels of relative motion if $\sigma_1\sigma_2\gamma_1\gamma_2>0$ and the highest if $<0$. They are made of bodies near rest iff $g_1g_2(\mu_1^{-1}+\mu_2^{-1})>0$; e.g. $\sigma_i=+1$ with $\gamma_1\gamma_2>0$ (lowest), or $\sigma_i=-1$ with $\gamma_1\gamma_2<0$ (highest, negative relative inertia).
- **(6.4)** Leading order in $\varepsilon$, uniform hopping, weak binding $F\ll\lvert\bar t(P)\rvert$, up to $O(\chi^{2/3})$ and $O(\xi_\nu/L)$: $E_b(P)=E_1(0)+E_2(0)-E_\varnothing+M(0)-2\bar t_0+s_e[-2\lvert\bar t(P)\rvert+a_\nu(4\lvert\bar t(P)\rvert F^2)^{1/3}]$, with $a_0=1.0188$, $a_1=2.3381$.
- **(6.5)** Definitions: bond weights $b_\nu$, bond overlap $\beta_\nu$ with $\lvert\beta_\nu\rvert<1$ for every bound level, local pair inertia $m(\rho)$.
- **(6.6)** Exact for the band contract (order $\tau^2$, B2): $E_b''(0)=\sum_\rho b_\nu(\rho)/m(\rho)$ plus a ring term that depends on the continuation and is exponentially small in $L/\xi_\nu$ (times $L^2$).
- **(6.7)** Order $\varepsilon^2$, error $O(\varepsilon^3)$: $m(\rho)=\mu_1+\mu_2+s_c\bar M^{(2)}(\rho)(c_1^{-2}+c_2^{-2})$; the mutual term enters each body's inertia once, over its own speed squared, with sign $+$ for (a) and $-$ for (c).
- **(6.8)** On the line, through order $\varepsilon^2$: $1/E_b''(0)-\mu_1-\mu_2=D^{\rm lat}_\nu+D^{\rm mark}_\nu$. The leading part $D^{\rm lat}_\nu=(\mu_1+\mu_2)(1-\beta_\nu)/\beta_\nu$, from the $P$-dependence of $K_1+K_2$ with the binding by $W$, is nonzero for every bound level and, in weak binding, grows with the level of relative motion. $D^{\rm mark}_\nu=O(\varepsilon^2)$ comes from the $\rho$-dependence of $K_i$. No description of the class has $D^{\rm lat}_\nu=0$; it vanishes only for unbound pairs ($\beta_\nu\to1$).
- **(6.9)** Order $\tau^2$, through $\varepsilon^2$: the cost of the bound pair is $\Delta E_\nu=\langle M\rangle_\nu+K_\nu$ with the kinetic cost $K_\nu=2\bar t_0(\beta_\nu-1)$.
- **(6.10)** Negative, leading order, both couplings: $1/E_b''(0)-\mu_1-\mu_2=\Delta E_\nu/c^2$ does not hold as an identity on the class. For equal speeds the coefficients of the contact value $M(0)$ differ ($2s_c/\beta_\nu$ against 1), and so do those of $K_\nu$ ($4\mu_1\mu_2/\beta_\nu$ against $c^{-2}$). For unequal speeds no single speed fits, neither $c_1$ nor $c_2$. For $\sigma_1=\sigma_2$, bodies near rest, $\langle\lvert r\rvert\rangle_\nu\ll L/6$ and $1-\beta_\nu\gg\lvert M(0)\rvert/R_i$, the difference of inertia and that of cost have opposite signs. No control in the class.
- **(6.11)** Definition: the weight of the bound level is the $f_\nu^2$-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, $n(g_1+g_2)\Delta\phi$, for every level.
- **(6.13)** (7.7) carries over to the bound level, with $E_b''(0)$ and $W'_\nu$, for envelopes broad against the range of the harmonics of $E_b(P)$, at first order in $g_s$, up to $O(\tau^4\gamma g_s)$.
- **(6.14)** Negative, leading order (order zero in $\varepsilon$ at fixed $\chi$), both couplings: $\Pi_\nu=\beta_\nu(\mu_1\Pi_1+\mu_2\Pi_2)/(\mu_1+\mu_2)$. Even under the named condition $\gamma_1c_1^2=\gamma_2c_2^2$ the bound level's fall factor is $\beta_\nu\Pi_1\neq\Pi_1$; the first order of the difference is $-(1-\beta_\nu)\Pi_1$, with corrections $O(\varepsilon\Pi_1)$. No control in the class.
- **(6.15)** Item 5, at fixed $n,\gamma_i,M_i,\tau_i$: $\chi\propto1/t_f$. The deviations of items 2 and 4 fall as $t_f^{-2/3}$ (weak binding) and the mark terms as $1/t_f$. B1 and R1 confine $t_f$ to the window of (6.15), with an upper bound $t_f<(2M_i-\delta_1)/(4N_f)$ for $\sigma_i=+1$. No $t_f$ inside the window removes the negatives (6.10), (6.14).

## Consistency checks

1. *Dimensions.* $P$ is dimensionless, so $\mu_i$, $m$, $1/E_b''$ have the dimension $[C]^{-1}$. So do $\bar M^{(2)}/c_i^2$ and $\mu_1\mu_2K_\nu$ ($c_i^2\sim[C]^2$), and $\Pi_\nu,\Pi_i\sim[C]$; $\beta_\nu$, $\chi$ are pure numbers. Passed.
2. *Unbound limit $\chi\to0$ on the line* (e.g. $\gamma_2\to0$): $\beta_0=1-0.270\chi^{2/3}\to1$ and $M^{(2)}\to0$, so $1/E_b''(0)\to\mu_1+\mu_2$. This agrees with minimizing $E_1(p_1)+E_2(p_2)$ at fixed $p_1+p_2=P$, whose curvature is $-8\bar\kappa_1\bar\kappa_2/(\bar\kappa_1+\bar\kappa_2)=1/(\mu_1+\mu_2)$. Passed.
3. *Tight binding $\chi\to\infty$.* Even class: $f_0\simeq\delta_{r0}$, $\beta_0\simeq1/\chi$, so $E_b''(0)\simeq8\bar\kappa_1\bar\kappa_2/F$. This agrees with direct second-order perturbation of the contact state, $-\lvert\bar t(P)\rvert^2/F$. Odd class: $f_0\simeq(\delta_{r,1}\pm\delta_{r,-1})/\sqrt2$, $\beta_0\to\frac12$, so $1/E_b''(0)=2(\mu_1+\mu_2)$. This agrees with the two-site level $V(1)-\lvert\bar t(P)\rvert$. Passed.

## Open issues

- At finite $L$ the curvature at $P=0$ depends on the continuation in $P$ through the ring term of (6.6); in the code it is 12% at $L=12$ and 3% at $L=16$ (set A). Only the line value is a property of the bound level.
- The closed forms (6.2), (6.4), (6.15) use $M^{(2)}$; at $\eta\approx0.3$ (code) the exact binding slope is about half of $F$. Statements in terms of $\beta_\nu$, $b_\nu$, $M$ 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 $O(\varepsilon)$ correction of (6.14) from the second-order weights is not evaluated (the bond function $B$ of (4.7) is not quoted); in the code it is 13–16%.
- For the even class in tight binding, $\beta_0\sim1/\chi$ and B3 fails when $\chi\gtrsim\delta_1\delta/\tau'^2$; the order-$\tau^4$ band contract is then needed.
- R1 for $\sigma_i=+1$ bounds $t_f$ by $(2M_i-\delta_1)/(4N_f)$, restrictive for many field objects; whether weakly coupled resonances with multiply excited field states can be tolerated 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/v1/check_inertia.py` (check). It treats $N_f=1$, $t_f=1$ and two parameter sets: A ($\sigma_i=+1$, $M=(3,4)$, $\tau_1=\tau_2=0.2$, $\gamma=(0.09,0.07)$, $r$ even, lowest level) and B ($\sigma_i=-1$, $M=(2.5,2)$, $\tau=(0.16,0.2)$, $\gamma=(0.10,-0.14)$, $r$ odd, highest level). For each set it runs $L=12,16$ and couplings (a), (c). It builds the exact sector contract of two bodies and the field object (body 1 as representative, continued in $P$) and finds the exact bound level by its weight on $\mathcal G$ (≥0.9996). It takes its curvature at $P=0$ by second-order perturbation in $P$ (cross-checked by finite differences), and builds the band contract by second-order Schrieffer–Wolff on the $\tau=0$ levels. Results: (6.6) with its ring term agrees with the exact curvature to $\le0.16\%$ in all eight cases (the rest is order $\tau^4$). The ring term is 12% ($L=12$) and 3% ($L=16$) for A, 6% and 0.7% for B. $\beta_\nu=0.69$–$0.79$; $1/E_b''(0)$ exceeds $\mu_1+\mu_2$ by 22–29%. $D\,c_i^2/\Delta E\approx-500$ to $-575$ (A, equal speeds) and $-208$ to $-363$ (B): negative and far from 1, as in (6.10). 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). The script also prints $\sigma_iR_i/(c_i^2\mu_i)=0.98$–$1.00$, a check of (2.14). Outputs: `code/v1/output/inertia_check.txt`, `code/v1/output/inertia_check.json`.
- `code/v1/check_weight.py` (check). It takes set A, couplings (a), (c), $L=16,20$, with a held source $g_s=-0.01$ (and $-0.005$ at $L=16$), and computes the exact eigenvalues of the full contract near the bottom of the bound band (weight on $\mathcal G$ ≥0.9997), where the pair sits in the weight well opposite the source. It compares them with the reduced description $E_b(\hat P)+W_\nu(\hat X)$ of Step 6, using exact sector levels $E_b(P_k)$ and (6.11) with exact held weights. The main level spacing agrees to 0.03% ($L=16$) and 0.17% ($L=20$); the same description with the pair inertia set to $\mu_1+\mu_2$ misses it by 16% and 10%. Level deviations scale as $g_s^2$. The levels come in nearly degenerate pairs because $E_b(P+\pi)=E_b(P)$. $E_b''(0)(\mu_1+\mu_2)=0.82$ ($L=16$) and $0.86$ ($L=20$), against $\beta_0=0.79$ and $0.85$; the difference is the ring term. The ratio $\Delta W_\nu/(\Delta W_1+\Delta W_2)$ is 0.87 at these couplings and tends to 1 as they are scaled down (0.997 at scale 0.03), as in (6.12). Outputs: `code/v1/output/weight_check.txt`, `code/v1/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; Airy functions (continuum limit of a linear potential)
- band projection, double commutator for the acceleration from rest
- exact diagonalization (dense and shift-invert sparse) for the code checks

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