Version 3 · earlier version; the current one is v4 · External review, round 3: major errors

# Inertia and weight of a pair bound through the field, full grip

- **Subproject:** 07-ilang-weight
- **Package:** 07-inertia-of-pair-full-grip
- **Version:** v3
- **Mode:** external regeneration
- **Date:** 2026-10-21

## Changes from previous version

- Setup S6, Step 5, new (7.14), (7.15): the source is the fixed held source $g_s\Phi_{x_s}$ in laboratory coordinates. Its first-order projection on the selected band is the centre contract (7.14). The $P=0$ symbol of that contract is related to the midpoint-conditioned density $\rho_0$ of v2, with remainders (7.15). The midpoint-following operator is now an auxiliary probe only (I1).
- Setup S4, S5, Step 3 (*Flux remainder*), (7.7), (7.8), (7.11), (7.12): the ring-flux remainder $\rho_L$ is kept explicit and bounded for $w\gg1$ and $w\lesssim1$. The leading fall factor carries $1+\rho_L$, and the mutual-order results carry $\varepsilon_L$. S5 implies $w\gg1$ (I2).
- Setup S6, (7.12): the hierarchy now contains $\delta_m\tau^2/d^2\ll\kappa_s$, and also $\tau^2/(d^2\eta_{\min}L)\ll\kappa_s^{3/2}$ from (7.15) (I3).
- Code: new `check_fixed_source.py`, which tests the fixed source exactly in the full laboratory space and in the band contract in laboratory coordinates. The main script labels the probe as auxiliary, reports $\rho_L$ and has the small-$w$ controls F (I1, I2).
- Result, Open issues and Code updated accordingly.

## Response to verification

- I1: Accepted and fixed in Setup S6, Step 5 (*Fixed source and centre contract*, *Relation to the internal density*, (7.14), (7.15)) and Code. At first order in $g_s$, the fixed source acts on the isolated selected band as the centre contract (7.14). The $P=0$ symbol $W_{\rm ex}$ of this contract is the weight of the centre at rest. By (7.15), $W_{\rm ex}'$ equals $W_0'=g_s\rho_0'$ up to stated remainders. The midpoint-following operator is labelled an auxiliary mathematical probe. Numerically:
  - **Exact.** The eigenvalues of the full laboratory contract with the fixed source ($L=12,16$) agree with (7.14) built from exact matrix elements to $10^{-4}$ of the bond effect. At these sizes they do not resolve $W_0$ itself, because the internal state already changes at $P=2\pi/L$.
  - **Exact first order.** From exact Bloch vectors ($L=64,128$), $W_{\rm ex}'$ of the bond agrees with $W_b'$ of $W_0$ to 0.2–1.3% for $s\ge8$. Without the kinetic part the difference is 3–14%.
  - **Band level.** The band contract in laboratory coordinates with the fixed source at the S5 sizes ($L=256$) agrees with $W_b'$ to 0.15–0.27%. $h_c$ with $W_0$ reproduces its lowest 21 eigenvalues to 0.4–0.8% of the bond effect.
- I2: Accepted and fixed in Setup S4, S5, Step 3 (*Flux remainder*), Step 4 ((7.7), (7.8)) and Step 5 ((7.11), (7.12)). $\rho_L=O(e^{-c(L/w)^{3/2}})$ for $w\gg1$ and $O(L^2(w^3/4)^{L/2-2})$ for $w\lesssim1$, which is $O(w^3)$ at $L=6$, as the verifier says. Where $w$ may be small, $\rho_L$ is kept explicit. S5 implies $w\gg1$, and (7.8), (7.12) list $\varepsilon_L\le\lvert\rho_L\rvert/(2\eta_{\min})$. Control F: $\rho_L=0.125$ at $L=6$, $w=0.50$.
- I3: Accepted and fixed in Setup S6 and (7.12). The condition $\delta_m\tau^2/d^2\ll\kappa_s$ is added to the hierarchy, not derived from the other assumptions.

## Setup and assumptions

**Description (M8).** The field, the two bodies, the full grip and the field state of `question.md`: ring of even length $L\ge6$, $N_f\ge1$ field objects with own term $C_1$, $n=N_f/L$; Model D bodies $b_1,b_2$ of different types, own contract the motion term $A$ with $\tau_i$, rest and clock term only in the pair term $\gamma_i\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes(m_i+D_i)\otimes\Phi_x$; no pair term between $b_1,b_2$; records at $\kappa\to0$. $D_i$ commutes with every term, so the clock level $k_i$ is conserved and on branch $\sigma$ the pair term is $\sigma\gamma_iM_i\Phi_x$; $g_i=\sigma_i\gamma_iM_i$ for the selected branches.

**Inputs** (quoted in `question.md`): (2.1), (2.10) of 07-ilang-weight/02-two-free-bodies; (4.2) of 06-ilang-source/04-profile; (4.2), (4.5), (4.7) of 07-ilang-weight/04-weight-of-bond (ring, $\mathcal X=L$); (5.1), (5.3), (5.4), (5.7), (5.14) of 07-ilang-weight/05-inertia-of-mark (FG: $\theta=0$, $K=T_-$, $v=\tau$); (6.1), (6.5), (6.6), (6.8), (6.9), (6.13) of 07-ilang-weight/06-inertia-of-pair. Their notation is used unchanged ($\ell,\ell_0,\ell^{(2)},\phi,B,\eta,\eta_i,\mu_i,\kappa_i,\bar\kappa_i,\bar t_0,d_i,U,f_0,b_0,\beta_0,m,\Delta E_0,K_0,\langle\cdot\rangle_0,W_{\rm pair}$). On $0\le x\le L$, $\ell(x)=\ell_0-\frac{x(L-x)}{2L}$, $\ell_0=\frac{L^2-1}{12L}$ (it solves $\Lambda\ell=\delta_0-\frac1L$, $\sum_x\ell=0$, by substitution). $F:=ng_1g_2/t_f$; $'$ is the central difference in the place.

**Assumptions** (M8, M9; each a choice of the package).
- **S1** Field state: the lowest state adapted to the bodies (A11): in the group $\mathcal G$ every field object is in the lowest one-object state of the held configuration.
- **S2** Band regime: $\max_i\tau_i\ll d$, where $d$ is the distance of the held levels of $\mathcal G$ from all other held levels of its parity class (Step 2): the field excitations of $\mathcal G$, the two single branch flips and the joint flip, each with all field states. At leading order in $\eta$, $d\simeq\min\big(2n\lvert g_1\rvert,2n\lvert g_2\rvert,d_\times,\varepsilon_1-2n(g_1^++g_2^+)\big)$, $g^+:=\max(g,0)$, with $d_\times\simeq2n\lvert g_1+g_2\rvert-\frac32\lvert B(0)\rvert$ the distance to the jointly flipped sector. S2 contains the nonresonance condition **(N×)** $d_\times\gg\max_i\tau_i$. Omitted: relative $O(\tau^2/d^2)$.
- **S3** Field: $\eta=\max_i\lvert g_i\rvert\ell_0/t_f\ll1$ and $N_f\eta\ll1$. Omitted: relative $O(\eta)$ in mutual terms, $O(N_f\eta^2)$ in field overlaps.
- **S4** Binding: $w\ll L$, $w:=(4\lvert\bar t_0\rvert/\lvert F\rvert)^{1/3}$. The flux term of (6.6) is kept as $\rho_L$ (Step 3): $O(e^{-c(L/w)^{3/2}})$ for $w\gg1$, $O(L^2(w^3/4)^{L/2-2})$ for $w\lesssim1$. $\varepsilon_L$ is its size against the mutual part of the inertia. $\langle V\rangle_0/\lvert B(0)\rvert=O(w/L)$.
- **S5** Regime of item 3 (ii): $\delta_m:=\dfrac{\lvert\mu_1+\mu_2\rvert(1-\beta_0)}{\lvert B(0)\rvert(\tau_1^{-2}+\tau_2^{-2})}\ll1$. By (7.5) this is $\lvert D^{\rm lat}/D^{\rm mark}\rvert$ up to relative $O(\eta,w/L)$. It implies $1-\beta_0\le4\eta\delta_m/c\ll\eta$ (Step 4), not conversely. Hence $w\gg1$ and $\varepsilon_L=O(e^{-c(L/w)^{3/2}}/\eta_{\min})$ up to powers of $L/w$.
- **S6** Source: the held source $g_s\Phi_{x_s}$ (a pair term of a held source with the field), added to $C$ in laboratory coordinates; weights to first order in $g_s$. The pair is at distance $s$ from $x_s$, measured from its midpoint $X=(x_1+x_2)/2$, with $w\ll s\le L/2$ and $u:=s/L$. For (7.12) in addition: $\eta\ll\kappa_s$, $w/L\ll\kappa_s$, $\delta_m\tau^2/d^2\ll\kappa_s$ and $\tau^2/(d^2\eta_{\min}L)\ll\kappa_s^{3/2}$, with $\kappa_s$ of (7.9) and $\eta_{\min}=\min_i\eta_i$. These imply $(1-2u)^2\gg\eta$, which excludes a neighbourhood of the antipode, so $\phi'\neq0$.
- Out of the class (tuned): $\lvert\tau_1^2g_2+\tau_2^2g_1\rvert<c\max(\tau_1^2\lvert g_2\rvert,\tau_2^2\lvert g_1\rvert)$ for a fixed $c\in(0,1)$ (this is $\lvert\mu_1^{-1}+\mu_2^{-1}\rvert<c\max_i\lvert\mu_i^{-1}\rvert$, by Step 1); in item 4, $\lvert g_1+g_2\rvert$ small against $\max\lvert g_i\rvert$; descriptions that violate (N×).

## Derivation

### Step 1. Held levels, half-gaps and odd parts

By S1, non-interacting objects in the lowest state all occupy the lowest one-object level, so the held level of $\mathcal G$ is $E(j,l)=N_f\varepsilon_0(g_1\,\text{at}\,j,\,g_2\,\text{at}\,l)$ and the mutual term $M(r)$, $r=l-j$, is (4.2) of 06-ilang-source/04-profile. Define, with the other coupling held fixed,
$$\sigma_iR_i:=\tfrac{N_f}2\big[\varepsilon_0(g_i)-\varepsilon_0(-g_i)\big],\quad c_i^2:=\frac{\sigma_iR_i}{\mu_i},\quad M_i^{o}(r):=\tfrac12\big[M(g_i)-M(-g_i)\big](r),\quad M_i^e:=M-M_i^o,\quad \bar f(\rho):=\tfrac14\big[f(\rho-1)+2f(\rho)+f(\rho+1)\big].\tag{7.1}$$
$\sigma_iR_i$ is the branch-odd part of the cost (5.3) ($\theta=0$). Rayleigh–Schrödinger perturbation of $\varepsilon_0$ in $\sum_ag_a\delta_{x_a}$, with $\varepsilon_q-\varepsilon_0=2t_f(1-\cos q)$ and $\frac1L\sum_{q\ne0}\frac{e^{iqx}}{2(1-\cos q)}=\ell(x)$, gives $N_f\varepsilon_0=-2N_ft_f+n\sum_ag_a-\frac n{t_f}\sum_{a,b}g_ag_b\ell(x_a-x_b)+O(n\lvert g\rvert\eta^2)$. Hence $\sigma_iR_i=ng_i[1+O(\eta^2)]$; $M=B(r)[1+O(\eta)]$ with $B$ of (4.2) of 04-weight-of-bond, and since $B\propto g_1g_2$ is odd in each coupling, $M_1^o=M_2^o=B[1+O(\eta)]$, $M_i^e=O(\eta B)$. In (5.3) the field state of the flipped branch differs at first order in $g_i$ by a vector of squared norm $O(\eta^2)$ per object, so $c_i^2=\tau_i^2[1+O(N_f\eta^2)]$ and $\mu_i=ng_i\tau_i^{-2}[1+O(N_f\eta^2)]$; in particular $\operatorname{sign}\mu_i=\operatorname{sign}g_i$.

### Step 2. Band contract and selected level (item 1)

*Group and distances.* Each motion term conserves $x_i$ plus the branch index mod 2, so $r+s_1+s_2$ mod 2 is conserved ($s_i=0$ on the selected, $1$ on the flipped branch). $\mathcal G$ ($r$ even, $s_1=s_2=0$) lies in the class with $r+s_1+s_2$ even, together with its field excitations, the single flips at odd $r$ and the joint flip at even $r$. The class with odd $r$ has its own selected level, and everything below holds for either. The motion terms connect $\mathcal G$ at first order only to the single flips; the joint flip is reached at second order. Its held levels are $E_{--}(r')=N_f\varepsilon_0(-g_1\,\text{at}\,j,-g_2\,\text{at}\,l)$, and by the expansion of Step 1, whose second-order terms are even under the joint flip, $E(r)-E_{--}(r')=2n(g_1+g_2)+B(r)-B(r')+O(n\lvert g\rvert\eta^2)$. $B$ ranges over an interval of length $\lvert F\rvert L/4=\frac32\lvert B(0)\rvert[1+O(L^{-2})]$ ($V$ below), so the two held spectra are at distance $d_\times=2n\lvert g_1+g_2\rvert-\frac32\lvert B(0)\rvert$ at leading order. The single flips are at $2n\lvert g_i\rvert[1+O(\eta)]$ ((5.4) for each body), and the excited field states of the flipped sectors at $\varepsilon_1-2n\sum_{\rm flipped}g_i$; this gives $d$ of S2.

Under S2 the derivation of (2.10) applies unchanged: it uses only that $\mathcal G$ is isolated in the held spectrum and that each motion term moves one body by one place and flips its branch; the full grip changes only the held levels ($\theta=0$ in (5.1)). So (2.10) holds with error $O(\tau^4/d^3)$; the terms through the joint flip are of fourth order, with denominators $\ge d$, and belong to it. Where (N×) fails, in particular for $g_1=-g_2$, where reflection maps the selected held spectrum onto the jointly flipped one, $\mathcal G$ is not isolated and the scalar band contract is not used; such descriptions are outside the class. With (2.1) the ansatz $\psi(j,l)=e^{iPj}g(l-j)$ gives (6.1) unchanged (the double step of $b_1$ carries the relative field coordinates: phase $e^{-2iP}$).

*Amplitudes.* The double step of $b_i$ across the relative bond $\rho$ passes through the held level with $b_i$ flipped at relative place $\rho$. Since $M(-g_i)=-M_i^o+M_i^e$, the gap from the held level at $r=\rho\mp1$ is
$$\Delta_i(r;\rho)=2\sigma_iR_i+M_i^o(r)+M_i^o(\rho)+M_i^e(r)-M_i^e(\rho).\tag{7.2}$$
Second-order Schrieffer–Wolff with symmetrized denominators gives $\kappa_i(\rho)=-\frac{\tau_i^2a_i^2}8\big[\Delta_i(\rho-1;\rho)^{-1}+\Delta_i(\rho+1;\rho)^{-1}\big]$, up to the excited-state term of (5.3); for $M=0$ this is $\bar\kappa_i=-1/(8\mu_i)$ of (5.3). The difference $-1/(8\kappa_i(\rho))-\mu_i$ vanishes with the partner's coupling $g_j$. Its first order is the shift of the denominators by $M_i^o=O(\sigma_iR_i\eta_j)$. The next terms are relative $O(\eta)$ to it: the second order of this expansion, relative $O(\eta_j)$, and the partner's change of the field overlaps, relative $O(\eta_i)$. The $M_i^e$ terms combine to a second difference, which is negligible. So the **local inertia** of $b_i$ is
$$m_i(\rho):=-\frac1{8\kappa_i(\rho)}=\mu_i+\frac{\bar M_i^o(\rho)}{c_i^2}+O(\lvert\mu_i\rvert\eta\eta_j)=\mu_i+\frac{\bar B(\rho)}{\tau_i^2}\big[1+O(\eta)\big].\tag{7.3}$$
The partner's mark changes each body's local inertia by the bond average of the mutual term over that body's own $\tau_i^2$. The same gaps give $d_i(r)=-2\bar\kappa_i[1-\bar M_i^o(r)/\sigma_iR_i]$. (6.5) and (6.6) hold: they use only the form (6.1), a real $f_0$ at $P=0$, and Hellmann–Feynman in the bond magnitudes and the flux.

*Bound level.* $U(r)-U(0)=V(r)[1+O(\eta,\tau^2/d^2)]$ with $V(r):=B(r)-B(0)=F\,r(L-r)/L$ on $0\le r\le L$. Near the edge $P=0$, $k=0$ of the relative band, (6.1) has relative hop $\bar t_0=-\frac18(\mu_1^{-1}+\mu_2^{-1})$: the selected level is the lowest if $\bar t_0<0$, the highest if $\bar t_0>0$. It is bound at $r\approx0$ iff the extremum of $V$ at $r=0$ pushes it out of the band: a minimum ($F>0$) for the lowest, a maximum ($F<0$) for the highest, i.e. $g_1g_2(\mu_1^{-1}+\mu_2^{-1})>0$. With Step 1, in the class (S2 with (N×)):
$$\text{selected level bound}\iff\tau_1^2g_2+\tau_2^2g_1>0,\qquad g_i=\sigma_i\gamma_iM_i .\tag{7.4}$$
Both $g_i>0$: bound, lowest. Both $g_i<0$: not bound. Opposite signs: bound (highest) iff the inequality holds. Otherwise the extreme level sits at $r\approx L/2$, where $V$ has its other extremum (control: case D of the code). In the continuum limit of (6.1), $-4\lvert\bar t_0\rvert\partial_r^2+\lvert F\rvert\lvert r\rvert$, the width is $w$ of S4 and $1-\beta_0=O(w^{-2})$ for $w\gg1$ (for $w\ll1$, $\beta_0=O(w^3)$); $f_0>0$ (Perron–Frobenius), so $b_0>0$ and $0<\beta_0\le1$.

### Step 3. Inertia of the pair (item 2)

By (7.3), $m(\rho)=\mu_1+\mu_2+\delta m(\rho)$, $\delta m=\sum_i\bar M_i^o/c_i^2+e(\rho)$, $e=O(\sum_i\lvert\mu_i\rvert\eta\eta_j)$. In (6.6), $1/\sum_\rho b_0/m$ is exact in the constant part of $\delta m$, which it divides by $\beta_0$: a constant $e$ shifts $1/E_b''(0)$ by $e/\beta_0$. The variation of $\delta m$ over the pair enters first, and at second order as $-\mathrm{Var}(\delta m)/((\mu_1+\mu_2)\beta_0)$ (variance in the weights $b_0/\beta_0$), relative $O(\eta w^2/L^2)$ to the first-order term. The band contract differs from the exact contract by relative $O(\tau^2/d^2)$ in the $r$-dependent part of the hops. In their constant part it differs as for each body alone, and the exact $\mu_i$ absorb this up to relative $O(\tau^2/d^2)$ of the lattice part. Hence
$$D_0=D^{\rm lat}\Big[1+O\Big(\frac{\tau^2}{d^2}\Big)\Big]+D^{\rm mark}\Big[1+O\Big(\eta,\frac{\tau^2}{d^2}\Big)\Big]-\frac{\rho_L}{E''_{\rm line}}+O\Big(\frac{\rho_L^2}{E''_{\rm line}}\Big),\quad D^{\rm lat}=(\mu_1+\mu_2)\frac{1-\beta_0}{\beta_0},\quad D^{\rm mark}=\frac1{\beta_0^2}\sum_\rho b_0\sum_i\frac{\bar M_i^o}{c_i^2}=\Big(\frac1{\tau_1^2}+\frac1{\tau_2^2}\Big)\frac{\sum_\rho b_0\bar B}{\beta_0^2}\big[1+O(\eta)\big].\tag{7.5}$$
Both parts carry the factor $1/\beta_0$: $\sum_\rho b_0\bar B=\beta_0B(0)+\sum_\rho b_0\bar V$ with $\lvert\sum_\rho b_0\bar V\rvert/\beta_0=O(\lvert F\rvert\max(w,1))$, so $\lvert D^{\rm mark}\rvert=(\tau_1^{-2}+\tau_2^{-2})\lvert B(0)\rvert\beta_0^{-1}[1+O(\eta,\max(w,1)/L)]$. Each remainder is relative to the part it corrects, so (7.5) holds for small $\beta_0$ as well. This is (6.8) with $s_c=+1$ and $c_i=\tau_i$, the full-grip speeds of (5.7), plus the flux remainder of (6.6). $D^{\rm lat}$ comes from the lattice form of the bands, $D^{\rm mark}$ from the mutual term. Each body's inertia counts the part of $M$ odd in its own coupling, and at leading order this is all of $M$ for each body.

*Flux remainder.* By (6.6), $E_b''(0)=E''_{\rm line}(1+\rho_L)$ with $\rho_L:=\tilde\Phi'(0)^2\partial^2_{\tilde\Phi}\epsilon_0/E''_{\rm line}$. This term comes from circuits of the relative motion around the ring, and $\tilde\Phi'(0)^2=O(L^2)$ outside the tuned case.
- For $w\gg1$ the continuum bound state (Step 2) decays as $\exp[-\frac23(\lvert r\rvert/w)^{3/2}]$, and a circuit passes $r\approx\frac L2$. So $\lvert\rho_L\rvert=O(e^{-c(L/w)^{3/2}})$ up to powers of $L/w$, with $c>0$.
- For $w\lesssim1$ the locator expansion about $r=0$ gives the lowest circuit. It has $\frac L2$ hops $\simeq\bar t_0$ and $\frac L2-1$ denominators $\lvert V(2k)\rvert\ge\lvert F\rvert$, so $\partial^2_{\tilde\Phi}\epsilon_0=O(\lvert\bar t_0\rvert^{L/2}/\lvert F\rvert^{L/2-1})$. With $E''_{\rm line}=O(\bar t_0^2/\lvert F\rvert)$ ($b_0(\pm1)\simeq\lvert\bar t_0/V(2)\rvert$), this gives $\lvert\rho_L\rvert=O(L^2(w^3/4)^{L/2-2})$. This is $O(w^3)$ at $L=6$ and not exponentially small (control F of the code).

Against the mutual part of the inertia the flux term is $\varepsilon_L:=\lvert\rho_L\rvert/(E''_{\rm line}\lvert D^{\rm mark}\rvert)$. Since $E''_{\rm line}=\beta_0(\mu_1+\mu_2)^{-1}[1+O(\eta)]$ and $\lvert D^{\rm mark}\rvert\ge2\eta_{\min}\lvert\mu_1+\mu_2\rvert\beta_0^{-1}[1+O(\eta,\max(w,1)/L)]$ (from $\lvert g_1g_2\rvert(\tau_1^{-2}+\tau_2^{-2})\ge\min_i\lvert g_i\rvert\sum_i\lvert g_i\rvert\tau_i^{-2}$), we get $\varepsilon_L\le\lvert\rho_L\rvert/(2\eta_{\min})[1+O(\eta,\max(w,1)/L)]$. At small $L$ these corrections are not small (control F at $L=6$: $\varepsilon_L=1.9$). Wherever $w$ may be small, $\rho_L$ and $\varepsilon_L$ are kept explicit.

### Step 4. Cost of the bond and inertia against cost (item 3)

With $f_0$ real, $E_b(0)=\sum_rf_0^2U+2\sum_\rho b_0\,t(\rho;0)$, $E_i(0)=E_i^{\rm held}+\bar d_i+2\bar\kappa_i$, $E^{\rm held}_{\rm pair}(r)=E_1^{\rm held}+E_2^{\rm held}-E_\varnothing+M(r)$ (definition (4.2)). The partner shifts $d_i$ and $\kappa_i$ by the relative amount $-\bar M_i^o/\sigma_iR_i$ (Step 2), and $\bar d_i=-2\bar\kappa_i$, so these shifts cancel up to the factor $1-\beta_0$:
$$\Delta E_0=\langle M\rangle_0+K_0\big[1+O(\eta)\big],\qquad K_0=2\bar t_0(\beta_0-1)=\tfrac14(\mu_1^{-1}+\mu_2^{-1})(1-\beta_0).\tag{7.6}$$
**(i)** Since $(\mu_1+\mu_2)/(\mu_1^{-1}+\mu_2^{-1})=\mu_1\mu_2$, $D^{\rm lat}=4\mu_1\mu_2K_0/\beta_0$ exactly. By Step 3, $\sum_\rho b_0\bar B/\beta_0=\langle M\rangle_0+O(\eta\lvert B(0)\rvert,\lvert F\rvert w)$. With $v=\tau_i$, the speed of (5.7) of either body, and $j\neq i$:
$$D_0-\frac{\Delta E_0}{\tau_i^2}=\frac{4\mu_1\mu_2}{\beta_0}K_0\Big[1-\frac{\beta_0}{4\mu_1\mu_2\tau_i^2}\Big]+\frac{\langle M\rangle_0}{\beta_0}\Big[\frac1{\tau_j^2}+\frac{1-\beta_0}{\tau_i^2}\Big]-\frac{\rho_L}{E''_{\rm line}}+O\Big(\Big(\eta+\frac wL+\frac{\tau^2}{d^2}\Big)\lvert D^{\rm mark}\rvert+\frac{\tau^2}{d^2}\lvert D^{\rm lat}\rvert+\frac{\rho_L^2}{E''_{\rm line}}\Big),\qquad\lvert4\mu_1\mu_2\tau_i^2\rvert=\frac{(2n\lvert g_1\rvert)(2n\lvert g_2\rvert)}{\tau_j^2}\ge\frac{d^2}{\tau_j^2}\gg1.\tag{7.7}$$
The lattice part exceeds $K_0/v^2$ by a factor $\ge d^2/\tau_j^2$, and the mark part exceeds $\langle M\rangle_0/v^2$ by $\langle M\rangle_0[\tau_j^{-2}+(1-\beta_0)\tau_i^{-2}]/\beta_0$. For $g_1g_2<0$ both displayed terms are positive ($\mu_1\mu_2K_0>0$, $\langle M\rangle_0>0$). So (i) fails unless the remainder and the flux term are as large as both of them, which needs $\tau_i^2/\tau_j^2=O(\eta+w/L+\tau^2/d^2)$ and $\lvert D^{\rm lat}\rvert$ small against $\lvert D^{\rm mark}\rvert$: the limit of (7.8). For $g_1,g_2>0$ the two terms have opposite signs, and (i) holds only on the surface where they cancel, a tuned relation between $1-\beta_0$ and $\eta$. Elsewhere, no.

**(ii)** In S5, by (7.5), $\lvert D^{\rm lat}\rvert=\delta_m\lvert D^{\rm mark}\rvert[1+O(\eta,w/L)]$: the $\beta_0$ factors cancel, and the mutual term dominates the inertia. With the tuned case excluded, $\lvert\mu_1+\mu_2\rvert=\lvert\mu_1^{-1}+\mu_2^{-1}\rvert\lvert\mu_1\mu_2\rvert\ge c\max_i\lvert\mu_i\rvert\ge\frac c2\sum_i\lvert\mu_i\rvert$. With $\lvert\mu_i\rvert=n\lvert g_i\rvert\tau_i^{-2}$ and $\lvert B(0)\rvert=2n\min_i\lvert g_i\rvert\,\eta$ this gives $1-\beta_0\le4\eta\delta_m/c\ll\eta$; so $w\gg1$ and $\beta_0=1-O(\eta\delta_m)$. The converse fails: for $\tau_1=\tau_2$, $0<g_2\ll g_1$, $\delta_m\simeq(1-\beta_0)/(4\eta_2)$, which needs $1-\beta_0\ll\eta_2$. By (7.7), $\lvert K_0\rvert=\beta_0\lvert D^{\rm lat}\rvert/\lvert4\mu_1\mu_2\rvert\le2\delta_m(\tau_{\max}^2/d^2)\lvert B(0)\rvert[1+O(\eta)]$, so $\Delta E_0=\langle M\rangle_0[1+O(\delta_m\tau^2/d^2)]$. For $w\gg1$ the weights $b_0/\beta_0$ and $f_0^2$ differ by relative $O(w^{-2})$, so $\sum_\rho b_0\bar B/\beta_0=\langle M\rangle_0[1+O(\eta)]$. Against the mutual scale $\lvert D^{\rm mark}\rvert$ the omitted terms are then: the lattice part, $O(\delta_m)$; its remainder, $O(\delta_m\tau^2/d^2)$; the mark remainder, $O(\eta,\tau^2/d^2)$; the ring remainder, $\varepsilon_L$, which is $O(e^{-c(L/w)^{3/2}}/\eta_{\min})$ since S5 implies $w\gg1$; $1/\beta_0-1=O(\eta\delta_m)$; and $K_0$ in $\Delta E_0$, $O(\delta_m\tau^2/d^2)$. Hence
$$D_0=\Big(\frac1{\tau_1^2}+\frac1{\tau_2^2}\Big)\Delta E_0\Big[1+O\Big(\eta,\delta_m,\frac{\tau^2}{d^2},\varepsilon_L\Big)\Big];\qquad\text{equal speeds }\tau_1=\tau_2=v:\ D_0=\frac{2\Delta E_0}{v^2}.\tag{7.8}$$
The binding cost has inertia, but $(\tau_1^{-2}+\tau_2^{-2})\Delta E_0\ne\Delta E_0/\tau_i^2$: the mutual term is odd in both couplings, so it enters both half-gaps (7.3) but the cost only once. It approaches $\Delta E_0/\tau_i^2$ only for $\tau_i/\tau_j\to0$, with relative deviation $\tau_i^2/\tau_j^2$.

### Step 5. Weight and fall (item 4)

*Fixed source and centre contract.* The source $g_s\Phi_{x_s}$ of S6 breaks (2.1), so it has no $P$ sectors. Let $\Psi_P$, $P\in\frac{2\pi}L\mathbb Z_L$, be the eigenvectors of the selected level without source. Write them as $\Psi_P=L^{-1/2}\sum e^{iPX}u_P$, with the midpoint $X=j+h/2$ ($h\in(-\frac L2,\frac L2]$ the representative of $r$) and the internal state $u_P$ (branches, $r$, field place relative to $X$). The phases are fixed by $\langle u_P|u_0\rangle>0$. The selected band is isolated from the other levels of the relative motion by a gap $\Delta_{\rm rel}>0$. At first order in $g_s$, projection on it gives the centre contract
$$h_c=E_b(\hat P)+V,\qquad\langle X|V|X'\rangle=g_s\langle X|\Phi_{x_s}|X'\rangle,\qquad|X\rangle:=L^{-1/2}\textstyle\sum_Pe^{-iPX}\Psi_P,\tag{7.14}$$
up to $O(g_s^2/\Delta_{\rm rel})$ from the other levels. The $P=0$ symbol $W_{\rm ex}(X):=\sum_{X'}\langle X|V|X'\rangle=g_s\sum_Pe^{iPX}\langle\Psi_P|\Phi_{x_s}|\Psi_0\rangle$ is the potential of the centre at rest. The rest of $V$ comes from the $P$-dependence of $u_P$ and acts as a position-dependent inertia. It changes the acceleration from rest of a start with real envelope of width $w_X\gg w$ at relative $O(w^2/w_X^2)$. So (5.14) and (6.13) apply to $h_c$, with $W_{\rm ex}$ as the weight.

*Relation to the internal density.* Inserting $\Psi_P$ gives $W_{\rm ex}(X)=g_s\sum_aG(a;x_s-X-a)$, where $G(a;k):=\frac1L\sum_Pe^{-iPa}\sum u_P^*u_0\vert_k$ (sum over branches and $r$, field place $k$ from the midpoint). The $P=0$ part is $\sum_aG(a;k)=\rho_0(k)$, the field density at distance $k$ from the midpoint in the $P=0$ eigenvector, i.e. the internal density conditioned on the midpoint. Since $u_P$ is smooth in $P$, $G$ has a range of a few places in $a$. For $\lvert k\rvert\gg w$, a Taylor expansion in $a$ gives $W_{\rm ex}=g_s[\rho_0-\alpha'+\frac12\beta''-\dots](x_s-X)$ with $\alpha:=\sum_aaG$ and $\beta:=\sum_aa^2G$.

At first order in the couplings, each component of $u$ (fixed branches and $r$) has the far density $n-\frac{2n}{t_f}\sum_bc_b\ell(k-z_b)$ of its held field (Step 1), with couplings $c_b=\pm g_i$ at $z_b$. This density is quadratic in $k$ ($\ell''=\frac1L$). So $\beta''$ is constant and drops from the gradient, and $\alpha''=-\frac{2n}{t_fL}\sum_{\rm comp}(\sum_bc_b)\,a_1^{\rm comp}$, where $a_1^{\rm comp}$ is the first moment of $G$ in that component. On $\mathcal G$ every component has total coupling $g_1+g_2$, and $\sum a_1=0$ by the phase choice. So only the flipped components contribute: their weight is $O(\tau^2/d^2)$ and their $a_1=O(1)$. Hence, with $W_0(X):=g_s\rho_0(x_s-X)$,
$$W_{\rm ex}'(X)=W_0'(X)+O\Big(\frac{\tau^2}{d^2}\frac{n\lvert g_s\rvert\max_i\lvert g_i\rvert}{t_fL}\Big)+O\Big(\frac{w^2}{L^2}\frac{n\lvert g_s\rvert\max_ig_i^2\,\ell_0}{t_f^2}\Big).\tag{7.15}$$
The last term is the second order in the couplings, whose far density is quartic. Relative to the bond part $W_b'$ below ($\lvert W_b'\rvert\simeq4\kappa_s^{3/2}n\lvert g_sg_1g_2\rvert\ell_0/t_f^2$), the two remainders are $O(\tau^2/(d^2\eta_{\min}L\kappa_s^{3/2}))$ and $O((\eta/\eta_{\min})(w/(L\kappa_s))^2)$. The same holds for one body, with its place as centre. $W_0$ is a property of the $P=0$ eigenvector. The code computes it from the density, and cross-checks it with the Hellmann–Feynman shift under the operator $g_s\sum_{j,l}\lvert j,l\rangle\langle j,l\rvert\otimes\Phi_{X(j,l)+s}$. This operator is an auxiliary mathematical probe on the sector space: it couples both bodies and the field and is not a contract of the description.

*Content of $W_0$.* $\rho_0$ (for odd $r$, the two places at distance $k\mp\frac12$ averaged) contains the held densities and, through the flipped components, the change of the hops: the kinetic contribution. In the band contract in laboratory coordinates the source enters (2.10) through the held blocks at first order in $g_s$, so (6.13) carries over: $W_0'=\sum_rf_0^2(\partial_1+\partial_2)W_{\rm pair}$ plus the kinetic part. The normalized weights $b_0/m$ of the first form of (6.13) give the same constant part as $f_0^2$, and in the $r$-dependent part $\langle V\rangle_0$ they differ by relative $O(1-\beta_0,\eta)$. By (7.3) with the source in place of the partner, $\kappa_i,d_i$ change by the relative amount $-\phi$ ($M^o_{is}=ng_i\phi$). So the kinetic energies weigh $-(\text{kinetic energy})\,\phi'$. A body alone at $P=0$ has none ($\bar d_i+2\bar\kappa_i=0$), and the bond part is $-K_0\phi'$. All such terms are relative $O(\tau^2/d^2)$ against $ng\phi'$.

*Lattice identities.* For $1\le x\le L-1$: $\ell'(x)=-\frac12(1-\frac{2x}L)$; $(\ell^2)'=(2\ell+\frac1L)\ell'$ (as $\Lambda\ell=-\frac1L$ there); and since $\Lambda\ell^{(2)}=\ell$, $\ell^{(2)}$ is even about $0$ and $L/2$, the cubic $-\frac{Lx}{12}+\frac{x^2}4-\frac{x^3}{6L}=\frac{x(L-x)}{6}\ell'(x)$ is $(\ell^{(2)})'$. Inserting these into the derivative of $2\ell_0\ell+\ell^2-\frac2L\ell^{(2)}$ in (4.5) gives $4\ell_0\kappa_s\ell'$ with $\kappa_s$ below, and $\phi'=-\frac{2g_s}{t_f}\ell'$:
$$W_i'(X)=\phi'\big[ng_i+2\kappa_sC_i^{(2)}\big],\qquad C_i^{(2)}:=-\frac{ng_i^2\ell_0}{t_f},\qquad\kappa_s:=\frac{L^2(1-2u)^2+2}{L^2-1}=(1-2u)^2+O(L^{-2}),\tag{7.9}$$
up to relative $O(\eta^2)$. $C_i^{(2)}$ is the second-order even part of the cost; it is weighed with $2\kappa_s$ but has no inertia by (5.3).

*Bond.* In (4.7) at $x_{1,2}=X\mp r/2$: $f_0$ is even in $r$ at $P=0$ (as $V$ and $\kappa_i$ are), so $\langle r\rangle_0=0$. As $\phi$ is quadratic away from $x_s$, $\partial_X[W_b(X;r)-W_b(X;0)]=V(r)\phi'(X)+O(\lvert F\phi'\rvert r^2/L)$ for $\lvert r\rvert<s$, and $W_b(X;0)$ is (4.5) with $g_i^2\to2g_1g_2$. So, with the kinetic part,
$$W_b':=W_0'-W_1'-W_2'=\phi'\big[2\kappa_sB(0)+\langle V\rangle_0-K_0\big]+O\big(\eta\lvert B(0)\phi'\rvert\big)+O\big(\lvert F\phi'\rvert w^2/L\big).\tag{7.10}$$
The first remainder is the next order of the bond weight in the couplings. Like $W_b'$ it is odd about the antipode, hence proportional to $\phi'$. $\langle V\rangle_0=O(\lvert B(0)\rvert w/L)$ and $K_0$ are kept; (7.10) needs no relation between $\kappa_s$ and the small parameters.

*Fall factors.* By (5.14), (6.13) applied to $h_c$ (7.14), with $W_{\rm ex}'=W_0'$ up to (7.15), $\Gamma_i:=E_i''(0)W_i'=W_i'/\mu_i$ and $\Gamma_{\rm pair}:=E_b''(0)W_0'$ set the acceleration from rest $-\Gamma$ of a smooth start in the fixed source. With $\Gamma_b:=W_b'/D_0$ and $1/E_b''=\mu_1+\mu_2+D_0$, exactly,
$$\Gamma_{\rm pair}=\frac{\mu_1\Gamma_1+\mu_2\Gamma_2+D_0\Gamma_b}{\mu_1+\mu_2+D_0},\qquad\Gamma_i=\tau_i^2\phi'\Big[1-\frac{2\kappa_sg_i\ell_0}{t_f}+O\Big(\eta^2,N_f\eta^2,\frac{\tau^2}{d^2}\Big)\Big].\tag{7.11}$$
*Leading order.* $W_b'$ and $D^{\rm mark}$ are $O(\eta)$; the lattice inertia has weight $-K_0\phi'=-\beta_0D^{\rm lat}\phi'/(4\mu_1\mu_2)$, which is $O(\tau^2/d^2)$ against $\tau^2\phi'D^{\rm lat}$. Hence, with $E_b''=E''_{\rm line}(1+\rho_L)$ (Step 3), $\Gamma_{\rm pair}=(1+\rho_L)\beta_0\bar\Gamma[1+O(\eta,\tau^2/d^2)]$ for every $w$ of S4, with $\bar\Gamma:=\frac{\mu_1\Gamma_1+\mu_2\Gamma_2}{\mu_1+\mu_2}=\frac{\mu_1\tau_1^2+\mu_2\tau_2^2}{\mu_1+\mu_2}\phi'[1+O(\eta)]$. So $\Gamma_{\rm pair}=\Gamma_1=\Gamma_2$ at leading order (with $1-\beta_0\to0$, hence $w\gg1$ and $\rho_L\to0$) iff $\tau_1=\tau_2$. At $w\lesssim1$ and small $L$, the factor $1+\rho_L$ is not close to 1 (control F). For $\tau_1\ne\tau_2$ the bodies already fall differently; $1-\beta_0$ lowers the pair's fall, since the internal kinetic energy has inertia but no weight at this order.

*Order of the mutual term (S5, S6).* By (7.10), (7.8), with $\langle V\rangle_0/(2\kappa_sB(0))=O(w/(L\kappa_s))$, $K_0/(\kappa_sB(0))=O(\delta_m\tau^2/(\kappa_sd^2))$ (Step 4) and $\Delta E_0=B(0)[1+O(\eta,w/L)]$:
$$\Gamma_b=\frac{2\kappa_s\,\phi'}{\tau_1^{-2}+\tau_2^{-2}}\Big[1+O\Big(\frac\eta{\kappa_s},\frac w{L\kappa_s},\delta_m,\frac{\tau^2}{d^2},\frac{\delta_m\tau^2}{\kappa_sd^2},\frac{\tau^2}{d^2\eta_{\min}L\kappa_s^{3/2}},\varepsilon_L\Big)\Big],\qquad \Gamma_{\rm pair}-\bar\Gamma\overset{\tau_1=\tau_2=\tau}{=}4(1-\kappa_s)\frac{g_1g_2\ell_0}{t_f(g_1+g_2)}\tau^2\phi'.\tag{7.12}$$
The sixth remainder is the first one of (7.15) for the fixed source. The second one, $O((\eta/\eta_{\min})(w/(L\kappa_s))^2)$, is below $w/(L\kappa_s)$ unless the couplings differ by a factor $\gtrsim L\kappa_s/w$. Under the hierarchy of S6, and with $\varepsilon_L\ll1$ (S5 gives $\varepsilon_L=O(e^{-c(L/w)^{3/2}}/\eta_{\min})$), every remainder is small; the $f_0^2$ and $b_0/m$ forms of $W_0'$ differ by relative $O((1-\beta_0)w/(L\kappa_s))$ of $\Gamma_b$. For equal speeds the bond's weight and inertia both carry the factor 2 of (7.8). The weight also carries $\kappa_s=1-\frac{4s(L-s)-3}{L^2-1}<1$ for $1\le s\le L/2$, so $\Gamma_b=\kappa_s\tau^2\phi'$ against $\Gamma_i=\tau^2\phi'$ at leading order. At leading order the binding cost is weighed in the same proportion as its inertia only in the limit $u\to0$ ($w\ll s\ll L$), where $\Gamma_b/\Gamma_i=1-4u+O(u^2)$ up to the remainders of (7.11), (7.12). At finite $u$, where $1-\kappa_s\simeq4u$ exceeds these remainders, it is weighed less. The second equation of (7.12) is the first order of the difference, from (7.11) with $D_0\simeq2B(0)/\tau^2$, $\Gamma_b-\bar\Gamma\simeq(\kappa_s-1)\tau^2\phi'$, $\mu_1+\mu_2\simeq n(g_1+g_2)/\tau^2$. Besides this, $\bar\Gamma-\Gamma_i=O(\eta)$ for $g_1\ne g_2$ from the self terms of (7.11).

*One body and both bodies.* By (7.11), (7.12), $\Gamma_b/\Gamma_i=2\kappa_s/(1+\tau_i^2/\tau_j^2)$ at leading order. Equality with both bodies at once ($\Gamma_b=\Gamma_1=\Gamma_2$, one ratio of weight to inertia for bond and bodies) needs $\Gamma_1=\Gamma_2$, i.e. $\tau_1=\tau_2$ at leading order, and then $\kappa_s=1$, i.e. $u\to0$. Equality with body $i$ alone holds at $\kappa_s=\frac12(1+\tau_i^2/\tau_j^2)$. This value lies in $(\frac12,1)$, and is reached at one source distance, $u\simeq\frac12(1-\sqrt{\kappa_s})$, iff $\tau_i<\tau_j$ (inside S6 if $1-\tau_i^2/\tau_j^2\gg w/L$); for $\tau_i\ge\tau_j$ it is not reached at any $u>0$. There $\Gamma_b/\Gamma_j=\tau_i^2/\tau_j^2<1$: the bond shares the proportion of the slower body only, at a tuned source position.

### Step 6. The hopping of the field (item 5)

At fixed $n,\gamma_i,M_i,\tau_i$ (so fixed $g_i$, $\mu_i$) and leading order: $B,\langle M\rangle_0,D^{\rm mark},W_b'/\phi'\propto t_f^{-1}$; $F\propto t_f^{-1}$, so $w\propto t_f^{1/3}$, and for $w\gg1$, $1-\beta_0\propto t_f^{-2/3}$, $K_0,\langle V\rangle_0,D^{\rm lat}\propto t_f^{-2/3}$; $\phi'\propto g_s/t_f$, $\Gamma_i/\phi'$ and $\mu_i$ are independent of $t_f$. Hence
$$\frac{D^{\rm lat}}{\lvert D^{\rm mark}\rvert}\propto t_f^{1/3},\qquad \frac{D^{\rm mark}}{\Delta E_0}\to\frac1{\tau_1^2}+\frac1{\tau_2^2},\quad \frac{\Gamma_b}{\Gamma_i}\to\frac{2\kappa_s}{1+\tau_i^2/\tau_j^2}\quad\text{independent of }t_f.\tag{7.13}$$
A stiffer field weakens the bond ($\propto t_f^{-1}$) and widens it, and S5 lies at the small-$t_f$ end, bounded by $\eta\propto t_f^{-1}\ll1$ and by $\varepsilon_1\propto t_f$ in S2. No value of $t_f$ restores the proportions of (7.8) and (7.12).

## Result

Class: full grip for both bodies, $N_f\ge1$, even $L\ge6$, S1–S4 (S2 including (N×); S5, S6 where stated), without the tuned cases.
- **(7.1)** Definitions: half-gap $\sigma_iR_i$, $c_i^2=\sigma_iR_i/\mu_i=\tau_i^2[1+O(N_f\eta^2)]$, the part $M_i^o$ of the mutual term odd in $g_i$, and the bond average $\bar f$.
- **(7.2), (7.3)** Under S2 and S3, the band contract is (2.10)/(6.1) with error $O(\tau^4/d^3)$ and gaps (7.2). Here $d$ is the distance of $\mathcal G$ from all other held levels of its parity class, the joint flip included, and $\max_i\tau_i\ll d$. Where (N×) fails (e.g. $g_1=-g_2$) it is not used. Each body's local inertia is $m_i(\rho)=\mu_i+\bar M_i^o(\rho)/c_i^2+O(\lvert\mu_i\rvert\eta\eta_j)=\mu_i+\bar B(\rho)/\tau_i^2\,[1+O(\eta)]$. (6.5), (6.6) hold for the full grip. Control: coupling (c) has the opposite sign of the mark part ($s_c=-1$ in (6.8)).
- **(7.4)** In the class (S2 with (N×)), the selected level is bound at $r\approx0$ iff $\tau_1^2g_2+\tau_2^2g_1>0$: lowest if both $g_i>0$, highest for opposite signs, never for both $g_i<0$. Otherwise it sits at $r\approx L/2$ (control).
- **(7.5)** $D_0=1/E_b''(0)-\mu_1-\mu_2=D^{\rm lat}[1+O(\tau^2/d^2)]+D^{\rm mark}[1+O(\eta,\tau^2/d^2)]$ plus the ring remainder $-\rho_L/E''_{\rm line}+O(\rho_L^2/E''_{\rm line})$. Here $\rho_L=O(e^{-c(L/w)^{3/2}})$ for $w\gg1$ and $O(L^2(w^3/4)^{L/2-2})$ for $w\lesssim1$ (control F: $\rho_L=0.125$ at $L=6$, $w=0.50$). $D^{\rm lat}=(\mu_1+\mu_2)(1-\beta_0)/\beta_0$, and $D^{\rm mark}=(\tau_1^{-2}+\tau_2^{-2})\beta_0^{-2}\sum_\rho b_0\bar B\,[1+O(\eta)]$. Both carry $1/\beta_0$, and the estimate holds for small $\beta_0$. It is (6.8) with $s_c=+1$, $c_i=\tau_i$.
- **(7.6)** $\Delta E_0=\langle M\rangle_0+K_0[1+O(\eta)]$, $K_0=\frac14(\mu_1^{-1}+\mu_2^{-1})(1-\beta_0)$.
- **(7.7)** Item 3 (i), negative: $D_0-\Delta E_0/\tau_i^2=(4\mu_1\mu_2K_0/\beta_0)[1-\beta_0/(4\mu_1\mu_2\tau_i^2)]+(\langle M\rangle_0/\beta_0)[\tau_j^{-2}+(1-\beta_0)\tau_i^{-2}]-\rho_L/E''_{\rm line}$, up to $O((\eta+w/L+\tau^2/d^2)\lvert D^{\rm mark}\rvert+(\tau^2/d^2)\lvert D^{\rm lat}\rvert+\rho_L^2/E''_{\rm line})$. The lattice term exceeds $K_0/v^2$ by a factor $\ge d^2/\tau_j^2$. For $g_1g_2<0$ both terms are positive, and (i) fails except in the limit of (7.8). For $g_1,g_2>0$ it holds only on a tuned surface.
- **(7.8)** Item 3 (ii), negative: in S5 ($\delta_m\ll1$, which implies $1-\beta_0\ll\eta$ and $w\gg1$), $D_0=(\tau_1^{-2}+\tau_2^{-2})\Delta E_0[1+O(\eta,\delta_m,\tau^2/d^2,\varepsilon_L)]$, i.e. $2\Delta E_0/v^2$ for $\tau_1=\tau_2=v$; here $\varepsilon_L\le\lvert\rho_L\rvert/(2\eta_{\min})[1+O(\eta,w/L)]$. It approaches $\Delta E_0/\tau_i^2$ only as $\tau_i/\tau_j\to0$.
- **(7.14)** At first order in $g_s$, the fixed held source $g_s\Phi_{x_s}$ in laboratory coordinates acts on the isolated selected band as the centre contract $h_c=E_b(\hat P)+V$, up to $O(g_s^2/\Delta_{\rm rel})$. The $P=0$ symbol $W_{\rm ex}$ of $V$ is the weight of the centre at rest; the rest of $V$ changes the acceleration from rest of a start of width $w_X\gg w$ at relative $O(w^2/w_X^2)$.
- **(7.15)** For $s\gg w$: $W_{\rm ex}'=W_0'+O(\frac{\tau^2}{d^2}\frac{n\lvert g_s\rvert\max\lvert g_i\rvert}{t_fL})+O(\frac{w^2}{L^2}\frac{n\lvert g_s\rvert\max g_i^2\ell_0}{t_f^2})$, with $W_0(X)=g_s\rho_0(x_s-X)$ and $\rho_0$ the field density at distance $x_s-X$ from the midpoint in the $P=0$ eigenvector. Relative to the bond weight the remainders are $O(\tau^2/(d^2\eta_{\min}L\kappa_s^{3/2}))$ and $O((\eta/\eta_{\min})(w/(L\kappa_s))^2)$. The same holds for one body. The midpoint-following operator of the code is an auxiliary probe, not a contract of the description.
- **(7.9), (7.10)** In S6: $W_i'=\phi'[ng_i+2\kappa_sC_i^{(2)}]$ up to relative $O(\eta^2)$, with $\kappa_s=(L^2(1-2u)^2+2)/(L^2-1)$. $W_b'=\phi'[2\kappa_sB(0)+\langle V\rangle_0-K_0]+O(\eta\lvert B(0)\phi'\rvert)+O(\lvert F\phi'\rvert w^2/L)$. Here $W_0=g_s\rho_0(x_s-X)$ of (7.15), kinetic contribution included.
- **(7.11)** Exact: $\Gamma_{\rm pair}=(\mu_1\Gamma_1+\mu_2\Gamma_2+D_0\Gamma_b)/(\mu_1+\mu_2+D_0)$, and $\Gamma_i=\tau_i^2\phi'[1-2\kappa_sg_i\ell_0/t_f+O(\eta^2,N_f\eta^2,\tau^2/d^2)]$. For every $w$ of S4, $\Gamma_{\rm pair}=(1+\rho_L)\beta_0\bar\Gamma[1+O(\eta,\tau^2/d^2)]$. For $1-\beta_0\to0$ (so $w\gg1$, $\rho_L\to0$) this equals $\Gamma_1=\Gamma_2$ iff $\tau_1=\tau_2$. Controls: $\tau_1\ne\tau_2$; for $w\lesssim1$ at small $L$, $\rho_L$ is not negligible (F).
- **(7.12)** In S5 and S6 ($\eta\ll\kappa_s$, $w/L\ll\kappa_s$, $\delta_m\tau^2/d^2\ll\kappa_s$, $\tau^2/(d^2\eta_{\min}L)\ll\kappa_s^{3/2}$), for the fixed source: $\Gamma_b=2\kappa_s\phi'/(\tau_1^{-2}+\tau_2^{-2})\,[1+O(\eta/\kappa_s,w/(L\kappa_s),\delta_m,\tau^2/d^2,\delta_m\tau^2/(\kappa_sd^2),\tau^2/(d^2\eta_{\min}L\kappa_s^{3/2}),\varepsilon_L)]$. For $\tau_1=\tau_2$, at leading order, the binding cost is weighed in the same proportion as its inertia only as $u\to0$. At finite $u$, where $1-\kappa_s$ exceeds the remainders of (7.11), (7.12), it is weighed less, by $\kappa_s<1$, and $\Gamma_{\rm pair}-\bar\Gamma=4(1-\kappa_s)g_1g_2\ell_0\tau^2\phi'/(t_f(g_1+g_2))$. For $\tau_1\neq\tau_2$, $\Gamma_b=\Gamma_i$ for one body only at the tuned source distance $\kappa_s=(1+\tau_i^2/\tau_j^2)/2$, which exists iff $\tau_i<\tau_j$. $\Gamma_b=\Gamma_1=\Gamma_2$ only for $\tau_1=\tau_2$ and $u\to0$. Control: any $u>0$ (e.g. $u=1/4$).
- **(7.13)** At fixed $n,\gamma_i,M_i,\tau_i$: mark quantities $\propto t_f^{-1}$, lattice quantities $\propto t_f^{-2/3}$ ($w\gg1$), and $D^{\rm mark}/\Delta E_0$, $\Gamma_b/\Gamma_i$ are independent of $t_f$ at leading order.

## Consistency checks

1. **Dimensions.** $g,\tau,t_f,M,K_0$ are energies (rates in $\lambda$); $P,r,u,\phi,\kappa_s,\delta_m$ are dimensionless. So $\mu,D$ are inverse energies, and (7.3), (7.5), (7.7), (7.8) are consistent ($M/\tau^2$, $\mu_1\mu_2K_0$). $\Gamma=E''W'$ and $\tau^2\phi'$ are both energy squared per place.
2. **$g_2\to0$** (no binding): $B,M_i^o,W_b',D^{\rm mark}\to0$ linearly, (7.3) gives $m_1\to\mu_1$, $F\to0$ so $w\to\infty$ (S4 fails, as it must), and (7.9) is the single-body weight of (4.5).
3. **Coincident held pair $r=0$** ($\langle V\rangle_0=K_0=0$). (7.9) plus (7.10) give $\phi'[n(g_1+g_2)+2\kappa_s(C_1^{(2)}+C_2^{(2)}+B(0))]=\phi'[nG+2\kappa_sC_G^{(2)}]$, $G=g_1+g_2$, as must hold: two point couplings at one place are one coupling $G$.

## Open issues

- The code has $N_f=1$, so S5 is reached only at $\delta_m=0.09$ (S1) and $0.13$ (S2), with $L=256$. The remainders have sizeable coefficients: $W_b'-(7.10)$ is up to $3.4\,\eta\lvert B(0)\phi'\rvert$, and the band contract misses 5–10% of $D_0$. The leading form of (7.12) is reproduced to 4–12% in S1 inside the S6 hierarchy, and to 28–44% in S2. At $u=\frac1{16}$ the deficit $1-\kappa_s$ is not resolved.
- The kinetic weight $-K_0\phi'$ accounts for the difference between the exact selected level and the held-field average to 4–14% (S2), 9–30% (S1) and 16–45% (R); the excess grows with $u$ and $\eta$.
- The fall itself (time evolution) is not simulated; $\Gamma$ is taken from (5.14), (6.13) applied to the centre contract (7.14). The exact eigenvalues with the fixed source ($L=12,16$) confirm (7.14) but do not resolve $W_0$: at $P=2\pi/L$ the internal state already differs from $u_0$. $W_0$ is confirmed through the $P=0$ symbol instead. The $b_0/m$ form of (6.13) differs from $W_0$ at relative $O(1-\beta_0)$ in $\langle V\rangle_0$.
- The remainders of (7.15) are order estimates. Near the pair they are larger: at $s=4\approx2w$ ($L=64$, case A) $W_{\rm ex}'$ differs from $W_0'$ by 5.9% of $W_b'$, against 0.8–1.3% at $s\ge8$.
- The bound $O(L^2(w^3/4)^{L/2-2})$ on $\rho_L$ for $w\lesssim1$ is an order estimate without constants. Control F lies below it ($\rho_L=0.125$ at $L=6$; $7.6\cdot10^{-3}$ at $L=8$).
- Descriptions that violate (N×) ($g_1+g_2$ near 0, e.g. $g_1=-g_2$) would need a band contract on $\mathcal G$ together with the jointly flipped sector. They are outside the class.
- $\kappa_s$ is a property of the ring (zero mode). Other geometries are out of scope (tori).
- $c_i^2=\tau_i^2[1+O(N_f\eta^2)]$ is an estimate of the overlap corrections of (5.3), not derived in detail.

## Code

- `code/v3/check_pair_full_grip.py` (role: check; $N_f=1$, $t_f=1$). This is the v2 script, with three changes: the midpoint probe is labelled auxiliary, $\rho_L$ and $\varepsilon_L$ are reported, and the controls F are added.
  - **Band contract and exact level.** It builds the band contract (2.10) bond by bond from the exact held blocks (second-order Schrieffer–Wolff over all field states of the flipped sectors). It also builds the exact sector contract of two full-grip bodies and one field object, continued in $P$ and restricted to the parity class of the selected level. It finds the selected level by its overlap with the band contract and takes $E_b''(0)$ by Richardson differences.
  - **Items 1–3.** From the exact held spectra of the class it computes the distances of $\mathcal G$ from the field excitations and single flips, and from the joint flip (S2). It compares $D_0$ with (7.5), $m_i(\rho)$ with (7.3), and $\Delta E_0$ with (7.6), (7.8). It reports $\rho_L=E''_{\rm band}/E''_{\rm line}-1$ and $\varepsilon_L$.
  - **Item 4.** It takes $W_0'$ from $\rho_0$ (7.15), and $W_i'$ from the body densities, at distance $s\pm1$ in the exact $P=0$ eigenvectors. In cases A ($L=64$, $s=8$) and S1 ($s=32$) it cross-checks $W_0'$ with the auxiliary midpoint probe: sector eigenvalues with $\pm g_s$, $g_s=10^{-5}$, which is not a held source. It forms $\Gamma_i=W_i'/\mu_i$, $\Gamma_{\rm pair}=E_b''(0)W_0'$ and $\Gamma_b=W_b'/D_0$ with the exact $D_0$, and compares them with (7.10)–(7.12). For comparison it gives the held-field averages of v1 ($f_0^2$-average of exact held weights, no kinetic part).
  - **Cases.** A ($\eta=0.15$, $L=32,64$), A′ ($\eta=0.05$), A″ ($\eta=0.05$, smaller $\tau/d$), B (unequal $g_i$ and $\tau_i$), R ($L=128$, $\eta=0.15$), S1 ($L=256$, $\eta=0.1$, $\delta_m=0.09$), S2 ($L=256$, $\eta=0.05$, $\delta_m=0.13$), C (opposite signs), D (both negative), F ($L=6,8$, $\tau/d=0.1$, $w\approx0.5$).
- `code/v3/check_fixed_source.py` (role: check; $N_f=1$, $t_f=1$; it uses the contract builders of the first script). The source is the fixed term $g\Phi_0$ on the field, in laboratory coordinates. Parts:
  - **exact.** Full laboratory space $(s_1,s_2,j,l,y)$ in the parity class (dimension $2L^3$), $L=12,16$, cases EA (equal bodies) and EB (unequal), with the parameters of A, B. The Bloch vectors built from the exact sector vectors are eigenvectors of the laboratory contract (residual $\le6\cdot10^{-15}$). The script computes the eigenvalues of $C+g\Phi_0$, $g=10^{-7}$, in the selected band. It compares them with $h_c$ (7.14) for four choices of $V$: from exact matrix elements (ME), from $W_0$ (mid), from the held-field average (held) and without bond (nobond). It also compares the row sums $W_{\rm ex}$ with $W_0$.
  - **me64, me128.** Exact first order from the exact Bloch vectors of all $P$ ($L=64$: A, B; $L=128$: R). Alongside, the band contract (2.10) in laboratory coordinates with the fixed source: exact held blocks with the source, and their first derivative in $g$ in resolvent form. This derivative agrees with finite differences $\pm10^{-5}$ of the band contract with the source to $6\cdot10^{-9}$ (relative).
  - **bandS1, bandS2.** The same band contract with the fixed source at $L=256$. It gives $W_{\rm ex}'$ of the pair and of each body, and the lowest 21 band eigenvalues ($g=10^{-6}$), compared with $h_c$.
- Outputs: `code/v3/output/check_log.txt` and `code/v3/output/check_results.json` (first script), and `code/v3/output/fixed_source_<part>.txt` and `.json` for the parts exact, me64, me128, bandS1 and bandS2.
- Findings (from the logs). The numbers of the first script for cases A–D are those of v2.
  - **Item 1, distances.** The distance to the joint flip in the exact held spectra agrees with $2n\lvert g_1+g_2\rvert-\frac32\lvert B(0)\rvert$ to 0.4–3.7%. In case C (opposite signs) the joint flip is the nearest sector ($6.22\cdot10^{-4}$, against $6.58\cdot10^{-4}$ for the single flips and excitations). $\max_i\tau_i/d=0.33$–$0.48$ (0.16 in A″).
  - **Item 1, levels.** The selected levels have overlap $\ge0.983$ with the band contract at every $P$ used. $E_b''(0)$ of (2.10) agrees with the exact value to 0.7–4.6%. Case C is bound at the top ($\langle\lvert r\rvert\rangle=0.14$). In case D the extreme level sits at the antipode ($\langle\lvert r\rvert\rangle=21.4$, $L=48$), as (7.4) states.
  - **Flux remainder (I2).** In cases A–D, $\lvert\rho_L\rvert\le1.6\cdot10^{-6}$, which is the resolution of the Richardson curvature, and $\lvert\varepsilon_L\rvert\le1.7\cdot10^{-5}$. In control F, $\rho_L=0.125$ and $\varepsilon_L=1.9$ at $L=6$, $w=0.50$; at $L=8$, $w=0.55$, $\rho_L=7.6\cdot10^{-3}$ and $\varepsilon_L=0.063$.
  - **Item 2.** (7.3) holds near $r=0$ to 0.27–2.5% in the odd-part form, and to 2–25% in the form with $\bar B$. (7.5) reproduces $D_0$ of the band contract to 0.1–2% of $\lvert D^{\rm lat}\rvert+\lvert D^{\rm mark}\rvert$, and the exact $D_0$ to 3–11% of it (the band-versus-exact part).
  - **Item 3.** $\Delta E_0$ agrees with $\langle M\rangle_0+K_0$ to 0.5–3.2%. $D_0/[(\tau_1^{-2}+\tau_2^{-2})\Delta E_0]$ is 1.016 in R ($\delta_m=0.16$) and 1.022 in S1 ($\delta_m=0.09$). In S2 ($\delta_m=0.13$) it is 0.876; there $D^{\rm lat}/D^{\rm mark}=-0.16$, and the band contract misses 9.8% of $D_0$.
  - **Fixed source, exact eigenvalues (I1).** The eigenvalues of the full laboratory contract with the fixed source agree with $h_c$[ME] to $\le3\cdot10^{-8}$ in units of the first-order shifts, i.e. $10^{-4}$ of the bond effect. This confirms (7.14) exactly at first order. $h_c$[mid] misses them by 0.29–1.08 of the bond effect, held by 0.26–1.01 and nobond by 0.76–1.38. At $L\le16$, $P=2\pi/L$ is not small, so these eigenvalues do not resolve $W_0$.
  - **Fixed source, exact first order (I1).** $W_{\rm ex}'$ of the bond agrees with $W_b'$ of $W_0$ to 0.05–7% for $3\le s\le6$ ($L=12,16$; 11–19% at $s=2\approx w$). At $L=64$ the agreement is 0.2–1.3% for $s\ge8$, and 1.2% (B) and 5.5% (A) at $s=4$. At $L=128$ (R) it is 1.1%, 0.5% and 0.2% at $s=8,16,32$. Without the kinetic part (held-field average), the bond weight differs from $W_b'$ by 4–14% at $L=64$ and 3.2–8.3% at $L=128$. For single bodies, $\lvert W_{\rm ex,i}'/W_i'-1\rvert\le1.3\%$ at $L\le16$ and $\le6.4\cdot10^{-4}$ at $L\ge64$.
  - **Fixed source, band contract in laboratory coordinates (I1).** At $L=64,128$ it agrees with the exact first order to $\le0.3\%$ of $W_b'$ for $s\ge8$ (0.8% at $s=4$). At $L=256$ its bond weight equals $W_b'$ of the exact $\rho_0$ to 0.15–0.19% (S1) and 0.20–0.27% (S2), against a kinetic part of 2.7–7.4% (S1) and 3.4–10% (S2). This is $\Gamma_b$ for the fixed source within 0.3% of the values below. Its lowest 21 eigenvalues are reproduced by $h_c$[mid] to 0.4% (S1) and 0.8% (S2) of the bond effect, by $h_c$[held] to 5.2% and 7.0%, and by $h_c$[ME] to $2\cdot10^{-9}$.
  - **Item 4, weight of the selected level.** The auxiliary probe agrees with the density form of $W_0'$ to $6\cdot10^{-9}$ (A) and $5\cdot10^{-8}$ (S1). The difference between the exact selected level and the held-field average, over $\phi'$, is 1.04–1.14 (S2), 1.09–1.30 (S1) and 1.16–1.45 (R) times the kinetic weight $-K_0$ of (7.10). $W_b'-(7.10)$ is $c\,\eta\lvert B(0)\phi'\rvert$, with $c\approx3.4$, $1.6$–$1.7$ and $-(0.5$–$0.8)$ at $u=\frac1{16},\frac18,\frac14$, the same in S1 and S2. It is linear in $\eta$, as the first remainder of (7.10) states.
  - **Item 4, single bodies.** $\Gamma_i/(\tau_i^2\phi')$ is $0.846,0.880,0.933$ (S1) and $0.923,0.942,0.971$ (S2), against $1-2\kappa_s\eta=0.847,0.888,0.950$ and $0.923,0.944,0.975$ from (7.11).
  - **Item 4, bond.** In S1, $\Gamma_b=W_b'/D_0$ over the leading form of (7.12) is 1.044 and 1.119 at $u=\frac1{16},\frac18$ ($\eta/\kappa_s=0.13,0.18$). At $u=\frac14$, where $\eta/\kappa_s=0.40$ is outside the S6 hierarchy, it is 1.376. In S2 it is 1.28–1.44, dominated by $\delta_m$ and by the band-versus-exact part of $D_0$. The same structure with the exact held-field weights, $-K_0\phi'$ and $D^{\rm lat}+D^{\rm mark}$ of (7.5) reproduces $\Gamma_b$ to 5.5–7.3% (S1), 10–11% (S2) and 5.5–7.8% (R). The residual is the band-versus-exact part of $D_0$ (5.5%, 9.8%, 5.1%).
  - **Item 4, bond against bodies.** $\Gamma_b/\Gamma_1$ is $0.944,0.715,0.369$ (S1), $1.065,0.790,0.370$ (S2) and $1.039,0.798,0.437$ (R), at $\kappa_s=0.766,0.563,0.250$. At $u\ge\frac18$ the bond is weighed clearly below the bodies' proportion. At $u=\frac1{16}$ the deficit $1-\kappa_s=0.23$ lies within the remainders.

## Methods used

- Rayleigh–Schrödinger and Schrieffer–Wolff perturbation theory
- Bloch sectors of the joint translation, band contract
- Bloch decomposition with a fixed source, centre (Wannier) representation, symbol at $P=0$
- Hellmann–Feynman theorem, Perron–Frobenius
- Lattice Green function on the ring (pseudo-inverse of the Laplacian)
- Continuum limit of the relative motion (linear potential)
- Exact diagonalization, shift-invert Lanczos, Richardson extrapolation (code)

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