Version 4 · current

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

- **Subproject:** 07-ilang-weight
- **Package:** 07-inertia-of-pair-full-grip
- **Version:** v4
- **Mode:** external regeneration
- **Date:** 2026-10-22

## Changes from previous version

- Step 2, S4: the narrow-pair asymptotics are stated per parity class (even: $\beta_0\simeq\frac{w^3}4\frac L{L-2}$, $\langle V\rangle_0=O(w^6\lvert F\rvert)$; odd: $\beta_0\to\frac12$, $\langle V\rangle_0\to F(1-\frac1L)$), with the bond-weighted $\bar B_0$ and its distance from $\langle M\rangle_0$ (I1, I2).
- Step 3, (7.5): the estimate of $\rho_L$ for $w\lesssim1$ is restricted to the even class (I2).
- Step 4 (i), (7.7): $\bar B_0$ is kept, and the remainder no longer contains $w/L$; Step 4 (ii) replaces $\bar B_0$ by $\langle M\rangle_0$ only for $w\gg1$ (I1).
- Step 5, (7.14)–(7.17), S6, (7.12): the fixed-source reduction is derived in the band contract with the source. New (7.16) gives the internal state, new (7.17) the symbol and the kernel moments, and (7.15) now gives the acceleration of the projected midpoint with the start-width hierarchy. The v3 remainder $\tau^2/(d^2\eta_{\min}L\kappa_s^{3/2})$ is replaced by the band-contract error $\tau^4/(d^4\eta_{\min}\kappa_s)$ (I3).
- Code: new `check_centre_fall.py` (parity classes; internal state and the fall in the band contract with the source). The checks are labelled EXACT and BAND. The two v3 scripts are unchanged, and their outputs are copied (I1–I4).
- Result, Open issues and Code updated accordingly.

## Response to verification

- I1: Accepted and fixed in Step 2 (*Broad pairs*, *Narrow pairs*), Step 4 (i), (7.7), and Step 4 (ii). (7.7) keeps $\bar B_0=\sum_\rho b_0\bar B/\beta_0$ in its mark terms, for every $w$ and both classes. $\bar B_0-\langle M\rangle_0$ tends to $F(1-\frac3{2L})$ in the even class for $w\to0$, as the verifier states, and is $O(\eta\lvert B(0)\rvert)+O(\lvert F\rvert/w)$ for $w\gg1$. Only there, in (7.8), is $\bar B_0$ replaced by $\langle M\rangle_0$. The sign argument of (i) is redone with $\bar B_0$. The model check (`centre_fall_classes`) gives $0.750F$, $0.977F$ and $0.994F$ at $L=6,64,256$, $w=0.2$, and $w(\bar B_0-\langle M\rangle_0)/F=0.50$–$0.63$ for $4\le w\le16$.
- I2: Accepted and fixed in Step 2, S4, Step 3 (*Flux remainder*) and (7.5). The narrow-pair asymptotics are stated separately: even class $\beta_0\simeq\frac{w^3}4\frac L{L-2}$; odd class $\beta_0\to\frac12$, $\langle V\rangle_0\to F(1-\frac1L)=O(F)$. The estimate of $\rho_L$ for $w\lesssim1$ is restricted to the even class; for the odd class $\rho_L$ is kept explicit and listed as open. Model check: odd class $\beta_0=0.501$–$0.506$ at $w=0.2$.
- I3: Accepted and fixed in Step 5 (*Fixed source and band contract*, *Internal state* (7.16), *Centre contract* (7.14), (7.17), *Acceleration from rest*, *Fall of the pair* (7.15)), S6 and (7.12). In the band contract with the source:
  - **Internal state.** A gauge transformation of the relative chain gives $u_P=e^{i\Theta_P}v_P$, with phase slope $\arctan(\nu\tan P)$, $\nu$ from the local inertias (7.3), and width change $-(1-\bar\nu^2)P^2/6$ from the Airy scaling.
  - **Kernel.** The source is sampled at the midpoint displaced by $\frac12\bar\nu h$: first moment $O(\lvert\bar\nu\rvert w)$ per relative place, zero in total; second moment $O(\bar\nu^2w^2)$. Unequal parameters enter through $\bar\nu$.
  - **Observable.** The projected midpoint is the centre (zero Berry connection). The relative gap $\Delta_{\rm rel}\simeq1.32\lvert F\rvert w$ bounds the interband terms.
  - **Acceleration.** It follows from Ehrenfest with the Weyl symbol, with $\epsilon$ from the momentum spread, the gradient variation over the start and the displacement. The hierarchy $(\eta_{\min}\kappa_s)^{-1/2}\ll w_X\ll\min(s,L\kappa_s^{1/2}(\eta_{\min}/\eta)^{1/2})$ puts the errors below the bond's part.
  - **Numbers.** In the band contract at $L=256$ the fall of the midpoint (S1) and of the centre (S1u) equals $-\langle E_b''\rangle\langle W_{\rm ex}'\rangle$ to 0.3–1.2% of the bond part. The leading form misses by 4–24% of it, as the start entries of $\epsilon$ predict. What is not derived is listed in Open issues: the relation of the band-contract $W_0$ to the exact $\rho_0$ beyond its order, and the ring closure for $w\lesssim1$.
- I4: Accepted in the form fixed by the question's note of 2026-10-22 (Code, *Labels*). Item 4 is checked against EXACT eigenvalues with the held source at $L=12,16$, where the full laboratory space ($2L^3$) is feasible. In the regime of item 3 (ii) (S1, S2 at $L=256$) it is checked against BAND, the band contract in laboratory coordinates with the held source, now including the fall itself. Each check is labelled; no exact check in that regime is claimed.

## 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 $w\lesssim1$ in the even class. $\varepsilon_L$ is its size against the mutual part of the inertia. $\langle V\rangle_0/\lvert B(0)\rvert=O(\max(w,1)/L)$ (Step 2).
- **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 $\tau^4/d^4\ll\eta_{\min}\kappa_s$, 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 $\phi'\neq0$. For the fall of a start, (7.15) adds $w\gg1$ and the start-width hierarchy stated there.
- 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. Everything below holds for either class, except the narrow-pair asymptotics ($w\lesssim1$), which differ (end of this step). 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; $f_0>0$ (Perron–Frobenius), so $b_0>0$ and $0<\beta_0\le1$. Write $\bar B_0:=\sum_\rho b_0\bar B/\beta_0$ for the bond-weighted mutual term.
- *Broad pairs* ($w\gg1$, either class): $1-\beta_0=O(w^{-2})$. The bond weights $b_0/\beta_0$ differ from $f_0^2$ by relative $O(w^{-2})$ in shape, and $V$ has slope $\le\lvert F\rvert$ with a kink at $r=0$, where $f_0^2=O(1/w)$. Hence $\bar B_0-\langle M\rangle_0=O(\eta\lvert B(0)\rvert)+O(\lvert F\rvert/w)$, and $\langle V\rangle_0=O(\lvert F\rvert w)$.
- *Narrow pairs, even class* ($w\ll1$): $f_0$ concentrates at $r=0$, $f_0(\pm2)\simeq\lvert\bar t_0/V(2)\rvert$, so $\beta_0\simeq\frac{w^3}4\frac L{L-2}$, $\langle V\rangle_0=O(w^6\lvert F\rvert)$, and $\bar B_0\to\bar B(\pm1)=B(0)+F(1-\frac3{2L})$.
- *Narrow pairs, odd class* ($w\ll1$): the sites $r=\pm1$ are degenerate and joined by the hop across $\rho=0$, so $f_0(\pm1)\to2^{-1/2}$, $\beta_0\to\frac12$, $\langle V\rangle_0\to V(1)=F(1-\frac1L)$, and $\bar B_0\to\bar B(0)=B(0)+F\frac{L-1}{2L}$.

So $\langle V\rangle_0/\lvert B(0)\rvert=O(\max(w,1)/L)$ in both classes, and $\bar B_0-\langle M\rangle_0$ is $O(\lvert F\rvert)$, not small against $\lvert F\rvert$, for narrow pairs.

### 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 $w\lesssim1$ in the even class 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). In the odd class at $w\lesssim1$ the state sits on $r=\pm1$ with $\beta_0\to\frac12$; $\rho_L$ is not estimated there and is kept explicit.

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, and by (7.5) $D^{\rm mark}=(\tau_i^{-2}+\tau_j^{-2})\bar B_0/\beta_0\,[1+O(\eta)]$ with $\bar B_0$ of Step 2. With $v=\tau_i$, the speed of (5.7) of either body, and $j\neq i$:
$$D_0-\frac{\Delta E_0}{\tau_i^2}=\frac{4\mu_1\mu_2}{\beta_0}K_0\Big[1-\frac{\beta_0}{4\mu_1\mu_2\tau_i^2}\Big]+\frac{\bar B_0}{\beta_0\tau_j^2}+\frac1{\tau_i^2}\Big[\frac{\bar B_0}{\beta_0}-\langle M\rangle_0\Big]-\frac{\rho_L}{E''_{\rm line}}+O\Big(\Big(\eta+\frac{\tau^2}{d^2}\Big)\lvert D^{\rm mark}\rvert+\frac{\tau^2}{d^2}\lvert D^{\rm lat}\rvert+\frac{\rho_L^2}{E''_{\rm line}}\Big),\qquad\lvert4\mu_1\mu_2\tau_i^2\rvert=\frac{(2n\lvert g_1\rvert)(2n\lvert g_2\rvert)}{\tau_j^2}\ge\frac{d^2}{\tau_j^2}\gg1.\tag{7.7}$$
This holds for every $w$ of S4 and both classes; $\bar B_0$ is not replaced by $\langle M\rangle_0$, since they differ by $O(\lvert F\rvert)$ for narrow pairs (Step 2). The lattice part exceeds $K_0/v^2$ by a factor $\ge d^2/\tau_j^2$. The mark part exceeds $\langle M\rangle_0/v^2$ by the two middle terms; the first of them is $\frac{\tau_i^2}{\tau_i^2+\tau_j^2}D^{\rm mark}$. For $g_1g_2<0$ the lattice term and $\bar B_0/(\beta_0\tau_j^2)$ are positive ($\mu_1\mu_2K_0>0$; $\bar B_0>0$, as $B(0)\simeq\lvert F\rvert L/6$ and $\lvert\bar B_0-B(0)\rvert=O(\lvert F\rvert\max(w,1))$ by S4). The last term is $[\langle M\rangle_0(1-\beta_0)+\bar B_0-\langle M\rangle_0]/\beta_0$; by Step 2 it is positive for $w\ll1$ (even class: $\beta_0\to0$; odd class: $\bar B_0-\langle M\rangle_0\to\lvert F\rvert\frac{L-1}{2L}$) and for $1\ll w\ll L$ ($\langle M\rangle_0(1-\beta_0)\sim\lvert F\rvert L/w^2$ against $\lvert F\rvert/w$). There (i) fails unless the remainder and the flux term cancel the displayed terms, which needs $\tau_i^2/\tau_j^2=O(\eta+\tau^2/d^2)$ and the other two terms small against $\lvert D^{\rm mark}\rvert$: the limit of (7.8). For $g_1,g_2>0$ the lattice and mark 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 $w\gg1$, Step 2 gives $\bar B_0=\langle M\rangle_0+O(\eta\lvert B(0)\rvert)+O(\lvert F\rvert/w)$. As $\lvert F\rvert/\lvert B(0)\rvert=O(1/L)$ and $w^{-2}=O(1-\beta_0)=O(\eta\delta_m)$, this is $\bar B_0=\langle M\rangle_0[1+O(\eta,\delta_m)]$; only here is $\bar B_0$ replaced by $\langle M\rangle_0$. 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 band contract.* The source $g_s\Phi_{x_s}$ of S6 breaks (2.1), so it has no $P$ sectors. It is a held object, so S2 and the derivation of (2.10) apply with the source among the held objects: the band contract in laboratory coordinates is $H_0+g_sH_1$, with $H_0$ of (2.10) and $H_1$ its first derivative in $g_s$. $H_1$ is the held weight $\partial_{g_s}E(j,l)$ (diagonal) plus the change of $K_1,K_2$, the kinetic part. The source changes each gap by the relative amount $O(\phi)$, as the partner does in (7.3). So the $O(\tau^4/d^3)$ error of (2.10) has a gradient $O(\tau^4\lvert\phi'\rvert/d^3)$, relative $O(\tau^4/(d^4\eta_{\min}\kappa_s))$ to $W_b'$ of (7.10) below ($\lvert W_b'\rvert\simeq2\kappa_s\lvert B(0)\phi'\rvert$, $\lvert B(0)\rvert\ge d\,\eta_{\min}$). The fall in the fixed source is derived in this band contract.

*Internal state.* Use the midpoint $X=j+h/2$ and the representative $h\in(-\frac L2,\frac L2]$ of $r$; the formulas are written for the even class (in the odd class $\Theta_P$ starts at the bond $\rho=0$). The eigenvectors of the selected level of $H_0$ are $\Psi_P=L^{-1/2}e^{iPX}u_P(h)$, $P\in\frac{2\pi}L\mathbb Z_L$, where $u_P$ is the selected eigenvector of (6.1) in this gauge. Its hop from $h+2$ to $h$ is $e^{iP}t(\rho;P)=\kappa_1(\rho)e^{-iP}+\kappa_2(\rho)e^{iP}=t(\rho;0)[\cos P-i\nu(\rho)\sin P]$, $\rho=h+1$. On the line (the chart of $h$ cut at $h\approx\frac L2$) a gauge transformation makes the hops real:
$$u_P(h)=e^{i\Theta_P(h)}v_P(h),\qquad\Theta_P(h)=\sum_{\rho\in(0,h)}\theta(\rho;P),\qquad\theta=\arctan[\nu(\rho)\tan P],\qquad\nu(\rho):=\frac{m_2(\rho)-m_1(\rho)}{m_1(\rho)+m_2(\rho)},\qquad w_P=w\Big[1-\frac{1-\bar\nu^2}6P^2\Big(1+O\Big(\frac{\tau^2}{d^2}\Big)\Big)+O(P^4)\Big].\tag{7.16}$$
The sum runs over the bonds between $0$ and $h$, with the sign of $h$, so $\Theta_P$ is odd in $h$. $v_P$ is the real positive selected state with hops $t(\rho;0)[1-(1-\nu(\rho)^2)\sin^2P]^{1/2}$; it is even in $h$, as $U$ and the $\kappa_i$ are. On the ring the phase must close at $h\approx\frac L2$ (its total is the flux $\tilde\Phi$ of Step 3), where $v_P=O(e^{-c(L/w)^{3/2}})$ for $w\gg1$; this is the error of the line. By (7.3), $\nu(\rho)=\bar\nu+O(\eta\lvert\rho\rvert/L)$ with $\bar\nu:=\nu(\pm1)=\frac{\mu_2-\mu_1}{\mu_1+\mu_2}+O(\eta)$, and $\lvert\bar\nu\rvert\le2/c+O(\eta)$ outside the tuned case. For $w\gg1$ the continuum limit (Step 2) has a width $\propto\lvert\text{hop}\rvert^{1/3}$; this gives the width $w_P$ of $v_P$. The $O(\tau^2/d^2)$ comes from the $r$-dependence of the hops, which adds to $V$ a part of relative size $O(\tau^2/d^2)$ that scales with the hop.

*Centre contract.* By (7.16), $\Psi_P=L^{-1/2}e^{iPY}v_P(h)$ up to phases $O(P^3h,\eta Ph^2/L)$, with the centre $Y:=X+\frac12\bar\nu h=\frac{m_1x_1+m_2x_2}{m_1+m_2}$ ($m_i:=m_i(\pm1)$). The selected band is separated from the other levels of the relative motion by $\Delta_{\rm rel}$ ($\simeq1.32\lvert F\rvert w$ for $w\gg1$: $\lvert a_1\rvert-\lvert a_1'\rvert$ of the Airy function, the lowest odd and even levels of $-\partial_\xi^2+\lvert\xi\rvert$). At first order in $g_s$, projection on the band gives the centre contract
$$h_c=E_b(\hat P)+V,\qquad V_{PP'}=g_s\langle\Psi_P|H_1|\Psi_{P'}\rangle.\tag{7.14}$$
The other levels enter at $O(g_s^2/\Delta_{\rm rel})$. The interband parts of $H_1$ and of $\hat X$ give bounded oscillations of frequency $\ge\Delta_{\rm rel}$ and amplitude $O(g_s)$; they do not accumulate. The projected midpoint is $\Pi\hat X\Pi=i\partial_P+A$ with $A=i\langle u_P|\partial_Pu_P\rangle=-\sum_hv_P^2\,\partial_P\Theta_P=0$, as $v_P^2$ is even and $\Theta_P$ odd. So it is the centre $\hat Y$ conjugate to $P$, and $\Pi\hat h\Pi=0$. The band transformation adds to $\hat X$ an internal operator of size $O(\tau^2/d^2)$; it commutes with $E_b(\hat P)$ and changes the acceleration below by a relative $O(\tau^2/(d^2w_X^2))$. Let $K:=\frac{P+P'}2$, $q:=P-P'$; then $\Theta_P-\Theta_{P'}=\frac q2\nu_Kh+O(q^3h)$ with $\nu_K:=\partial_K\theta=\bar\nu[1+(1-\bar\nu^2)K^2+O(K^4)]$. Hence $V$ is the Weyl operator in $(\hat Y,\hat P)$ of the symbol
$$\mathcal W(Y,K)=\sum_hv_K(h)^2\,H_1\big(Y-\tfrac12\nu_Kh,\;h\big),\tag{7.17}$$
up to terms with $\partial_Y^2H_1$ times $\partial_q^2(v_Pv_{P'})$, which are constant in $Y$ at first order in the couplings. The kinetic part of $H_1$, a hop, enters at the centre of the hop. (7.17) makes the kernel of $V$ explicit: for each relative place $h$ the source is sampled at the midpoint displaced by $\frac12\nu_Kh$. Its first moment is $-\frac12\bar\nu h$ per relative place, $O(\lvert\bar\nu\rvert w)$, and zero in total ($v_K^2$ is even). Its second moment is $\frac14\bar\nu^2\langle h^2\rangle_0=O(\bar\nu^2w^2)$, so its range is $O(\lvert\bar\nu\rvert w)$. $\mathcal W$ is even in $K$: $\mathcal W=\mathcal W_0(Y)+K^2\mathcal W_2(Y)+O(K^4)$. The row sum $W_{\rm ex}$ of $V$ (its $P=0$ symbol in the gauge $\langle u_P|u_0\rangle>0$, which is that of (7.16)) has $W_{\rm ex}'=g_s\mathcal W_0'$ up to $\partial_Y^3\mathcal W_2$, which vanishes at first order in the couplings.

*Acceleration from rest.* For the symbol $E_b(K)+g_s\mathcal W$ of $h_c$, Ehrenfest's theorem with the Moyal bracket gives, at first order in $g_s$, $\frac{d^2}{d\lambda^2}\langle\hat Y\rangle=g_s\langle-E_b''(K)\,\partial_Y\mathcal W+E_b'(K)\,\partial_Y\partial_K\mathcal W\rangle$, up to $\partial_Y^3\mathcal W$ times odd derivatives of $E_b$, whose average vanishes for the start below. Take a start at rest in the selected band, with real envelope of width $w_X$ in $Y$, centred at $\bar X$ (the mean projected midpoint). Its Wigner function is even in $K$, with $\langle K^2\rangle=1/(4w_X^2)$. Then
$\ddot{\bar X}=-g_sE_b''(0)\langle\mathcal W_0'\rangle-g_s\langle K^2\rangle\big[\tfrac12E_b''''(0)\langle\mathcal W_0'\rangle-E_b''(0)\langle\mathcal W_2'\rangle\big]+O(w_X^{-4})$.
- The bracket is relative $O(w_X^{-2})$. $E_b''''/E_b''=O(1)$ in the class, since the band depends on $P$ through the hop magnitude of (7.16). The kinetic part of $\mathcal W_2$ is the source's change of the inertias by the relative amount $-\phi$ ((7.3) with the source in place of the partner), $g_s\mathcal W_2\simeq-\frac12E_b''\phi$; it is relative $O(E_b''/(n\lvert g_1+g_2\rvert))=O(\tau^2/d^2)$. The $v_K$ part changes $\langle V\rangle_0$ by $-\frac16(1-\bar\nu^2)K^2\langle V\rangle_0$.
- Over the start, $\langle\mathcal W_0'\rangle=\mathcal W_0'(\bar X)+\frac{w_X^2}2\mathcal W_0'''(\bar X)+\dots$. At first order in the couplings $\mathcal W_0$ is quadratic away from $x_s$ ($\ell''=\frac1L$). So $\mathcal W_0'''$ comes from the second-order terms of (7.9), (7.10), through $\kappa_s\simeq(1-2u)^2$ and $\phi'\propto1-2u$: $W_i'''/W_i'\simeq-48g_i\ell_0/(t_fL^2)$ and $W_b'''/W_b'\simeq24/(L^2\kappa_s)$. The tail of the start at the source gives $O(e^{-s^2/(2w_X^2)})$.

*Fall of the pair.* Let $W_0(Y):=g_s\sum_hv_0(h)^2H_1(Y,h)$ be the midpoint-conditioned weight of the $P=0$ state, kinetic part included. In the exact contract it is $g_s\rho_0(x_s-Y)$, with $\rho_0$ the field density at distance $x_s-Y$ from the midpoint in the $P=0$ eigenvector. By (7.17), $g_s\mathcal W_0-W_0=g_s\sum_hv_0^2[H_1(Y-\frac12\bar\nu h,h)-H_1(Y,h)]$; the variation of $\nu(\rho)$ adds a displacement odd in $h$, which drops out in the same way. Expand in $\bar\nu h$. The first-order terms in the couplings, $n[g_1\phi(x_1)+g_2\phi(x_2)]/g_s$ and the kinetic part, have $\partial_Y^2H_1$ even in $h$ and $\partial_Y^3H_1=0$, so they give nothing. In the second-order terms of (4.5), (4.7) use $Q:=2\ell_0\ell+\ell^2-\frac2L\ell^{(2)}$, $Q'''=\frac4L\ell'$, $(\phi(x_1)\phi(x_2))'''\simeq6\phi''\phi'$ and $(\ell^{(2)})'''=\ell'$. At first order in $\bar\nu$ the self terms give $-\bar\nu\langle h^2\rangle_0(g_2^2-g_1^2)\frac{ng_s\ell'}{t_f^2L}$; the bond terms start at $\bar\nu^2$. Compare with $\lvert W_b'\rvert\simeq\frac{L\kappa_s^{3/2}}3\frac{n\lvert g_sg_1g_2\rvert}{t_f^2}$, use $L\kappa_s^{1/2}>1$, and use $\lvert W_b'\rvert\ge2\eta_{\min}\kappa_s\lvert W_0'\rvert$ at leading order ((7.9), (7.10)). Then the acceleration of the projected midpoint from rest is
$$\ddot{\bar X}=-E_b''(0)\big[W_0'(\bar X)+\epsilon\,\lvert W_b'\rvert\big],\qquad\epsilon=O\Big(\frac1{w_X^2\eta_{\min}\kappa_s},\ \frac\eta{\eta_{\min}}\frac{w_X^2}{L^2\kappa_s},\ \lvert\bar\nu\rvert(1+\lvert\bar\nu\rvert)\frac{g_1^2+g_2^2}{\lvert g_1g_2\rvert}\frac{w^2}{L^2\kappa_s}\Big),\tag{7.15}$$
for $w\gg1$, $s\gg w$ and $w_X\ll s$, up to $O(e^{-c(L/w)^{3/2}},e^{-c(s/w)^{3/2}},e^{-s^2/(2w_X^2)})$ and the $O(\tau^4/(d^4\eta_{\min}\kappa_s))$ of the band contract. The first two entries of $\epsilon$ come from the start (momentum spread, and variation of the gradient over the start), the third from the displacement of the centre from the midpoint; for equal bodies $\bar\nu=0$. By (7.11) below, $E_b''(0)W_b'$ is the bond's part of the acceleration. So $\epsilon\ll1$ puts every error below the mutual-order effect. This is the start-width hierarchy $(\eta_{\min}\kappa_s)^{-1/2}\ll w_X\ll\min\big(s,\,L\kappa_s^{1/2}(\eta_{\min}/\eta)^{1/2}\big)$, together with $\lvert\bar\nu\rvert(1+\lvert\bar\nu\rvert)(g_1^2+g_2^2)w^2\ll\lvert g_1g_2\rvert L^2\kappa_s$. A body alone has no internal coordinate, and (5.14) applies directly.

*Content of $W_0$.* $W_0$ contains the held densities and, through the kinetic part of $H_1$, the change of the hops: $W_0'=\sum_hf_0^2(\partial_1+\partial_2)W_{\rm pair}$ plus the kinetic part ($v_0=f_0$). 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 $ng\phi'$. The code computes $W_0$ from the exact density $\rho_0$ (for odd $h$, the two places at distance $k\mp\frac12$ averaged), 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, not a contract of the description.

*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 $\phi'$. $\langle V\rangle_0=O(\lvert B(0)\rvert\max(w,1)/L)$ (Step 2) and $K_0$ are kept; (7.10) needs no relation between $\kappa_s$ and the small parameters.

*Fall factors.* $\Gamma_i:=E_i''(0)W_i'=W_i'/\mu_i$ and $\Gamma_{\rm pair}:=E_b''(0)W_0'$ are the fall factors: by (5.14) for each body alone and by (7.15) for the pair, the acceleration from rest in the fixed source is $-\Gamma$, for the pair up to $\epsilon\lvert E_b''W_b'\rvert$ of (7.15). 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 $\Delta E_0=B(0)[1+O(\eta,w/L)]$:
$$\Gamma_b=\frac{2\kappa_s\,\phi'}{\tau_1^{-2}+\tau_2^{-2}}\Big[1+O\Big(\frac\eta{\kappa_s},\frac w{L\kappa_s},\delta_m,\frac{\tau^2}{d^2},\frac{\delta_m\tau^2}{\kappa_sd^2},\frac{\tau^4}{d^4\eta_{\min}\kappa_s},\varepsilon_L\Big)\Big],\qquad \Gamma_{\rm pair}-\bar\Gamma\overset{\tau_1=\tau_2=\tau}{=}4(1-\kappa_s)\frac{g_1g_2\ell_0}{t_f(g_1+g_2)}\tau^2\phi'.\tag{7.12}$$
The sixth remainder is the error of the band contract with the source (Step 5). The fall of a start follows $\Gamma_b$ up to the further relative $\epsilon$ of (7.15). 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. 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 $-\rho_L/E''_{\rm line}+O(\rho_L^2/E''_{\rm line})$, in both parity classes. Here $\rho_L=O(e^{-c(L/w)^{3/2}})$ for $w\gg1$, and $O(L^2(w^3/4)^{L/2-2})$ for $w\lesssim1$ in the even class (control F: $\rho_L=0.125$ at $L=6$, $w=0.50$); in the odd class at $w\lesssim1$ it is not estimated. $D^{\rm lat}=(\mu_1+\mu_2)(1-\beta_0)/\beta_0$, 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 $s_c=+1$, $c_i=\tau_i$. Narrow pairs ($w\ll1$): $\beta_0\simeq\frac{w^3}4\frac L{L-2}$ in the even class, $\beta_0\to\frac12$ in the odd class.
- **(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, for every $w$ of S4 and both classes: $D_0-\Delta E_0/\tau_i^2=(4\mu_1\mu_2K_0/\beta_0)[1-\beta_0/(4\mu_1\mu_2\tau_i^2)]+\bar B_0/(\beta_0\tau_j^2)+(\bar B_0/\beta_0-\langle M\rangle_0)/\tau_i^2-\rho_L/E''_{\rm line}$, with $\bar B_0=\sum_\rho b_0\bar B/\beta_0$, up to $O((\eta+\tau^2/d^2)\lvert D^{\rm mark}\rvert+(\tau^2/d^2)\lvert D^{\rm lat}\rvert+\rho_L^2/E''_{\rm line})$. $\bar B_0-\langle M\rangle_0$ is $O(\eta\lvert B(0)\rvert)+O(\lvert F\rvert/w)$ for $w\gg1$, but tends to $F(1-\frac3{2L})$ (even class) and $-F\frac{L-1}{2L}$ (odd class) for $w\to0$. The lattice term exceeds $K_0/v^2$ by a factor $\ge d^2/\tau_j^2$. For $g_1g_2<0$ the displayed terms are positive for $w\ll1$ and for $1\ll w\ll L$, and there (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)** In the band contract (2.10) with the fixed held source $g_s\Phi_{x_s}$ in laboratory coordinates (error of its gradient relative $O(\tau^4/(d^4\eta_{\min}\kappa_s))$ to $W_b'$), at first order in $g_s$, the source acts on the isolated selected band as the centre contract $h_c=E_b(\hat P)+V$, $V_{PP'}=g_s\langle\Psi_P|H_1|\Psi_{P'}\rangle$, up to $O(g_s^2/\Delta_{\rm rel})$ and bounded interband oscillations; $\Delta_{\rm rel}\simeq1.32\lvert F\rvert w$ for $w\gg1$.
- **(7.16), (7.17)** For $w\gg1$, on the line chart of $h$ (closure error $O(e^{-c(L/w)^{3/2}})$): the internal state is $u_P=e^{i\Theta_P}v_P$ with $\Theta_P(h)=\sum_{\rho\in(0,h)}\arctan[\nu(\rho)\tan P]$, $\nu(\rho)=(m_2(\rho)-m_1(\rho))/(m_1(\rho)+m_2(\rho))=\bar\nu+O(\eta\lvert\rho\rvert/L)$, $v_P$ real and even with width $w_P=w[1-(1-\bar\nu^2)(1+O(\tau^2/d^2))P^2/6+O(P^4)]$. The projected midpoint is the centre $Y=X+\bar\nu h/2$, and $V$ has the Weyl symbol $\mathcal W(Y,K)=\sum_hv_K^2H_1(Y-\frac12\nu_Kh,h)$: the kernel's first moment is $-\frac12\bar\nu h$ per relative place ($O(\lvert\bar\nu\rvert w)$, zero in total) and its second moment $\frac14\bar\nu^2\langle h^2\rangle_0=O(\bar\nu^2w^2)$.
- **(7.15)** In the same band contract, for $w\gg1$, $s\gg w$ and a start at rest in the selected band with real envelope of width $w_X\ll s$: the acceleration of the projected midpoint is $-E_b''(0)[W_0'(\bar X)+\epsilon\lvert W_b'\rvert]$ with $\epsilon=O(1/(w_X^2\eta_{\min}\kappa_s),\ (\eta/\eta_{\min})w_X^2/(L^2\kappa_s),\ \lvert\bar\nu\rvert(1+\lvert\bar\nu\rvert)\frac{g_1^2+g_2^2}{\lvert g_1g_2\rvert}\frac{w^2}{L^2\kappa_s})$, up to exponentially small terms. $W_0$ is the midpoint-conditioned weight of the $P=0$ state, $g_s\rho_0(x_s-X)$ in the exact contract. The errors are below the bond's part $E_b''W_b'$ of the acceleration for $(\eta_{\min}\kappa_s)^{-1/2}\ll w_X\ll\min(s,L\kappa_s^{1/2}(\eta_{\min}/\eta)^{1/2})$ and a small third entry of $\epsilon$ (zero for equal bodies). 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 $W_0$ is the midpoint-conditioned weight 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 ($\eta\ll\kappa_s$, $w/L\ll\kappa_s$, $\delta_m\tau^2/d^2\ll\kappa_s$, $\tau^4/d^4\ll\eta_{\min}\kappa_s$), for the fixed source: $\Gamma_b=2\kappa_s\phi'/(\tau_1^{-2}+\tau_2^{-2})\,[1+O(\eta/\kappa_s,w/(L\kappa_s),\delta_m,\tau^2/d^2,\delta_m\tau^2/(\kappa_sd^2),\tau^4/(d^4\eta_{\min}\kappa_s),\varepsilon_L)]$; the fall of a start follows it up to the further relative $\epsilon$ of (7.15). 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 is simulated only in the band contract with the source (S1, S1u at $L=256$). The exact laboratory contract is feasible only at $L\le16$; there its eigenvalues with the source confirm (7.14), but do not resolve $W_0$, since at $P=2\pi/L$ the internal state already differs from $u_0$. That the exact density $\rho_0$ gives the band-contract $W_0$ is derived only to the order $O(\tau^4/(d^4\eta_{\min}\kappa_s))$ of $W_b'$. Numerically the band contract agrees with the exact first order to $\le0.3\%$ of $W_b'$ ($L=64,128$, $s\ge8$).
- (7.15) is derived for $w\gg1$, and $\epsilon$ is an order estimate without constants. The code finds the coefficients of its start entries large: at $s=64$, $w_X=12$ (S1) the variation of the gradient over the start is 16% of the bond part, at $w_X^2/(L^2\kappa_s)=8.8\cdot10^{-3}$. Near the pair ($s\approx2w$, outside $s\gg w$) the reduction is worse: at $s=4$ ($L=64$, case A) $W_{\rm ex}'$ differs from $W_0'$ by 5.9% of $W_b'$, against 0.8–1.3% at $s\ge8$. For $w\lesssim1$ the gauge (7.16) still holds on the line, but the closure on the ring is not estimated.
- The bound $O(L^2(w^3/4)^{L/2-2})$ on $\rho_L$ for $w\lesssim1$ (even class) is an order estimate without constants. Control F lies below it ($\rho_L=0.125$ at $L=6$; $7.6\cdot10^{-3}$ at $L=8$). In the odd class at $w\lesssim1$, $\rho_L$ is not estimated.
- 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

- Labels. **EXACT**: the full laboratory contract with the held source, or its exact first order from exact Bloch vectors. **BAND**: the band contract (2.10) in laboratory coordinates with the held source. As the question's note of 2026-10-22 fixes, item 4 is checked against EXACT eigenvalues with the held source at $L=12,16$ (the full space has dimension $2L^3$), and in the regime of item 3 (ii) (S1, S2, $L=256$) against BAND, including the fall itself. No EXACT check in that regime is claimed.
- `code/v4/check_pair_full_grip.py` (role: check; $N_f=1$, $t_f=1$; unchanged from v3, outputs copied). It labels the midpoint probe as auxiliary, reports $\rho_L$ and $\varepsilon_L$, and has the controls F (even class).
  - **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/v4/check_fixed_source.py` (role: check; $N_f=1$, $t_f=1$; it uses the contract builders of the first script; unchanged from v3, outputs copied). 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 $h_c$.
- `code/v4/check_centre_fall.py` (role: check; $N_f=1$, $t_f=1$; new in v4; it uses the builders of both scripts). Parts:
  - **classes** (MODEL). The relative contract (6.1) with constant hop $\bar t_0=-w^3F/4$ and potential $B$, on the even and on the odd class, $L=6,8,64,256$, $0.2\le w\le16$. It gives $\beta_0$, $\langle V\rangle_0$ and $\bar B_0-\langle M\rangle_0$ against the narrow limits of Step 2.
  - **fallS1, fallS1u** (BAND, $L=256$). S1 (equal bodies, $\bar\nu=0$) and S1u ($g_2=g_1/2$, $\tau_2=\tau_1/2$, $\eta=0.1$, $\bar\nu=0.285$). It checks (7.16) at $P=\frac{2\pi}L k$, $k=1,2,4$: the phase slope of $u_P$ against $\arctan(\nu(1)\tan P)$, and $\langle h^2\rangle_P$ against the continuum width. Then it evolves a start at rest in the selected band (real Gaussian envelope of width $w_X$ in the centre) with $H_0\pm gH_1$, $g=10^{-6}$, and takes the secular acceleration of the mean midpoint $j+h/2$ (odd part in $g$). The midpoint is fitted with an offset over $[\lambda_{\max}/4,\lambda_{\max}]$, the centre $Y$ through the origin. Both are compared with $-E_b''(0)W_{\rm ex}'(\bar X)$, the leading form of (7.15), and with $-\langle E_b''(P)\rangle\langle W_{\rm ex}'(Y)\rangle$, the same with the two start averages; $W_{\rm ex}$ is the BAND row sum.
- Outputs: `code/v4/output/check_log.txt` and `code/v4/output/check_results.json` (first script), `code/v4/output/fixed_source_<part>.txt` and `.json` for the parts exact, me64, me128, bandS1 and bandS2, and `code/v4/output/centre_fall_<part>.txt` and `.json` for the parts classes, fallS1 and fallS1u.
- 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.** 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.** 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.** $W_{\rm ex}'$ 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.** 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.
  - **Parity classes, I1, I2 (MODEL).** Even class at $w=0.2$: $\beta_0$ equals $\frac{w^3}4\frac L{L-2}$ to 0.2%, and $\bar B_0-\langle M\rangle_0=0.750F,0.977F,0.994F$ at $L=6,64,256$, against $F(1-\frac3{2L})$. Odd class at $w=0.2$: $\beta_0=0.501$–$0.506$, $\langle V\rangle_0/F=(L-1)/L$, and $\bar B_0-\langle M\rangle_0=-0.407F,-0.489F,-0.495F$, against $-F\frac{L-1}{2L}=-0.417F,-0.492F,-0.498F$. For $4\le w\le16$, $w(\bar B_0-\langle M\rangle_0)/F$ is $0.50$–$0.63$ (even) and $0.05$–$0.07$ (odd), i.e. $O(\lvert F\rvert/w)$. Over all $w$, $\lvert\bar B_0-\langle M\rangle_0\rvert\le0.99\lvert F\rvert$.
  - **Internal state, (7.16) (BAND).** S1: the phase slope vanishes ($<3\cdot10^{-11}$). S1u: it equals $\arctan(\nu(1)\tan P)$ to 0.45%, with the local $\nu(1)=0.285$ against $(\mu_2-\mu_1)/(\mu_1+\mu_2)=0.328$, as (7.16) states. The width change is 0.93 (S1u) and 0.49 (S1) of the continuum value. In S1 ($\tau/d=0.48$) the $r$-dependence of the hops, which scales with the hop, adds 47% to the slope of $V$; with uniform hops the continuum value is reproduced to 0.3%. This is the $O(\tau^2/d^2)$ of $w_P$.
  - **Fall, (7.15) (BAND).** S1, $(s,w_X)=(32,6),(32,8),(64,8),(64,12)$: the secular acceleration of the midpoint equals $-\langle E_b''\rangle\langle W_{\rm ex}'\rangle$ to $+0.4$–$0.8\%$ of the bond part $E_b''W_b'$. It differs from the leading form $-E_b''(0)W_{\rm ex}'(\bar X)$ by $-4.3$, $-4.8$, $-9.8$ and $-17\%$ of the bond part; the start averages account for this: $\langle E_b''\rangle$ is $-0.09$ to $-0.35\%$, and $\langle W_{\rm ex}'\rangle$ is $-0.20$ to $-0.87\%$ of the total. The second entry of $\epsilon$ dominates at $w_X=12$, $s=64$ ($\kappa_s=0.25$). S1u, $(32,8),(64,8),(64,12)$: the centre's acceleration equals $-\langle E_b''\rangle\langle W_{\rm ex}'\rangle$ to 0.5%, 1.2% and 0.3% of the bond part. The midpoint differs from it by $-2.2$, $-4.7$ and $-1.4\%$ of the bond part, from the interband oscillation of $\bar\nu h/2$ left in the fit. The leading form misses by $-10$, $-20$ and $-24\%$ of the bond part.

## 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, gauge transformation of the relative chain
- Weyl symbol, Ehrenfest theorem with the Moyal bracket, Airy scaling of the bound state
- 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, time evolution by Krylov exponentials (code)

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