07-ilang-weight / 05-inertia-of-mark
05inertia of markverified
Determines, for a moving body with its own mark and three couplings, whether its inertia is its cost over the square of one speed for all levels, and whether weight over inertia is level-independent.
# Inertia of a body with its own mark against its cost
- **Subproject:** 07-ilang-weight
- **Package:** 05-inertia-of-mark
- **Version:** v3
- **Mode:** external regeneration
- **Date:** 2026-10-21
## Changes from previous version
- Step 7, "(7.7) for FG", parts 3–4: the acceleration is now that of the mean position in a chart cut at a bond away from the start (question note (ii)). The ring identity $i[C,\hat x_c]=\hat v-Lv_c$ separates a boundary term $\epsilon_{\rm cut}$ with a norm bound. The bond-velocity part is mapped to the band by Hellmann–Feynman, which replaces the band position operator of v2 (I1). (5.14) now carries $\epsilon_{\rm cut}$.
- Step 6, (a) under (5.12): the root lists are numerical findings only. A new exact statement: the solution set has empty interior. The order $\gamma^3$ of the tuned descriptions is out of scope (question note (i)) (I2, I3).
- Step 8: the large-$t_f$ and small-$t_f$ exclusions for (a) are removed (I3). The large-$L$ statement for (a), (c) now uses the held-block spectra and the sign of $\sigma\gamma$ (I4).
- Result (5.11), (5.12), (5.14), Open issues and Code updated. `check_fall_full_grip.py` is rewritten for the chart position. The wording of `check_inertia_ac.py` is corrected; its numbers are unchanged.
## Response to verification
- I1: Accepted and fixed in Step 7, parts 3–4. The observable is the chart position $\hat x_c$. Exactly, $a=\langle i[C,\hat v]\rangle+\epsilon_{\rm cut}$, with $\hat v$ the oriented-bond velocity and $|\epsilon_{\rm cut}|\le L\tau(\tau+N_f|g|)p_c$, where $p_c$ is the weight of the start near the cut. The band image of $\langle i[C,\hat v]\rangle$ uses only translation covariance and Hellmann–Feynman, with no position commutator. The code now computes $d^2\langle\hat x_c\rangle/d\lambda^2$, and it checks the split and the bound.
- I2: Accepted in the form fixed by the question's note of 2026-10-21 (i). For the tuned (a) descriptions that keep one speed through order $\gamma^2$, only the locating criterion (5.12) and the numerical findings are asked; higher orders are out of scope. Step 6 and the Result now say so explicitly and do not present their order $\gamma^3$ as settled.
- I3: Accepted and fixed in Step 6, Step 8 and the Result. (5.12) is kept as the necessary and sufficient criterion. The double points are stated as numerical findings of a search whose completeness is not established. The $N\ge3$, large-$t_f$ and small-$t_f$ exclusions are removed. They are replaced by an exact statement: the solution set has empty interior, since $s$ is a non-constant rational function.
- I4: Accepted and fixed in Step 8 ("Isolation, (a) and (c)", "$L\to\infty$") and in (5.11). From the held-block spectra, the branch with $\sigma\gamma<0$ (attractive block) becomes the lowest level at large $L$ and stays isolated, with $d\to\sqrt{4t_f^2+\gamma^2M_k^2}-2t_f$. The branch with $\sigma\gamma>0$ has $d\to0$.
## Setup and assumptions
**Description (assumption, M8).** The ring of $L\ge5$ places carries $N_f\ge1$ identical field objects ($c=+1$) with own term $C_1$. $\Phi_x$ is their number at $x$, and $n=N_f/L$. One Model D body sits on the same ring, at clock level $k$ with half-gap $M_k=m+2\pi k/(N\lambda_0)$. It is coupled by (a), (c) or the full grip (FG), with real $\gamma\neq0$; records at $\kappa\to0$. As in 03-weight-of-mark, $\theta=1$ for (a) and (c), and $\theta=0$ for FG. The total contracts of FG and (a) differ by the rest and clock term $B$, so they are different descriptions, not M6 replacements.
**Notation.**
- $g:=\gamma M_k$, $g_\sigma:=\sigma g$ (the $g_b$ of (3.4)), and $\varkappa_k:=|g|L/t_f$.
- $\varepsilon_q=2t_f(1-\cos q)$ for $q\in Q:=\frac{2\pi}L\{1,\dots,L-1\}$, $\varepsilon_1:=\varepsilon_{2\pi/L}$, and $\langle f\rangle_q:=\frac1L\sum_{q\in Q}f(q)$.
- $s_1:=\langle\varepsilon_q^{-1}\rangle_q=\ell_0/t_f$ with $\ell_0=\Lambda^+(0)=(L^2-1)/(12L)$, $s_2:=\langle\varepsilon_q^{-2}\rangle_q$, and $S_2:=\langle(1+\cos q)\varepsilon_q^{-2}\rangle_q=2s_2-s_1/(2t_f)$.
- For one field object at place $r$ relative to the body: $h(g):=C_1+gP_0$, with lowest eigenvalue $\varepsilon_0(g)$ and positive ground vector $\varphi_g$ (Perron–Frobenius). $\Omega_\sigma:=\varphi_{g_\sigma}^{\otimes N_f}$.
- Inertia $\mu_{k,\sigma}:=1/E''_{k,\sigma}(0)$. Held cost (3.4): $\mathrm{Cost}_{k,\sigma}=\theta\sigma M_k+N_f[\varepsilon_0(g_\sigma)-\varepsilon_0(0)]$.
**Assumptions.**
- **A1 (M9, field state).** At $\tau=0$ the field is in $\Omega_\sigma$, the lowest state of the branch-$\sigma$ block (the lowest state adapted to the body). The dressed level is its continuation in $\tau$ and $P$.
- **A2 (A13, regime).**
- The curvature is computed at order $\tau^2$, under the isolation condition (5.4) for FG and (6.4) at $\tau=0$ for (a) and (c).
- FG is expanded in the weak-mark regime **R1**: $N_f\varkappa_k\ll1$.
- (a) and (c) are expanded in $\gamma$ at fixed $L,N_f$, at leading order in $\tau$.
- Statements at small $t_f$ concern the coefficient of $\tau^2$ unless an order of limits is stated (Step 8).
- **A3 (standard mathematics).**
- Rayleigh–Schrödinger (RS) perturbation theory for a nondegenerate isolated eigenvalue: for $\lVert V\rVert<d/2$, with $d$ the distance to the rest of the unperturbed spectrum, the eigenvalue continues analytically (Kato).
- Schrieffer–Wolff (SW, direct rotation) for an isolated group of eigenvalues: for $\lVert V\rVert<d/2$ a unitary $e^{S}$, $S$ block-off-diagonal with $\lVert S\rVert=O(\lVert V\rVert/d)$, maps the unperturbed band onto the perturbed spectral subspace. Weyl's inequality.
- Fourier sums on the ring, for example $\sum_{j=1}^{L-1}\sin^{-2}(\pi j/L)=(L^2-1)/3$; Perron–Frobenius.
**Inputs used:** (6.1), (6.4), (6.10), (6.12) of 06-ilang-source/06-own-mark; (2.14) of 02-two-free-bodies; (3.4), (3.12), (3.13) of 03-weight-of-mark; (7.7) of 06-ilang-source/07-dressed-fall, for (a) and (c) (its FG form is derived in Step 7). (6.5) of 06-ilang-source/06-own-mark is quoted but not needed, because (5.2) is exact in $\gamma$.
## Derivation
### Step 1. Sector contract; the full grip
Place the body at $x$ and the field objects at $x+r_i$. The vectors $L^{-1/2}\sum_xe^{iPx}\lvert x,s\rangle\lvert\{x+r_i\}\rangle$ span the sector of total label $P$, since the total contract commutes with the joint translation. In these coordinates the field hopping acts on the $r_i$ unchanged, and the pair term becomes $g_\sigma\Phi_0$ on branch $\sigma$ ($\Phi_r$ counts the objects at relative place $r$). The hop $\lvert x+1\rangle\langle x\rvert$ maps the sector vector with $\{r_i\}$ to $e^{-iP}$ times the one with $\{r_i-1\}$, and the bond factor $\frac12(\Phi_x+\Phi_{x+1})$ of (c) acts before the hop, at relative places $0$ and $1$.
On level $k$ (the clock label commutes with the total contract), (6.1) holds with
$$
\hat C^{(0)}=\sum_{\sigma=\pm}\lvert\sigma\rangle\langle\sigma\rvert\otimes\hat C^{(0)}_\sigma,\quad \hat C^{(0)}_\sigma=\theta\sigma M_k+C_1+g_\sigma\Phi_0,\qquad \hat C^{(+)}=\frac\tau2(-i\sigma_1)\otimes K,\quad K=\begin{cases}T_-&\text{(a), FG}\\ T_-\bigl(1+\frac\gamma2(\Phi_0+\Phi_1)\bigr)&\text{(c)}\end{cases}
\tag{5.1}
$$
with $T_\pm$ shifting all relative places by $\pm1$. For FG, (6.1) applies unchanged: only $\hat C^{(0)}$ changes ($\theta=0$).
The reflection $\mathcal R:r\to-r$ leaves $\hat C^{(0)}_\sigma$ and $\Omega_\sigma$ invariant and maps $K$ to $K^\dagger$ (since $T_+^\dagger\Phi_rT_+=\Phi_{r-1}$). All these operators are real in the place basis.
### Step 2. Curvature at order $\tau^2$ and the branch splitting
$V(P):=e^{-iP}\hat C^{(+)}+e^{iP}\hat C^{(+)\dagger}$ flips the branch and has $\lVert V\rVert\le\tau\lVert K\rVert$. At $\tau=0$ the level is $E^0_\sigma=\theta\sigma M_k+N_f\varepsilon_0(g_\sigma)$, with vector $\lvert\sigma\rangle\otimes\Omega_\sigma$ (A1); $d_{k,\sigma}$ is its distance to the rest of the spectrum of $\hat C^{(0)}$.
The first RS order vanishes. The second uses $R:=(E^0_\sigma-\hat C^{(0)}_{-\sigma})^{-1}$, $(-i)^2=i^2=-1$ and expectation values in $\Omega_\sigma$:
$$E_{k,\sigma}(P)=E^0_\sigma+\frac{\tau^2}4\bigl[\langle KRK^\dagger\rangle+\langle K^\dagger RK\rangle-2J\cos2P\bigr]+O\Bigl(\frac{\tau^4\lVert K\rVert^4}{d^3_{k,\sigma}}\Bigr).$$
Here $J:=\langle KRK\rangle=\langle K^\dagger RK^\dagger\rangle$ is real, by $\mathcal R$ and reality. Hence, exactly in $\gamma$:
$$E''_{k,\sigma}(0)=2\tau^2J_{k,\sigma}+O\Bigl(\frac{\tau^4\lVert K\rVert^4}{d^3_{k,\sigma}}\Bigr),\qquad J_{k,\sigma}=\langle\Omega_\sigma\vert K\,(E^0_\sigma-\hat C^{(0)}_{-\sigma})^{-1}K\vert\Omega_\sigma\rangle .\tag{5.2}$$
The lowest eigenvector of $\hat C^{(0)}_{-\sigma}$ is $\Omega_{-\sigma}$, and $E^0_\sigma-E^0_{-\sigma}=\mathrm{Cost}_{k,\sigma}-\mathrm{Cost}_{k,-\sigma}$ exactly. Define:
- $R_k:=\frac12(\mathrm{Cost}_{k,+}-\mathrm{Cost}_{k,-})$ and $C^{\rm ev}_k:=\frac12(\mathrm{Cost}_{k,+}+\mathrm{Cost}_{k,-})$;
- $a_{k,\sigma}:=\langle\Omega_{-\sigma}\vert K\vert\Omega_\sigma\rangle$, equal to $\langle\Omega_{-\sigma}\vert K^\dagger\vert\Omega_\sigma\rangle$ by $\mathcal R$.
Separating the term of $\Omega_{-\sigma}$ in the spectral sum of $R$ gives
$$\mathrm{Cost}_{k,\sigma}=\sigma R_k+C^{\rm ev}_k,\qquad J_{k,\sigma}=\frac{a_{k,\sigma}^2}{2\sigma R_k}+J^{\rm exc}_{k,\sigma},\qquad \mu_{k,\sigma}=\frac{\sigma R_k}{\tau^2\bigl(a^2_{k,\sigma}+2\sigma R_kJ^{\rm exc}_{k,\sigma}\bigr)}\quad\text{at order }\tau^{-2},\tag{5.3}$$
where $J^{\rm exc}$ is the sum over the excited eigenvectors of $\hat C^{(0)}_{-\sigma}$. The inertia is set by the branch-odd part $R_k$ of the cost. The even part $C^{\rm ev}_k$ enters only through $a$ and $J^{\rm exc}$.
**M6.** $\mu_{k,\sigma}$, $R_k$ and the level differences of $C^{\rm ev}_k$ are curvatures and eigenvalue differences of one total contract. $C^{\rm ev}_k$ itself is fixed only up to the level-independent reference, the description without the body.
**Dressed level at $P=0$.** $E_{k,\sigma}(0)-E^0_\sigma=\frac{\tau^2}4\langle w\vert R\vert w\rangle$ with $w=(K^\dagger-K)\Omega_\sigma$, and $w\perp\Omega_{-\sigma}$ by $\mathcal R$. So it is $O(\tau^2)$, without the factor $1/R_k$.
### Step 3. Isolation for the full grip (item 1)
At $\tau=0$, $\hat C^{(0)}_\sigma=C_1+g_\sigma\Phi_0$ acts on $N_f$ non-interacting bosons, and its eigenvalues are sums of eigenvalues of $h(g_\sigma)$. Call $\sigma_l:=-\operatorname{sign}\gamma$ the lower branch ($g_{\sigma_l}=-|g|$) and $\sigma_u:=-\sigma_l$ the upper one.
- **Splitting.** $\hat C^{(0)}_{\sigma_u}-\hat C^{(0)}_{\sigma_l}=2|g|\Phi_0\ge0$, and $\Omega_{\sigma_u}$ has positive density at $r=0$. Hence $\Delta_b:=E^0_{\sigma_u}-E^0_{\sigma_l}=2|R_k|>0$.
- **Lower level.** By Perron–Frobenius, $E^0_{\sigma_l}$ is the nondegenerate lowest eigenvalue of $\hat C^{(0)}$.
- **Distances.** With $\varepsilon^\pm_1$ the first excitation energy of $h(\pm|g|)$: $d_{k,\sigma_l}=\min(\Delta_b,\varepsilon^-_1)$, and, if $\Delta_b<\varepsilon^-_1$, $d_{k,\sigma_u}=\min(\Delta_b,\varepsilon^-_1-\Delta_b,\varepsilon^+_1)$.
For FG $\lVert K\rVert=1$, so by Kato (5.2) holds for $\tau<d/2$. In R1, $\Delta_b=2n|g|(1+O(\varkappa_k^2))$ by (5.5), and $\varepsilon^\pm_1=\varepsilon_1(1+O(\varkappa_k))$. The condition reads
$$\tau<\tfrac12d_{k,\sigma},\qquad d_{k,\sigma}\simeq\begin{cases}\min\bigl(2n|\gamma|M_k,\ \varepsilon_1\bigr),&\sigma\gamma<0,\\ \min\bigl(2n|\gamma|M_k,\ \varepsilon_1-2n|\gamma|M_k\bigr),&\sigma\gamma>0,\ 2n|\gamma|M_k<\varepsilon_1,\end{cases}\qquad \varepsilon_1=2t_f\Bigl(1-\cos\frac{2\pi}L\Bigr).\tag{5.4}$$
Where $d_{k,\sigma}$ vanishes or tends to zero (then (5.4) fails at every fixed $\tau>0$; this concerns the $\tau=0$ spectrum and the sufficient criterion, not the exact level at $\tau>0$):
1. **$\gamma\to0$, both branches.** $d\to0$: without coupling the FG body is gapless, with band $\pm\tau\lvert\sin P\rvert$.
2. **Upper branch, $N_f\ge2$.** As $2n|\gamma|M_k$ approaches $\varepsilon_1$, the upper level crosses the lowest excited level of the other branch (the branches do not mix at $\tau=0$). Beyond that it crosses multi-object excitations at isolated values of $\gamma$.
3. **$N_f=1$: no crossing.** The even eigenvalues of $h(\pm|g|)$ solve $\pm1/|g|=\langle0\vert(E-C_1)^{-1}\vert0\rangle$, which decreases between consecutive poles; the odd ones are $\ge-2t_f\cos(2\pi/L)$. Hence $\varepsilon_0(|g|)$ lies strictly between the two lowest eigenvalues of $h(-|g|)$. The upper branch is isolated for every $\gamma\neq0$, with $d_{k,\sigma_u}\to0$ as $\varkappa_k\to\infty$.
4. **$L\to\infty$ at fixed $n,\gamma$.** $\varepsilon_1\to0$, so $d_{k,\sigma_u}\le\varepsilon_1^+\to0$. The lower branch stays isolated, with $d_{k,\sigma_l}\to\sqrt{4t_f^2+\gamma^2M_k^2}-2t_f$ (the binding gap of the infinite chain; $\Delta_b\propto N_f\to\infty$).
### Step 4. Weak-mark expansion
RS for $h(g)=h(0)+gP_0$ in plane waves $\lvert q\rangle$, with $\langle q\vert P_0\vert q'\rangle=1/L$ and $u:=\lvert q{=}0\rangle$, gives
$$
\begin{aligned}
&\varphi_g=u+g\varphi^{(1)}+O(g^2),\quad \langle q\vert\varphi^{(1)}\rangle=-\frac1{L\varepsilon_q},\qquad \varepsilon_0(g)=-2t_f+\frac gL-\frac{g^2s_1}L+\frac{g^3}L\Bigl(s_1^2-\frac{s_2}L\Bigr)+O(g^4),\\
&R_k=\theta M_k+n\gamma M_k\bigl[1+\gamma^2M_k^2(s_1^2-s_2/L)\bigr]+O(\gamma^5),\qquad C^{\rm ev}_k=-n\gamma^2M_k^2\,\ell_0/t_f+O(\gamma^4),\\
&a^2_{k,\sigma}=(1+\theta_c\gamma n)^2-2ng^2S_2+O(\gamma^3),\qquad J^{\rm exc}_{k,\sigma}=2n\Bigl\langle\frac{\cos q\,(1+\cos q)\,u_q^2}{2\sigma R_k-\varepsilon_q}\Bigr\rangle_q+O(\gamma^3),
\end{aligned}\tag{5.5}
$$
with $\theta_c=1$ for (c) and $0$ otherwise. The vertex is $u_q=-g_\sigma/\varepsilon_q$ for (a) and FG, and $u_q=\gamma/2-g_\sigma/\varepsilon_q$ for (c). Reasons:
- **$\varepsilon_0(g)$.** In the third RS order all matrix elements equal $g/L$, so $E^{(3)}=(g/L)^3[(\sum_q\varepsilon_q^{-1})^2-\sum_q\varepsilon_q^{-2}]$. $R_k$ and $C^{\rm ev}_k$ are the odd and even parts in $g$ of (3.4).
- **$a$ for (a) and FG.** $a=\alpha^{N_f}$ with
$$\alpha:=\langle\varphi_{-g_\sigma}\vert T_-\varphi_{g_\sigma}\rangle=1-g^2\bigl(\lVert\varphi^{(1)}\rVert^2+\langle\varphi^{(1)}\vert T_+\varphi^{(1)}\rangle\bigr)+O(g^4)=1-g^2S_2/L+O(g^4).$$
$\alpha$ is even in $g$ by $\mathcal R$. The $O(g)$ terms vanish because $T_\pm u=u$ and $\varphi^{(1)}\perp u$.
- **$a$ for (c).** It gains $\frac\gamma2N_f\alpha^{N_f-1}\bigl[\varphi_{-g_\sigma}(-1)\varphi_{g_\sigma}(0)+\varphi_{-g_\sigma}(0)\varphi_{g_\sigma}(1)\bigr]=\gamma n+O(\gamma^3)$. The $O(g)$ parts cancel because $\varphi^{(1)}$ is even in $r$.
- **$J^{\rm exc}$.** The component of $K\Omega_\sigma$ with one object excited is $N_f^{1/2}$ times a one-object vector $\eta$, with $\langle q\vert\eta\rangle=L^{-1}(1+e^{iq})u_q+O(\gamma^2)$. Its sources:
- $T_-\varphi_{g_\sigma}-\alpha\varphi_{-g_\sigma}=g_\sigma(1+T_-)\varphi^{(1)}$;
- for (c), also $\frac\gamma2T_-(P_0+P_1)u$, with $\langle q\vert T_-(P_0+P_1)\vert u\rangle=(1+e^{iq})/L$.
The corresponding component of $K^\dagger\Omega_\sigma$ is its $\mathcal R$-image, and the excitation energy is $\varepsilon_q+O(\gamma)$. For even $f$, $\sum_q(1+e^{iq})^2f(q)=\sum_q2\cos q(1+\cos q)f(q)$. Two-object excitations contribute $O(\gamma^4)$.
The one-object denominators $2\sigma R_k-\varepsilon_q$ stay away from zero by R1 ($|2R_k|<\varepsilon_1$) for FG, and for (a) and (c) by the one-object part of (6.4) at $\tau=0$ ($2M_k\neq\varepsilon_q$). (6.4), assumed in A2, also excludes the many-object resonances. They enter $J$ only at $O(\gamma^4)$, but they set the distance $d_{k,+}$ (Step 8).
### Step 5. Item 1: curvature of the full grip
Put $\theta=\theta_c=0$ in (5.5):
- $2\sigma R_k=2ng_\sigma[1+g^2(s_1^2-s_2/L)]$;
- $a^2=1-2ng^2S_2$;
- $2\sigma R_kJ^{\rm exc}=O(n^2|g|^3\langle\varepsilon_q^{-3}\rangle_q)=O(N_f^2\varkappa_k^3)$.
By (5.2) and (5.3), under (5.4) and R1:
$$E''_{k,\sigma}(0)=\frac{\tau^2}{n\sigma\gamma M_k}\Bigl[1-\gamma^2M_k^2K_L+O(N_f^2\varkappa_k^3)\Bigr]+O\Bigl(\frac{\tau^4}{d_{k,\sigma}^3}\Bigr),\qquad K_L=2nS_2+s_1^2-\frac{s_2}L\simeq\frac{(N_f+1)L^2}{180\,t_f^2}.\tag{5.6}$$
So $\gamma^2M_k^2K_L\simeq(N_f+1)\varkappa_k^2/180$. The leading term is the curvature $\tau^2/(\sigma\tilde M)$ of a Model D body with rest half-gap $\tilde M=n\gamma M_k$: in FG the mean density of the field supplies the rest term. Beyond R1, (5.2) still gives the order-$\tau^2$ curvature wherever $d_{k,\sigma}>2\tau$; for the lower branch this holds at every $\gamma\neq0$ for small enough $\tau$. The check of (5.2) and (5.6) against the exact dressed level is under "Code".
### Step 6. Item 2: inertia against cost
**Which cost.** The held cost (3.4) enters every comparison. By Step 2, $E_{k,\sigma}(0)-N_f\varepsilon_0(0)-\mathrm{Cost}_{k,\sigma}=O(\tau^2)$, without the factor $1/R_k$. Divided by $v^2$ it is $O(\tau^0)$, below the order $\tau^{-2}$ of $\mu$. So the dressed level gives the same answers here, and the bracket of (6.12) reduces to the held cost as $\tau\to0$.
**Criterion (M6).** One speed $v$ for all levels and branches means that $D_{k,\sigma}:=\mu_{k,\sigma}-\mathrm{Cost}_{k,\sigma}/v^2$ does not depend on $(k,\sigma)$; the constant is the reference of the cost. Equivalently, every difference of inertia between levels and branches equals the difference of cost divided by $v^2$.
**Leading order.**
- FG: by (5.3) and (5.5), $a^2=1+O(\gamma^2)$ and $2\sigma R_kJ^{\rm exc}=O(\gamma^3)$, so $\mu=\sigma R_k/\tau^2+O(\gamma^3)$.
- (a), (c): (2.14), with $\mathrm{Cost}=\sigma R_k+O(\gamma^2)$.
Together:
$$\mu_{k,\sigma}=\frac{\mathrm{Cost}_{k,\sigma}}{v^2}+O(\gamma^2)\ \text{ for every }k,\sigma,\qquad v=\tau\ \text{ (a), FG},\qquad v=\tau(1+\gamma n)\ \text{ (c)}.\tag{5.7}$$
For FG the leading term is of order $\gamma$: $\mu=n\sigma\gamma M_k/\tau^2$.
**Identification of $v$.** $v$ is the slope at $P\to0$ of the band of the motion term alone, with its mean-field factor: $\pm v\lvert\sin P\rvert$. This is the $M_k\to0$ limit of the mean-field bands $(M_k^2(1+\gamma n)^2+\tau^2\sin^2P)^{1/2}$ (a), $(1+\gamma n)(M_k^2+\tau^2\sin^2P)^{1/2}$ (c) and $(n^2\gamma^2M_k^2+\tau^2\sin^2P)^{1/2}$ (FG).
- In the regime every level has a nonzero mean-field half-gap, so its group velocity stays below $v$; no level attains it.
- For FG, $\pm\tau\lvert\sin P\rvert$ is the band of the description at $\gamma=0$.
- $v$ does not depend on $t_f$, so it is not fixed by the field band $-2t_f\cos q$.
**Order $\gamma^2$.** Changing $v$ at order $\gamma^2$ ($v^2\to v^2(1+b\gamma^2)$) adds $b\gamma^2\sigma M_k/\tau^2+O(\gamma^3)$ to $D$ for (a) and (c), and only $O(\gamma^3)$ for FG.
*FG.* By (5.3) and (5.5):
$$D_{k,\sigma}=-\frac{C^{\rm ev}_k}{\tau^2}+O(\gamma^3)=\frac{n\ell_0\gamma^2M_k^2}{t_f\tau^2}+O(\gamma^3),\qquad (\mu_{k,\sigma}-\mu_{k',\sigma'})-\frac{\mathrm{Cost}_{k,\sigma}-\mathrm{Cost}_{k',\sigma'}}{\tau^2}=\frac{n\ell_0\gamma^2(M_k^2-M_{k'}^2)}{t_f\tau^2}+O(\gamma^3).\tag{5.8}$$
The branch differences at one level still agree through $O(\gamma^2)$, but the level differences do not. The inertia follows the branch-odd part of the cost, not its even part (the second-order mark).
*(a) and (c).* With $v$ of (5.7), (5.3) and (5.5) give
$$D_{k,\sigma}=-\frac{C^{\rm ev}_k}{\tau^2}+\frac{\sigma M_k}{\tau^2}\bigl(2ng^2S_2-2\sigma M_kJ^{\rm exc}_{k,\sigma}\bigr)+O(\gamma^3).$$
The bracket equals $\frac{n\gamma^2M_k^2}{t_f^2}\bigl\langle\frac{(1+\cos q)(\sigma M_k-t_f)}{(1-\cos q)(2\sigma M_k-\varepsilon_q)}\bigr\rangle_q$ for (a), using $1-\frac{2\sigma M\cos q}{2\sigma M-\varepsilon_q}=\frac{2(1-\cos q)(\sigma M-t_f)}{2\sigma M-\varepsilon_q}$. For (c) it equals $\frac{n\gamma^2M_k}{2t_f^2}\bigl\langle\frac{(1+\cos q)(M_k+\sigma t_f\cos q)}{1-\cos q}\bigr\rangle_q$: the vertex $(\varepsilon_q-2\sigma M_k)^2/(4\varepsilon_q^2)$ cancels the denominator. Hence
$$
\begin{aligned}
D^{(a)}_{k,\sigma}&=\frac{n\gamma^2M_k^2}{2t_f^2\tau^2}\Bigl\langle\frac1{1-\cos q}\Bigl[t_f+\frac{2M_k(1+\cos q)(M_k-\sigma t_f)}{2\sigma M_k-\varepsilon_q}\Bigr]\Bigr\rangle_q+O(\gamma^3),\\
D^{(c)}_{k,\sigma}&=\frac{n\gamma^2M_k^2}{2t_f^2\tau^2}\Bigl\langle\frac{t_f(1+\cos q+\cos^2q)+\sigma M_k(1+\cos q)}{1-\cos q}\Bigr\rangle_q+O(\gamma^3).
\end{aligned}\tag{5.9}
$$
For $\sigma=+$, this is the $\tau\to0$ value of $-[M_k^2\Delta''_{k,+}/\tau^2+\Delta_{k,+}]/\tau^2$, the term of (6.12) that the quoted inputs leave unevaluated. At $N_f=1$, (5.9) agrees with $\tau^2D$ from the exact $\tau^2$ coefficient (5.2) up to $O(\gamma^3)$ (code, `check_inertia_ac.py`).
Two limits of (5.9):
- **$t_f\gg M_kL^2$** (all $\varepsilon_q\gg M_k$). $D^{(a)}=\frac{n\ell_0\gamma^2M_k^2}{t_f\tau^2}(1+O(M_kL^2/t_f))$ and $D^{(c)}=\frac{n\gamma^2M_k^2}{2t_f\tau^2}\langle\frac{1+\cos q+\cos^2q}{1-\cos q}\rangle_q(1+O(M_kL^2/t_f))$. Both are branch-even and proportional to $M_k^2$.
- **$t_f\ll M_k$.** Both equal $\sigma\frac{n\gamma^2M_k^3}{2t_f^2\tau^2}\langle\frac{1+\cos q}{1-\cos q}\rangle_q(1+O(t_f/M_k))$: branch-odd and proportional to $M_k^3$.
**One speed at order $\gamma^2$, whole non-resonant class.** With $y:=\sigma M_k/t_f$, (5.9) reads
$$\tau^2D_{k,\sigma}=n\gamma^2t_f\,\mathbb F(y)+O(\gamma^3),\qquad \mathbb F_a(y)=\frac{y^2}2\Bigl\langle\frac{1}{1-\cos q}\Bigl[1+\frac{(1+\cos q)\,y(y-1)}{y-1+\cos q}\Bigr]\Bigr\rangle_q,\qquad \mathbb F_c(y)=\frac{y^2}2\Bigl\langle\frac{1+\cos q+\cos^2q+y(1+\cos q)}{1-\cos q}\Bigr\rangle_q.\tag{5.12}$$
A change of $v$ at order $\gamma$ adds a branch-odd term at order $\gamma$, which (5.7) excludes. A change $v^2\to v^2(1+b\gamma^2)$ adds $b\gamma^2\sigma M_k/\tau^2$. So one $v$ exists at order $\gamma^2$ iff the $2N$ points $(y,\mathbb F(y))$, $y=\pm M_k/t_f$, lie on one straight line. The condition depends only on $L$ and the ratios $M_k/t_f$, not on $N_f$, $\gamma$ or $\tau$.
- **(c).** The branch average $\frac12[\mathbb F_c(y)+\mathbb F_c(-y)]=\frac{y^2}2\langle\frac{1+\cos q+\cos^2q}{1-\cos q}\rangle_q$ is untouched by $b$. Its coefficient is positive ($1+\cos q+\cos^2q\ge\frac34$), so it differs between levels, since the $M_k$ are distinct and $N\ge2$. So (c) has no single $v$ at order $\gamma^2$, for every member of the non-resonant class.
- **(a).** $\mathbb F_a$ has poles at the one-object resonances $y=1-\cos q$ (where $\cos q(1+\cos q)\neq0$). Two levels share a line iff their chords coincide: with $s(x):=[\mathbb F_a(x)-\mathbb F_a(-x)]/(2x)$ and $i(x):=[\mathbb F_a(x)+\mathbb F_a(-x)]/2$, iff $s(M_0/t_f)=s(M_1/t_f)$ and $i(M_0/t_f)=i(M_1/t_f)$. These are the double points of the plane curve $x\mapsto(s(x),i(x))$.
- For $N$ levels every pair of levels must satisfy the two-level condition.
- **Exact.** $s$ is a non-constant rational function, since $\mathbb F_a(x)$ has a pole at $x=1-\cos\frac{2\pi}L$ and $\mathbb F_a(-x)$ has none. So $s(x_0)-s(x_1)$ vanishes only on a closed set with empty interior. Hence the ratios $(M_k/t_f)_k$ with one $v$ at order $\gamma^2$ form a set with empty interior, and one $v$ fails on an open dense subset of the class.
- **Numerical findings** (code). The search runs in $(u_0+u_1,\,u_0u_1)$, $u=x^2$, where both equations have their poles on straight lines. It finds double points for every $5\le L\le16$: 1 for $L=5,6,8$ and up to 21 for $L=15$, for example $(M_0,M_1)/t_f=(0.35551,1.35669)$ at $L=7$. The two grids used differ by one point at $L=15$, and the search is not shown to be complete. Each point found has a resonance value $1-\cos q$ between $M_0/t_f$ and $M_1/t_f$, and no ratio occurs in two of them. At the $L=7$ example the exact $\tau^2$ coefficient deviates from a common line at $O(\gamma^3)$, against $O(\gamma^2)$ at a generic pair. At such points $v^2=\tau^2(1+b\gamma^2)$, with $b$ fixed by the common line.
- **Scope.** For these tuned descriptions the question (note (i)) asks only for the criterion (5.12) and these findings. Their behaviour at order $\gamma^3$ and beyond is out of scope and is not settled here.
**Controls.**
- **(5.7).** It holds for every member of the class in the regime, so there is no control in the class. In the wider class with a branch-even clock term $\mathbb 1_2\otimes D'$ added to the body ($D'$ diagonal in the levels, not proportional to $\mathbb 1$), $\mathrm{Cost}_{k,\sigma}$ shifts by $D'_k$. $E_{k,\sigma}(P)$ shifts by the same $D'_k$ at every $P$, so $\mu$ is unchanged and (5.7) fails at order $\gamma^0$.
- **(5.8), (5.9), (5.12).** For FG and (c): a body with one half-gap ($D=0$, a wider class). Then $D_{k,\sigma}=D_\sigma$. For FG, $v=\tau$ holds through $O(\gamma^2)$. For (c), $b$ removes the odd part of $D_\sigma$ and the even part is a constant. For (a) the class itself contains controls: the two-level bodies at the double points of (5.12) found numerically.
### Step 7. Item 3: weight over inertia
By (3.12) and (3.13), $\Delta W_{k,\sigma}=\mathcal W(C^{\rm mark}_{k,\sigma})$ for (a) and (c) (they coincide at $\tau=0$) and for FG ($\theta=0$). The rest term does not depend on the place, so all three share
$$C^{\rm mark}_{k,\sigma}=\mathrm{Cost}_{k,\sigma}-\theta\sigma M_k=n\sigma\gamma M_k-n\ell_0\gamma^2M_k^2/t_f+O(\gamma^3).$$
For $E''$ use (6.10), $\sigma\tau^2(1\mp\gamma n)/M_k+O(\gamma^2)$ for (a) and (c), and (5.6) for FG. With $\mathcal W(C)=FC+F_2C^2+O(C^3)$:
$$E''_{k,\sigma}(0)\,\Delta W_{k,\sigma}=\tau^2F\,\Lambda\,\bigl[1+\sigma\gamma M_k\,\Xi+O(\gamma^2)\bigr]+O(g_s^2),\qquad \Xi:=\frac{nF_2}F-\frac{\ell_0}{t_f}=-\frac{[\ell^2+2\ell_0\ell-2\ell^{(2)}/\lvert X_f\rvert]}{2t_f\,[\ell]},\tag{5.10}$$
with $\Lambda=1$ for FG, $n\gamma(1-\gamma n)$ for (a) and $n\gamma(1+\gamma n)$ for (c).
- **Conditions.** (5.10) needs $g_s\neq0$ and $F\neq0$. On the ring $\ell(r)=\ell_0-r(L-r)/(2L)$. With $\rho:=r-L/2$, $\rho':=r'-L/2$ this gives $[\ell]=(\rho^2-\rho'^2)/(2L)$, so $F=0$ iff $\rho'=\pm\rho$: $x'=x$, or $x'$ is the mirror image of $x$ with respect to the source (reflection-equivalent places). For such places the reflection about $x_s$ maps the held description at $x$ onto that at $x'$. So $\Delta W_{k,\sigma}=0$ exactly, for every level and branch and to all orders: the product vanishes, is trivially universal, and $\Xi$ is not defined.
- **$\Xi$ in closed form.** $\ell^{(2)}$ solves $(2-T_+-T_-)\ell^{(2)}=\ell$ with zero mean. On $0\le r\le L$ it is therefore the quartic $a_2+\frac{L^2+4}{48L}\rho^2-\frac{\rho^4}{24L}$, since the second difference of $\rho^4$ is $12\rho^2+2$. Inserted into (5.10):
$$\Xi=-\frac{\rho^2+\rho'^2-1}{3Lt_f}.$$
So $\Xi\le0$, and $\Xi=0$ iff $\{\rho^2,\rho'^2\}=\{0,1\}$: $L$ even, one place antipodal to the source and the other next to it. (Code: agreement to $10^{-14}$ for $5\le L\le16$.)
- **Leading order.** The product is the same for all levels and both branches.
- **First dependence, $\Xi\neq0$.** At relative order $\gamma$, $\sigma\gamma M_k\Xi$, with one coefficient $\Xi$ for all three couplings, and $\lvert\gamma M_k\Xi\rvert\le\gamma M_kL/(6t_f)=O(\varkappa_k)$.
- **Next order, $\Xi=0$.** Insert $C^{\rm mark}=ng_\sigma-ns_1g^2+ns_3g_\sigma^3+O(g^4)$, $s_3:=s_1^2-s_2/L$ (from (5.5)), into (3.13) through $F_3$: $\Delta W=Fng_\sigma[1+\Xi g_\sigma+\Psi_0g^2+O(g^3)]$. For the curvature:
- FG: the bracket of (5.6) applies.
- (a), (c): by (5.3) and the expansion of $D$ above, $E''=\sigma\tau^2\frac{(1+\theta_c\gamma n)^2}{M_k(1+\gamma n)}[1-\sigma(\tau^2D_{k,\sigma}+C^{\rm ev}_k)/M_k+O(\gamma^3)]$.
Hence
$$\frac{E''_{k,\sigma}(0)\,\Delta W_{k,\sigma}}{\tau^2F\Lambda}=1+\sigma\gamma M_k\Xi+\gamma^2M_k^2\Psi_0-\begin{cases}\gamma^2M_k^2K_L&\text{FG}\\ \sigma(\tau^2D_{k,\sigma}+C^{\rm ev}_k)/M_k&\text{(a), (c)}\end{cases}+O(\gamma^3),\qquad \Psi_0:=s_3-2s_1\frac{nF_2}F+\frac{n^2F_3}F,\tag{5.15}$$
with $\Lambda=1$ (FG), $n\gamma/(1+\gamma n)$ (a), $n\gamma(1+\gamma n)$ (c), equal to the $\Lambda$ of (5.10) at its order. Where $\Xi=0$:
- **FG.** The first dependence is a level dependence $(\Psi_0-K_L)\gamma^2(M_k^2-M_{k'}^2)$; the branches differ at the earliest at $O(\gamma^3)$. For $N_f=1$, $(\Psi_0-K_L)t_f^2=-0.559,-1.025,-1.625,-2.359,-3.225,-4.225$ for $L=6,8,\dots,16$ (code, which also confirms (5.15) against the exact $\tau^2$ coefficient at $L=8$ for the three couplings).
- **(c).** A branch dependence at relative order $\gamma^2$: by (5.9) and $\langle\frac{\cos q(1+\cos q)}{1-\cos q}\rangle_q=\frac{(L-2)(L-4)}{3L}$, the branch-odd part is $-\sigma\gamma^2M_k\,n(L-2)(L-4)/(6Lt_f)\neq0$.
- **(a).** The branch-odd part is $\sigma\gamma^2M_k\frac{2nM_k^2}{t_f}\bigl\langle\frac{\cos q(1+\cos q)}{(1-\cos q)(4M_k^2-\varepsilon_q^2)}\bigr\rangle_q$. It is nonzero except where this average vanishes.
The mechanism: $E''\Delta W\propto\mathcal W(C^{\rm mark})/R_k$. The weight follows the whole mark, nonlinearly; the inertia follows only its branch-odd part.
**(7.7) for FG.** (7.7) was derived for (a) and (c) only; for FG it is derived here, for the mean position in a chart (note (ii) of the question). Fix the level $k$ and the branch $\sigma$.
1. *Held band with the source.* With the body at $j$, the held block is $h_{j,\sigma}=C_1+g_\sigma\Phi_j+g_s\Phi_{x_s}$ (absolute field places). Its lowest eigenvalue is $\mathcal E_\sigma(j)=E^0_\sigma+W_\sigma(j)+O(N_f^2g_s^2/d_{k,\sigma})$, with positive ground vector $\Omega_\sigma(j)$. Here $W_\sigma(j)=g_s\langle\Omega_\sigma(j)\vert\Phi_{x_s}\vert\Omega_\sigma(j)\rangle$ is the held weight, (3.12) with $\theta=0$. The vectors $\lvert j\rangle\rangle:=\lvert j,\sigma\rangle\otimes\Omega_\sigma(j)$ span the held band, with projector $\Pi$. Since $\lVert g_s\Phi_{x_s}\rVert=N_f|g_s|$, Weyl's inequality separates the band from the rest of the $\tau=0$ spectrum by $d_s\ge d_{k,\sigma}-2N_f|g_s|$. We require $N_f|g_s|\le d_{k,\sigma}/4$.
2. *Effective contract.* Let $\tau<d_s/2$, and let $e^{S}$ be the SW unitary (A3) from $\Pi$ to the band of the full contract. The motion term flips the branch, so the odd orders vanish:
$$H_{\rm eff}=\sum_jU(j)\lvert j\rangle\rangle\langle\langle j\rvert+\sum_jt(j)\bigl(\lvert j+1\rangle\rangle\langle\langle j-1\rvert+{\rm h.c.}\bigr)+O(\tau^4/d_s^3),$$
$$t(j)=-\frac{\tau^2}8\langle\Omega_\sigma(j+1)\vert R_j(\mathcal E_\sigma(j+1))+R_j(\mathcal E_\sigma(j-1))\vert\Omega_\sigma(j-1)\rangle,\qquad U(j)=\mathcal E_\sigma(j)+\frac{\tau^2}4\sum_\pm\langle\Omega_\sigma(j)\vert R_{j\pm1}(\mathcal E_\sigma(j))\vert\Omega_\sigma(j)\rangle,$$
with $R_y(E)=(E-h_{y,-\sigma})^{-1}$; the sign of $t$ is $(-i)^2$ from two forward hops. All entries are real. At $g_s=0$, translation by $j$ maps $t$ onto the relative coordinates of Step 2, so $t=-\tau^2J/4=-E''_{k,\sigma}(0)/8+O(\tau^4/d^3)$ by (5.2).
3. *Chart and boundary term.* Cut the ring at a bond $(c,c+1)$, and give the body place $x$ the label $x_c(x)\in\{c+1,\dots,c+L\}$, $x_c(x)\equiv x$ mod $L$. Then $x_c$ grows by $1$ along every bond except the cut, where it jumps by $1-L$.
- The mean position is $\langle\hat x_c\rangle$, with $\hat x_c:=\sum_xx_c(x)P_x$ a bounded observable. Its acceleration is $a_{k,\sigma}(0):=\frac{d^2}{d\lambda^2}\langle\hat x_c\rangle\big|_0=-\langle\Psi\vert[C,[C,\hat x_c]]\vert\Psi\rangle$, for the start $\Psi=e^S\Psi_0$, $\Psi_0=\sum_j\phi_j\lvert j\rangle\rangle$, $\phi_j$ real.
- Only the body motion term $A=\frac\tau2\sum_x(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+{\rm h.c.})$ moves the body, by one place. So
$$i[C,\hat x_c]=\hat v-Lv_c,\qquad \hat v:=-\frac\tau2\sum_x\bigl(\lvert x+1\rangle\langle x\rvert+\lvert x\rangle\langle x+1\rvert\bigr)\otimes\sigma_1 .$$
- $\hat v$ is the oriented-bond velocity. It is the same on every bond and commutes with the joint translation. $v_c$ is its term on the cut bond.
- Hence, exactly, $a_{k,\sigma}(0)=\langle\Psi\vert i[C,\hat v]\vert\Psi\rangle+\epsilon_{\rm cut}$, with $\epsilon_{\rm cut}:=-L\langle\Psi\vert i[C,v_c]\vert\Psi\rangle$.
- Only $A$ and the pair terms at $c,c+1$ fail to commute with $v_c$. So $i[C,v_c]$ acts only on the body places $c-1,\dots,c+2$, and $\lVert[C,v_c]\rVert\le2\lVert v_c\rVert(\lVert A\rVert+N_f|g|)=\tau(\tau+N_f|g|)$. This gives $|\epsilon_{\rm cut}|\le L\tau(\tau+N_f|g|)\,p_c$, with $p_c$ the probability that the start has the body at one of these places.
- The order-$m$ term of $e^S$ is $O((\tau/d_s)^m)$ and moves the body by at most $m$ places. So if the support of $\phi$ lies at distance $D_c$ from $\{c-1,\dots,c+2\}$, then $p_c=O((\tau/d_s)^{2D_c})$.
4. *Band image of the bulk term.* Since $e^{-S}Ce^S$ commutes with $\Pi$, exactly $\langle\Psi\vert i[C,\hat v]\vert\Psi\rangle=\langle\Psi_0\vert i[H_{\rm eff},V]\vert\Psi_0\rangle$, with $H_{\rm eff}=\Pi e^{-S}Ce^S\Pi$ and $V:=\Pi e^{-S}\hat ve^S\Pi$.
- *$g_s=0$.* $\Pi$ and $C$, hence $e^S$, commute with the joint translation $\mathcal T$, and $\mathcal T\lvert j\rangle\rangle=\lvert j+1\rangle\rangle$. So $e^S$ maps $\sum_je^{iPj}\lvert j\rangle\rangle$ onto the dressed level of label $P$. In that sector $\hat v$ acts as $\partial_PC(P)$: each hop $\lvert x+1\rangle\langle x\rvert$ carries $e^{-iP}$ in $C(P)$ and the factor $-i$ in $\hat v$. By Hellmann–Feynman, with $\mathcal T=e^{-i\hat P}$ on the band,
$$H^{(0)}_{\rm eff}=E_{k,\sigma}(\hat P),\qquad V^{(0)}=E'_{k,\sigma}(\hat P)=\frac{E''_{k,\sigma}(0)}2\sin2\hat P+O(\tau^4/d_s^3).$$
These two commute.
- *Source.* It changes $H_{\rm eff}$ by $\delta H=\sum_jW(j)\lvert j\rangle\rangle\langle\langle j\rvert+O(\tau^2N_f|g_s|/d_s^2)+O(N_f^2g_s^2/d_s)$ (part 2). It changes $V$ by $\delta V=O(\tau^2N_f|g_s|/d_s^2)$, since $\Pi\hat v\Pi=0$ makes $V=O(\tau^2/d_s)$.
- With $\sin2\hat P=\frac1{2i}(\mathcal T^{-2}-\mathcal T^2)$ and real $\phi$, $\langle\Psi_0\vert i[\sum_jW(j)\lvert j\rangle\rangle\langle\langle j\rvert,V^{(0)}]\vert\Psi_0\rangle=-E''\sum_j\phi_{j+1}\phi_{j-1}W'(j)$. Hence
$$a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}W'_{k,\sigma}(j)+\epsilon_{\rm cut}+O\Bigl(\frac{\tau^2N_f^2g_s^2}{d_s^2}\Bigr)+O\Bigl(\frac{\tau^4N_f|g_s|}{d_s^3}\Bigr),\qquad |\epsilon_{\rm cut}|\le L\tau(\tau+N_f|g|)\,p_c,\tag{5.14}$$
with $W'(j):=\frac12[W(j+1)-W(j-1)]$. The remainders come from:
- the $O(g_s^2)$ part of $\mathcal E$ and $[\delta H,\delta V]$: the first;
- $[H^{(0)}_{\rm eff},\delta V]$ ($H^{(0)}_{\rm eff}$ minus a constant is $O(\tau^2/d_s)$), the $\tau^2$ source part of $\delta H$, and the $O(\tau^4/d_s^3)$ part of $V^{(0)}$: the second.
Apart from $\epsilon_{\rm cut}$, every term carries $g_s$, because $[H^{(0)}_{\rm eff},V^{(0)}]=0$.
So (7.7) holds for FG, for the chart mean position. It holds with $E''$ of (5.2), the held weight with $\theta=0$, and the boundary term $\epsilon_{\rm cut}$, which is negligible for a start localized away from the cut. The fall from rest inherits the level dependence of (5.10).
- **(a) and (c).** Parts 3 and 4 use only the isolation of the held band and the fact that $C$ moves the body by at most one place. So for (a) and (c) they give the leading term of (7.7) for the same chart observable. The boundary term is again $-L\langle i[C,v_c]\rangle$, supported at the cut.
- **Remainder.** The FG remainder is set by the FG gap. In R1 the lower branch has $d_s\simeq2n|\gamma|M_k$, so the remainder grows as $\gamma\to0$, unlike the $O(\tau^4\gamma g_s)$ of (7.7).
- **Code** (`check_fall_full_grip.py`). It uses chart labels $0,\dots,L-1$ (cut opposite the start) and a Gaussian envelope of width 1 centred at $L/2-1$. The runs cover $N_f=1$, $L=16,20$, $\gamma M_k=0.1,0.3$, both branches, $\tau/d=0.1,0.05$ and $g_s=10^{-3},2\cdot10^{-3}$.
- The chart acceleration from the double commutator and from the joint evolution agree to $\le7\times10^{-5}$. It agrees with (5.14) to $\le1.8\times10^{-2}$.
- The deviation splits into a part $\propto(\tau/d)^2$ (coefficient $\le1.7$) and a part $\propto g_s$ ($\le2.4\times10^{-3}$ at $g_s=10^{-3}t_f$).
- Relative to the prediction, $|\epsilon_{\rm cut}|\le3\times10^{-9}$ and the bound is $\le1.8\times10^{-4}$.
- For the wider v2 start (width 1.5) at $L=12$, which reaches the cut ($p_c=3.5\times10^{-4}$), $\epsilon_{\rm cut}$ is $-8.3\%$ to $-8.7\%$ of the prediction, while $\langle i[C,\hat v]\rangle$ agrees with it to $2.8\times10^{-3}$. The split $a=\langle i[C,\hat v]\rangle+\epsilon_{\rm cut}$ holds to rounding, and the bound exceeds $|\epsilon_{\rm cut}|$ by a factor $3\times10^4$.
**Controls.**
- **Leading universality.** In the wider class of bodies whose motion term carries a level-dependent factor $A\otimes f(D)$, $E''$ scales with $f(D_k)^2$ while $\Delta W$ is unchanged, so universality fails at leading order.
- **(a), (c) with unequal couplings** of the $m$ and $D$ parts (M7) also fail. For FG such unequal couplings keep $\Lambda=1$.
- **The $O(\gamma)$ dependence.** In the wider class of a body with a position-dependent rest term and no field object (A10), $E''\Delta W$ is exactly universal.
### Step 8. Item 4: the hopping of the field
Take $n,\gamma,M_k,\tau$ fixed (and $L$, unless stated).
- **$v$.** (5.7) does not depend on $t_f$.
- **Isolation, FG.** The lower branch needs only $\tau<\frac12\min(\Delta_b,\varepsilon^-_1)$. For $N_f\ge2$, the upper branch lies below the excited lower-branch levels only if $\varepsilon_1>2n|\gamma|M_k$ (leading order in $\varkappa_k$), that is $t_f>n|\gamma|M_k/(1-\cos\frac{2\pi}L)\simeq N_f|\gamma|M_kL/(2\pi^2)$. In every case $d_{k,\sigma_u}\le\varepsilon^+_1<\varepsilon_1$. R1 needs $t_f\gg N_f|\gamma|M_kL$.
- **Isolation, (a) and (c).** For $R_k>0$ (as at weak mark, $R_k=M_k(1+\gamma n)+O(\gamma^3)$) the $\tau=0$ distances are $d_{k,-}=\min(\varepsilon^{(-)}_1,2R_k)$ and $d_{k,+}=\min\bigl(\varepsilon^{(+)}_1,\min_E|2R_k-E|\bigr)$. Here $\varepsilon^{(\pm)}_1$ is the first excitation energy of $\hat C^{(0)}_\pm$, and $E$ runs over all excitation energies of $\hat C^{(0)}_-$, including $0$.
- At $\gamma\to0$ the $E$ are the sums $\sum_{i=1}^j\varepsilon_{q_i}$ of up to $N_f$ one-object energies. So (6.4) at $\tau=0$ is $2M_k\neq\sum_{i=1}^j\varepsilon_{q_i}$ for $1\le j\le N_f$, up to $O(\gamma)$ shifts.
- (5.2) holds for $\tau\lVert K\rVert<\frac12d_{k,\sigma}$, with $\lVert K\rVert=1$ for (a) and $\le1+|\gamma|N_f/2$ for (c).
- The denominators of (5.9) need only the one-object part, $2M_k\neq\varepsilon_q$, which holds for $\varepsilon_1>2M_k$ or $4t_f<2M_k$. The full condition holds for $\varepsilon_1>2M_k$ or $4N_ft_f<2M_k$. In between it fails on a discrete set of $t_f$, for example $L=6$, $N_f\ge2$, $M_k=3t_f$: $2M_k=3t_f+3t_f$.
- **$L\to\infty$ at fixed $n,\gamma$** (held-block spectra, $g=\gamma M_k$). The block of the branch with $\sigma\gamma<0$ is attractive, with $\varepsilon_0(-|g|)\to-\sqrt{4t_f^2+g^2}$. The other block is repulsive, with $N_f[\varepsilon_0(|g|)+2t_f]=O(nt_f/L)$.
- So the branch with $\sigma\gamma>0$ lies above the other by $2\sigma M_k+N_f(\sqrt{4t_f^2+g^2}-2t_f)+o(N_f)\to\infty$. At large $L$ the branch with $\sigma\gamma<0$ is therefore the lowest level, for either sign of $\gamma$ (for $\gamma<0$ it is $\sigma=+$). It stays isolated, with $d\to\sqrt{4t_f^2+g^2}-2t_f$ (one object unbound).
- The branch with $\sigma\gamma>0$ has a repulsive block whose first excitation tends to $0$. So its $d\to0$, and it leaves the regime.
- **Size of the first level dependence.** It stays at order $\gamma^2$ in $\mu$, with relative size
$$\frac{\tau^2D_{k,\sigma}}{\sigma R_k}=\begin{cases}\sigma\gamma M_k\ell_0/t_f&\text{FG},\\ \sigma n\gamma^2M_k\ell_0/t_f\ \text{(a)},\quad \sigma n\gamma^2M_kL\,(1+O(L^{-1}))/(4t_f)\ \text{(c)},& t_f\gg M_kL^2,\\ n\gamma^2M_k^2\bigl\langle\frac{1+\cos q}{1-\cos q}\bigr\rangle_q/(2t_f^2)\simeq n\gamma^2M_k^2L/(6t_f^2)\ \text{(a), (c)},& t_f\ll M_k,\end{cases}\qquad \gamma M_k\Xi=O\Bigl(\frac{\gamma M_kL}{t_f}\Bigr).\tag{5.11}$$
- **$t_f$ large.** Every level and branch dependence of items 2 and 3 vanishes as $1/t_f$, and the body approaches the mean-field Model D body.
- **$t_f$ small against $M_k$** (with R1 kept by a small $\gamma$). FG keeps (5.8). In (a) and (c) the dependence grows as $1/t_f^2$ and becomes branch-odd.
- **One speed for (a) and $t_f$.** Varying $t_f$ moves $(M_0,M_1)/t_f$ along a ray. So a two-level body has one speed at order $\gamma^2$ at some $t_f$ iff its ray meets a double point of (5.12). By the exact statement under (5.12), this happens only on a parameter set with empty interior. For the double points found numerically, it happens at isolated $t_f$, with a resonance $\varepsilon_q$ between $2M_0$ and $2M_1$.
- **$t_f$ small, FG (order of limits).**
- *Upper branch.* $d_{k,\sigma_u}\le\varepsilon^+_1<\varepsilon_1\simeq4\pi^2t_f/L^2$. So for $t_f\lesssim\tau L^2/(2\pi^2)$ the criterion (5.4) fails and (5.2) is not controlled. This does not show that the exact level is not isolated; the upper branch is not followed there.
- *Lower branch, harmonics.* As $t_f\to0$, $d_{k,\sigma_l}\to|g|$, so (5.4) holds for $\tau<|g|/2$. In the sector contract $P$ enters only through the body hops, $e^{\mp iP}$ each. A term $\cos(DP)$ of the level therefore comes from closed paths with net body displacement $D$:
- $D$ is even, since each hop flips the branch;
- the path has at least $|D|$ body hops;
- it has at least $N_f\,{\rm dist}(D,L\mathbb Z)$ field hops, since every object must return to its relative place modulo $L$.
The $\tau^2$ term ($D=\pm2$) gives $J=O(t_f^{2N_f})$.
- *Lower branch at $t_f=0$.* Only windings $D\in L\mathbb Z$ survive. The lowest is $|D|=L$ (even $L$) or $2L$ (odd $L$) body hops in one direction. Away from the objects the cost is $0$, so the denominators are $-G$, $G:=N_f|\gamma|M_k$; for odd $L$ the path passes the objects once in the upper branch (denominator $-2G$). Hence
$$E''_{k,\sigma_l}(0)\big|_{t_f=0}=\begin{cases}2(-1)^{L/2}L^2\,\dfrac{\tau^L}{2^LG^{L-1}},&L\text{ even},\\[2mm]-4L^2\,\dfrac{\tau^{2L}}{2^{2L}G^{2L-1}},&L\text{ odd},\end{cases}\qquad\times\bigl(1+O(L\tau^2/G^2)\bigr)\tag{5.13}$$
(code: ratio of the exact curvature to (5.13) $0.97$–$0.98$ for $L=5,6,8$ at $\tau=0.1G$, and $0.85$ for $L=7$ at $\tau=0.2G$).
- *Order of limits.* The divergence of the inertia as $t_f\to0$ at finite cost ($\mathrm{Cost}_{k,\sigma_l}\to-N_f|\gamma|M_k$) holds for the coefficient of $\tau^2$, that is with $\tau\to0$ taken first. At fixed $\tau$ it holds while the $\tau^2$ term dominates every other harmonic: $(t_f/|g|)^{2N_f}\gg(\tau/|g|)^{L-2}$ for even $L$, and $(t_f/|g|)^{N_f}\gg(\tau/|g|)^{L-3}$ for odd $L$ (from $D=\pm(L-1)$), up to factors depending on $L$ and $N_f$. Over that range $\mu$ grows as $t_f^{-2N_f}$, and the inertia does not follow the cost.
- At fixed $\tau$ and $t_f\to0$ on a finite ring, the exact curvature tends to (5.13), so the inertia stays finite. It diverges only as $\tau/G\to0$, or as $L\to\infty$ at fixed $N_f$. (Code: $L=6$, $\tau=0.2|g|$, crossover at $t_f\approx0.01|g|$–$0.02|g|$.)
- **$L\to\infty$ at fixed $n$.** $N_f\varkappa_k=n|\gamma|M_kL^2/t_f\to\infty$: R1, the isolation of the branch with $\sigma\gamma>0$ and (5.6)–(5.10) fail. The order-$\gamma^2$ level dependences grow in proportion to $L$ ($\ell_0\simeq L/12$, $\langle(1-\cos q)^{-1}\rangle_q\simeq L/6$). (5.2) and (5.3) still hold for the branch with $\sigma\gamma<0$ (FG: the lower branch; (a), (c): the lowest level at large $L$).
## Result
Class: one Model D body on the ring field of $N_f\ge1$ objects, $L\ge5$, couplings (a), (c) and FG, field state A1, every level $k$ and both branches.
- **(5.1)** The sector contract has the form (6.1) with $\hat C^{(0)}_\sigma=\theta\sigma M_k+C_1+g_\sigma\Phi_0$ and $\hat C^{(+)}=\frac\tau2(-i\sigma_1)\otimes K$. For FG, (6.1) applies unchanged and only $\hat C^{(0)}$ changes ($\theta=0$).
- **(5.2)** Wherever the $\tau=0$ level is isolated at distance $d_{k,\sigma}$: $E''_{k,\sigma}(0)=2\tau^2J_{k,\sigma}+O(\tau^4\lVert K\rVert^4/d^3_{k,\sigma})$, exact in $\gamma$.
- **(5.3)** $\mathrm{Cost}_{k,\sigma}=\sigma R_k+C^{\rm ev}_k$ and, at order $\tau^{-2}$, $\mu_{k,\sigma}=\sigma R_k/[\tau^2(a^2_{k,\sigma}+2\sigma R_kJ^{\rm exc}_{k,\sigma})]$. The inertia is set by the branch-odd part $R_k$ of the held cost. The dressed level at $P=0$ differs from the held level by $O(\tau^2)$, without the factor $1/R_k$.
- **(5.4)** FG isolation: $\tau<\frac12d_{k,\sigma}$, with $d$ as displayed (at leading order in $\varkappa_k$).
- The lower branch ($\sigma\gamma<0$) is isolated for every $\gamma\neq0$, every $L$ and every $N_f$.
- For $N_f=1$ the upper branch is isolated for every $\gamma\neq0$.
- For $N_f\ge2$ the upper level lies below every excited lower-branch level iff $\Delta_b<\varepsilon^-_1$ (in R1: $2n|\gamma|M_k<\varepsilon_1$); beyond, it crosses lower-branch levels at isolated values of $\gamma$.
- $d_{k,\sigma}\to0$ as $\gamma\to0$ (both branches) and as $L\to\infty$ at fixed $n,\gamma$ (upper branch), so (5.4) fails there at every fixed $\tau>0$.
- **(5.5)** Weak-mark expansions of $\varphi_g$, $\varepsilon_0$, $R_k$, $C^{\rm ev}_k=-n\gamma^2M_k^2\ell_0/t_f+O(\gamma^4)$, $a^2$ and $J^{\rm exc}$, through the displayed orders, under R1 (FG) or (6.4) at $\tau=0$ ((a), (c)).
- **(5.6)** Item 1. Under (5.4) and R1, $E''_{k,\sigma}(0)=\frac{\tau^2}{n\sigma\gamma M_k}[1-\gamma^2M_k^2K_L+O(N_f^2\varkappa_k^3)]+O(\tau^4/d^3_{k,\sigma})$, with $\gamma^2M_k^2K_L\simeq(N_f+1)\varkappa_k^2/180$. The leading term is the curvature of a Model D body with half-gap $n\gamma M_k$.
- **(5.7)** Item 2 at leading order. $\mu_{k,\sigma}=\mathrm{Cost}_{k,\sigma}/v^2+O(\gamma^2)$ for every level and branch, with $v=\tau$ for (a) and FG and $v=\tau(1+\gamma n)$ for (c). The held cost enters; the dressed level gives the same answer. $v$ is the slope at $P\to0$ of the band $\pm v\lvert\sin P\rvert$ of the mean-field motion term alone. No level attains it, and it is independent of $t_f$, so it is not fixed by the field band.
- **(5.8)** Item 2 at order $\gamma^2$, FG. $\mu_{k,\sigma}-\mathrm{Cost}_{k,\sigma}/\tau^2=n\ell_0\gamma^2M_k^2/(t_f\tau^2)+O(\gamma^3)$. Branch differences agree through $O(\gamma^2)$, but level differences differ by $n\ell_0\gamma^2(M_k^2-M_{k'}^2)/(t_f\tau^2)$. There is no single $v$ at order $\gamma^2$ (in R1).
- **(5.9)** Item 2 at order $\gamma^2$, (a) and (c), non-resonant class ((6.4) at $\tau=0$). $D_{k,\sigma}=\mu_{k,\sigma}-\mathrm{Cost}_{k,\sigma}/v^2$ is given by (5.9), at leading order in $\tau$. It is checked against the exact $\tau^2$ coefficient at $N_f=1$.
- **(5.12)** Item 2, one speed at order $\gamma^2$, (a) and (c), non-resonant class, leading order in $\tau$. $\tau^2D_{k,\sigma}=n\gamma^2t_f\mathbb F(\sigma M_k/t_f)+O(\gamma^3)$. One $v$ exists at order $\gamma^2$ iff the points $(\pm M_k/t_f,\mathbb F(\pm M_k/t_f))$ of all levels lie on one line; this depends only on $L$ and $M_k/t_f$.
- (c): never, for any member of the class, since the branch average is $\propto M_k^2$ with a positive coefficient and Model D has $N\ge2$ distinct $M_k$.
- (a): the condition is that every pair $(M_k,M_{k'})/t_f$ is a double point of the chord curve $x\mapsto(s(x),i(x))$. The set of ratios satisfying it has empty interior, so one $v$ fails on an open dense subset of the class.
- (a), numerical findings (the search is not shown to be complete): for every $5\le L\le16$, isolated two-level double points are found. Each straddles a one-object resonance $2M_k=\varepsilon_q$, and no ratio is shared among them. For these tuned descriptions only (5.12) and these findings are asked (note (i) of the question); their orders beyond $\gamma^2$ are not settled.
- **(5.10)** Item 3, at first order in $g_s$, for $g_s\neq0$ and places $x,x'$ that are not reflection-equivalent with respect to the source ($F\neq0$). $E''_{k,\sigma}(0)\Delta W_{k,\sigma}=\tau^2F\Lambda[1+\sigma\gamma M_k\Xi+O(\gamma^2)]$, with $\Lambda=1$ (FG), $n\gamma(1\mp\gamma n)$ ((a), (c)), and $\Xi=-(\rho^2+\rho'^2-1)/(3Lt_f)$, $\rho=r-L/2$, $\rho'=r'-L/2$, $r=x_s-x$ and $r'=x_s-x'$ taken in $\{0,\dots,L-1\}$.
- The product is the same for all levels and both branches at leading order.
- If $\Xi\neq0$, the first level and branch dependence is $\sigma\gamma M_k\Xi$ at relative order $\gamma$, of size $O(\varkappa_k)$, with the same $\Xi$ for the three couplings.
- $\Xi=0$ iff $L$ is even and the places are the antipode of the source and a neighbour of it.
- For reflection-equivalent places $\Delta W_{k,\sigma}=0$ exactly.
- **(5.15)** Item 3 at relative order $\gamma^2$, same conditions: the display, with $\Lambda=1$, $n\gamma/(1+\gamma n)$, $n\gamma(1+\gamma n)$. Where $\Xi=0$, the first dependence is:
- FG: the level dependence $(\Psi_0-K_L)\gamma^2(M_k^2-M_{k'}^2)$, nonzero for $N_f=1$, $6\le L\le16$ even;
- (c): the branch dependence $-\sigma\gamma^2M_kn(L-2)(L-4)/(6Lt_f)\neq0$;
- (a): the branch dependence displayed in Step 7, nonzero unless its average vanishes.
- **(5.14)** FG form of (7.7), at first order in $g_s$, for the mean position $\langle\hat x_c\rangle$ in a chart cut at a bond $(c,c+1)$. It holds under $\tau<d_s/2$ and $N_f|g_s|\le d_{k,\sigma}/4$, with $d_s\ge d_{k,\sigma}-2N_f|g_s|$, for a start with real envelope in the dressed band:
- $a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}W'_{k,\sigma}(j)+\epsilon_{\rm cut}+O(\tau^2N_f^2g_s^2/d_s^2)+O(\tau^4N_f|g_s|/d_s^3)$, with $W'$ the central difference of the held weight ($\theta=0$);
- $|\epsilon_{\rm cut}|\le L\tau(\tau+N_f|g|)p_c$, where $p_c$ is the probability of the body at $c-1,\dots,c+2$ in the start. It is $O((\tau/d_s)^{2D_c})$ for an envelope at distance $D_c$ from these places.
The same argument gives the leading term of (7.7) for (a) and (c), for the same observable. The fall from rest inherits (5.10).
- **(5.11)** Item 4.
- $v$ is independent of $t_f$.
- For $N_f\ge2$, the FG upper level lies below the excited lower-branch levels only for $t_f>n|\gamma|M_k/(1-\cos\frac{2\pi}L)$ (leading order in $\varkappa_k$).
- For (a), (c) at weak mark ($R_k>0$), the isolation of $\sigma=+$ is (6.4) with sums of up to $N_f$ one-object energies. At leading order in $\gamma$ it holds for $\varepsilon_1>2M_k$ or $4N_ft_f<2M_k$, and in between it fails on a discrete set of $t_f$.
- The first level dependence stays at order $\gamma^2$, with the relative sizes displayed. It vanishes as $1/t_f$ for large $t_f$, and in (a) and (c) it grows as $1/t_f^2$ for $t_f\ll M_k$.
- For $t_f\lesssim\tau L^2/(2\pi^2)$ the criterion (5.4) fails for the FG upper branch, which is not followed there.
- FG lower branch, $\tau<|g|/2$: the $\tau^2$ coefficient of $E''$ is $O(t_f^{2N_f})$. So with $\tau\to0$ first, $\mu\to\infty$ as $t_f\to0$ at finite cost. At fixed $\tau$ this holds only while $(t_f/|g|)^{2N_f}\gg(\tau/|g|)^{L-2}$ ($L$ even), resp. $(t_f/|g|)^{N_f}\gg(\tau/|g|)^{L-3}$ ($L$ odd), up to factors depending on $L$ and $N_f$.
- At fixed $n$ and $\gamma$, (5.6)–(5.10) have no $L\to\infty$ limit, since R1 fails. (5.2) and (5.3) still hold for the branch with $\sigma\gamma<0$, whose held block is attractive. For (a), (c) this branch becomes the lowest level at large $L$ ($\sigma=+$ for $\gamma<0$) and stays isolated, with $d\to\sqrt{4t_f^2+\gamma^2M_k^2}-2t_f$. The branch with $\sigma\gamma>0$ has $d\to0$.
- **(5.13)** FG lower branch at $t_f=0$, $\tau<|\gamma|M_k/2$, $G=N_f|\gamma|M_k$: the curvature is the winding term displayed, of order $\tau^L$ ($L$ even) or $\tau^{2L}$ ($L$ odd). So at fixed $\tau$ on a finite ring the exact inertia stays finite as $t_f\to0$.
## Consistency checks
1. **$\gamma\to0$ and mean field for (a), (c) against (6.10).** With $a=1+\theta_c\gamma n$, $J^{\rm exc}=0$ and $R_k=M_k(1+\gamma n)$, (5.2)–(5.3) give $\sigma\tau^2/(M_k(1+\gamma n))$ for (a) and $\sigma\tau^2(1+\gamma n)/M_k$ for (c). These are the mean-field parts of (6.10), and both reduce to $\sigma\tau^2/M_k$ at $\gamma=0$. Passed.
2. **FG at mean field against a body with a rest term.** The leading term of (5.6), $\tau^2/(\sigma n\gamma M_k)$, is the curvature at $P=0$ of $\pm(\tilde M^2+\tau^2\sin^2P)^{1/2}$ with $\tilde M=n\gamma M_k$. That body's held cost $\sigma\tilde M$ equals the leading FG cost. Passed.
3. **Dimensions.** $\tau,t_f,M_k,g_s$, costs and $E''$ are contract units, and $\gamma,n,P,\ell,F$ are dimensionless. $K_L\propto t_f^{-2}$, and $\tau^2D/(\sigma R)$ and $\gamma M_k\Xi$ are dimensionless. $E''\Delta W=\tau^2F\Lambda$ has units of contract squared. Passed.
## Open issues
- FG beyond R1 (strong mark), including the lower branch as $L\to\infty$ at fixed $n$: (5.2) holds there, but items 2 and 3 are not evaluated.
- (5.9) and (5.12) are derived at $\tau\to0$ only; the finite-$\tau$ bracket of (6.12) is not evaluated.
- The double points of (5.12) for (a) are known only numerically, for $5\le L\le16$, from a search whose completeness is not established (two grids differ by one point at $L=15$). Their list, the case $L>16$, and whether solutions with $N\ge3$ levels exist are not settled.
- The FG upper branch for $t_f\lesssim\tau L^2/(2\pi^2)$, where (5.4) fails, is not followed.
- The prefactors of the order-of-limits conditions at small $t_f$ are not evaluated.
- (5.14) is checked at $N_f=1$ only. Its remainders and the cut bound are bounds, not leading terms; the cut bound exceeds the computed boundary term by about $3\times10^4$.
- $N_f\ge2$ and the FG upper branch beyond $2n|\gamma|M_k=\varepsilon_1$ are not checked numerically.
- The $O(\gamma)$ term of (5.10) has a control only in a wider class.
- The relation of $v=\tau$ to the maximal speed of 03-ilang-space is not quoted here and not used.
## Code
All scripts are checks. They run with $N_f=1$, and their outputs are in `code/v3/output/`, produced by this version's run.
`code/v3/check_full_grip.py` (unchanged since v1) does the following:
- builds the sector contract (5.1) for FG with $N_f=1$, and verifies it against the full contract at $L=6$ (eigenvalues agree to $4\times10^{-15}$);
- for $L=8,13$, $t_f=1$, $M_k\in\{1,1.4\}$ ($m=1$, $N=2$, $\lambda_0=5\pi/2$), $\gamma\in\{0.01,0.02,0.3\}$ and both branches, computes the exact curvature of the dressed level at $P=0$, by second-order perturbation in $P$ from a full diagonalization, at $\tau=10^{-5}$ and $\tau=d/5$, together with the exact isolation distance $d$;
- compares it with (5.2) and (5.6);
- compares $\tau^2\mu-\mathrm{Cost}$, its level and branch differences, and $E''\Delta W/(\tau^2F)$ with (5.8) and (5.10), and $\Delta W$ with (3.13).
Output: `code/v3/output/check_full_grip.txt`. Results:
- (5.2): agreement to $\le1.4\times10^{-12}$ at $\tau=10^{-5}$ ($\le5\times10^{-10}$ at $\gamma=0.3$, $\varkappa_k\le5.5$); at $\tau=d/5$ to $\le9\times10^{-9}$ ($\gamma\le0.02$) and $\le1.1\times10^{-3}$ ($\gamma=0.3$).
- (5.6): agreement to $\le1.2\times10^{-5}$ for $N_f\varkappa_k\le0.36$, with deviation $\propto\gamma^3$; up to $0.1$ at $\gamma=0.3$ (outside R1).
- (5.8): $\tau^2\mu-\mathrm{Cost}$ within $2.4\%$ (the deviation is branch-odd, $O(\gamma^3)$); the level difference $k=1$ minus $k=0$ within $3.2\%$; $\tau^2(\mu_+-\mu_-)$ equals $\mathrm{Cost}_+-\mathrm{Cost}_-$ to $\le7\times10^{-4}$ of the latter.
- (5.10): the branch-odd part of $E''\Delta W/(\tau^2F)-1$ equals $\gamma M_k\Xi$ to $\le0.2\%$ ($\Xi=-0.167$ for $L=8$, $-0.808$ for $L=13$, places $r=2,3$); the branch-even part is $O(\gamma^2)$; (3.13) reproduces $\Delta W$ to $O(C^3)$.
`code/v3/check_inertia_ac.py` → `check_inertia_ac.txt` (v3: wording of the search corrected; numbers unchanged):
- (5.9) for (a), (c) at $L=7,13$, $M_k\in\{1,1.4\}$, $\gamma\in\{0.01,0.02\}$, both branches, against $\tau^2D$ from the exact $\tau^2$ coefficient $J$ of (5.2). $(\text{exact}-(5.9))/\gamma^3$ is the same at both $\gamma$, between $-1.32$ and $-0.03$. The exact curvature at $\tau=10^{-4}$ equals $2\tau^2J$ to $\le2\times10^{-12}$.
- (5.12): the (c) branch-average coefficient for $5\le L\le16$ (all positive).
- (5.12): the double points of the (a) chord curve found for $5\le L\le16$, by a root search in $(u_0+u_1,u_0u_1)$ on two grids that agree except for one point at $L=15$. The completeness of the search is not established. It reports the resonance between the two ratios and that no ratio is shared.
- At the $L=7$ double point the residual of the four points from a line, divided by $\gamma^2$, is $1.0,2.1,4.2\times10^{-3}$ for $\gamma=0.002,0.004,0.008$, against $0.106$ at $(M_0,M_1)=(1,1.4)$.
`code/v3/check_small_hopping.py` (unchanged) → `check_small_hopping.txt`:
- (5.13) at $t_f=0$ for $L=5,\dots,8$: the ratio of the exact curvature to (5.13) tends to 1 with deviation $\approx(0.3$–$0.6)L\tau^2/G^2$.
- The crossover at $L=6$, $\tau=0.2G$: the exact curvature follows $2\tau^2J$ within $13\%$ (the $O(\tau^2/d^2)$ correction) for $t_f\ge0.05G$ and tends to the winding value as $t_f\to0$; $2\tau^2J=-8\tau^2t_f^2$ at small $t_f$.
`code/v3/check_fall_full_grip.py` (rewritten in v3) → `check_fall_full_grip.txt`:
- (5.14) for the chart position $\hat x_c={\rm diag}(0,\dots,L-1)$ of the body, in the joint description of body, branch and field object with a held source. The start is the dressed band state (direct rotation).
- $a(0)$ comes from the double commutator $-\langle[C,[C,\hat x_c]]\rangle$ and from the second difference of $\langle\hat x_c\rangle$ under the joint evolution.
- It checks the identity $i[C,\hat x_c]=\hat v-Lv_c$ (to $7\times10^{-18}$), the split $a=\langle i[C,\hat v]\rangle+\epsilon_{\rm cut}$, and the cut bound.
- Parameters and results are given in Step 7.
`code/v3/check_weight_next_order.py` (unchanged) → `check_weight_next_order.txt`:
- $F=0$ exactly on the reflection-equivalent pairs for $5\le L\le16$.
- $\Xi$ equals its closed form to $\le8\times10^{-15}$, and $\Xi=0$ pairs occur as stated.
- $\Psi_0-K_L$ at those pairs.
- (5.15) at $L=8$, $(r,r')=(3,4)$ for FG ($M_k=1,1.4$) and for (a), (c) ($M_k=1.2,1.6$, non-resonant): $(\text{exact}-\text{prediction})/\gamma^3$ is the same at $\gamma=0.01,0.02$.
## Methods used
- sector decomposition by joint translation; reflection symmetry
- Rayleigh–Schrödinger perturbation theory; Kato's isolation criterion
- Schrieffer–Wolff (direct rotation) effective band contract; Weyl's inequality; Hellmann–Feynman
- chart position on the ring and oriented-bond velocity; boundary term at the cut
- spectral decomposition of resolvents; Franck–Condon overlaps of product states
- path (hopping) expansion in $\tau$ and $t_f$; winding paths on the ring
- Perron–Frobenius; secular equation of a rank-one perturbation
- Fourier sums on the ring; discrete Poisson equation for $\ell^{(2)}$
- double points of a rational plane curve (numerical root search)
- exact diagonalization; second-order perturbation in $P$; joint evolution (numerical checks)