Version 2 · earlier version; the current one is v4 · External review, round 2: major errors

# Inertia of a body with its own mark against its cost

- **Subproject:** 07-ilang-weight
- **Package:** 05-inertia-of-mark
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-21

## Changes from previous version

- Setup: Schrieffer–Wolff (direct rotation) and Weyl's inequality added to A3; small-$t_f$ statements carry an order of limits (A2) (I1, I2).
- Step 3: "where it fails" now refers to the $\tau=0$ distance $d_{k,\sigma}$ and to the sufficient criterion, not to the exact level (I1).
- Step 4, last paragraph: the one-object denominators are separated from the full many-object condition (6.4) (I4).
- Step 6: the one-speed question at order $\gamma^2$ is settled over the non-resonant class by the line criterion (5.12): (c) is excluded by its branch average, and for (a) the solutions are determined numerically for $5\le L\le16$ (isolated two-level ratios) (I3). (5.9) is now checked numerically.
- Step 7: (5.10) states $F\neq0$ and $\Xi\neq0$; reflection-equivalent pairs are treated; $\Xi$ is given in closed form; the next order (5.15) is given where $\Xi=0$ (I5).
- Step 7: (7.7) for the full grip is derived as (5.14), with a remainder in the full-grip gap $d_s$ and a condition on the source (I2).
- Step 8: isolation of (a), (c) is stated with the full (6.4) and its Kato bound (I4). Small $t_f$: failure of the criterion is separated from non-isolation, the $\tau^2$ coefficient from the exact curvature, with the winding curvature (5.13) and an order of limits (I1).
- Result, Open issues, Code and Methods updated; four new scripts in `code/v2/`.

## Response to verification

- I1: Accepted and fixed in Step 8 ("$t_f$ small", (5.13)) and Step 3. The upper-branch statement is now a failure of the criterion (5.4), not a claim of non-isolation. The divergence of the inertia is restricted to the coefficient of $\tau^2$ ($\tau\to0$ first), with the range of $t_f$ at fixed $\tau$ in which it holds. At fixed $\tau$ the exact curvature tends to the winding value (5.13), which is checked numerically.
- I2: Accepted and fixed in Step 7 ("(7.7) for FG"). The source-perturbed effective band contract and the band position operator are derived through order $\tau^2g_s$. They give (5.14), with remainder $O(\tau^2N_f^2g_s^2/d_s^2)+O(\tau^4N_f|g_s|/d_s^3)$ under $\tau<d_s/2$, $N_f|g_s|\le d_{k,\sigma}/4$. The check uses the joint evolution with a held source.
- I3: Accepted and fixed in Step 6, (5.12). (c): the branch average excludes one speed for every non-resonant member. (a): the equality conditions are solved; solutions exist only for two-level bodies at isolated ratios $M_0/t_f,M_1/t_f$ (searched for $5\le L\le16$).
- I4: Accepted and fixed in Step 8 ("Isolation, (a) and (c)") and Step 4. The full condition (6.4) with sums of up to $N_f$ excitation energies is kept, together with its distance and the bound $\tau\lVert K\rVert<d_{k,\sigma}/2$, and distinguished from the one-object denominators of (5.9).
- I5: Accepted and fixed in Step 7. The section states $F\neq0$ (iff the places are not reflection-equivalent) and $\Xi\neq0$; for reflection-equivalent places $\Delta W=0$ exactly. It derives $\Xi=-(\rho^2+\rho'^2-1)/(3Lt_f)$ and gives the next dependence (5.15) where $\Xi=0$.

## 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 $x\mapsto(s(x),i(x))$.
  - In the variables $(u_0+u_1,\,u_0u_1)$, $u=x^2$, both equations have their poles on straight lines. A search over all roots (code) finds, for every $5\le L\le16$, isolated solutions: 1 for $L=5,6,8$ and up to 21 for $L=15$. For example $L=7$: $(M_0,M_1)/t_f=(0.35551,1.35669)$.
  - Every solution has a resonance value $1-\cos q$ between $M_0/t_f$ and $M_1/t_f$.
  - At this example the exact $\tau^2$ coefficient confirms one speed: the deviation from a common line is $O(\gamma^3)$, against $O(\gamma^2)$ at a generic pair (code).
  - No ratio occurs in two of the solutions found. So the search finds no body with $N\ge3$ levels and one $v$ at order $\gamma^2$ for $5\le L\le16$. (The search runs on two grids, which agree except for one point at $L=15$; its completeness is not proven.)

  So, for $5\le L\le16$, (a) has no single $v$ at order $\gamma^2$, except two-level bodies at isolated ratios $M_0/t_f$, $M_1/t_f$ that straddle a one-object resonance. There $v^2=\tau^2(1+b\gamma^2)$, with $b$ fixed by the common line.

**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 solutions of (5.12).
### 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. 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. *Position.* The body place $\hat x$ is diagonal in places and commutes with $\Pi$. In the band it is $X=\Pi e^{-S}\hat xe^{S}\Pi=\hat j+\frac12\Pi[[\hat x,S],S]\Pi+O(\tau^4/d_s^4)$, since $\Pi[\hat x,S]\Pi=0$. With $S$ at first order, $\langle\langle j'\vert[[\hat x,S],S]\vert j\rangle\rangle=\sum_mS_{j'm}S_{mj}(j+j'-2x_m)$. This vanishes for $j'=j\pm2$ (then $x_m=j\pm1$), and on the diagonal it gives $\xi(j)=w_+(j)-w_-(j)$, $w_\pm(j)=\frac{\tau^2}4\langle\Omega_\sigma(j)\vert R_{j\pm1}^2\vert\Omega_\sigma(j)\rangle$. At $g_s=0$, $\xi=0$ by $\mathcal R$, so $\xi=O(\tau^2N_f|g_s|/d_s^3)$.
4. *Acceleration.* Take the start $\Psi=e^{S}\sum_j\phi_j\lvert j\rangle\rangle$, $\phi_j$ real. Then $a_{k,\sigma}(0)=\frac{d^2}{d\lambda^2}\langle\hat x\rangle\big|_0=-\langle[H_{\rm eff},[H_{\rm eff},X]]\rangle$. For $X=\hat j$ the commutators give $4\sum_jt(j)[U(j+1)-U(j-1)]\phi_{j+1}\phi_{j-1}-\langle[T,[T,\hat j]]\rangle$, with $T$ the hop part; $[T,[T,\hat j]]$ vanishes for $j$-independent $t$. Inserting $t=-E''/8$ and $U(j+1)-U(j-1)=2W'(j)$ at leading order gives

$$a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}W'_{k,\sigma}(j)+O\Bigl(\frac{\tau^2N_f^2g_s^2}{d_s^2}\Bigr)+O\Bigl(\frac{\tau^4N_f|g_s|}{d_s^3}\Bigr),\qquad W'(j):=\tfrac12[W(j+1)-W(j-1)].\tag{5.14}$$

   The remainders come from:
   - the source part of $t$ ($|\partial_{g_s}t|\lesssim\tau^2N_f/d_s^2$) times $W'$ ($\le N_f|g_s|$), and the $O(g_s^2)$ part of $\mathcal E$: the first;
   - the source parts of the $\tau^2$ diagonal, of $t$ inside $[T,[T,\hat j]]$ and of $\xi$, the $O(\tau^4/d_s^3)$ terms of $H_{\rm eff}$, and $E''-2\tau^2J$: the second.

   Every remainder carries $g_s$: at $g_s=0$ the band contract and $X-\hat j$ are translation invariant, so the double commutator vanishes.

So (7.7) holds for FG with $E''$ of (5.2) and the held weight with $\theta=0$, and the fall from rest inherits the level dependence of (5.10). Its 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). The code checks it against the joint evolution with a held source (`check_fall_full_grip.py`). For $N_f=1$, $L=12,16$, $\gamma M_k=0.1,0.3$, both branches, $\tau/d=0.1,0.05$ and $g_s=10^{-3},2\cdot10^{-3}$, it agrees to $\le7.1\times10^{-3}$. The deviation splits into a part $\propto(\tau/d)^2$ (coefficient $\le0.53$) and a part $\propto g_s$ ($\le1.7\times10^{-3}$ at $g_s=10^{-3}t_f$).

**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).** 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 $L=6$, $N_f\ge2$, $M_k=3t_f$: $2M_k=3t_f+3t_f$.
  - At $L\to\infty$ with fixed $n$ the sums become dense in $[0,4N_ft_f]$, so $d_{k,+}\to0$ and the upper branch of (a), (c) 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 $\gamma^2$ for some $t_f$ only if $M_1/M_0$ equals $x_1/x_0$ of a double point of (5.12), and then at an isolated $t_f$ with a resonance $\varepsilon_q$ between $2M_0$ and $2M_1$. For $t_f$ so large that $2M_k<\varepsilon_1$ for all $k$, or so small that $2M_k>4t_f$ for all $k$, there is none (searched range $5\le L\le16$).
- **$t_f$ 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 upper-branch isolation and (5.6)–(5.10) fail, and 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 lower branch.

## 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): for two levels the condition is that $(M_0,M_1)/t_f$ is a double point of the chord curve $x\mapsto(s(x),i(x))$. In the searched range $5\le L\le16$ such points exist for every $L$; they are isolated, and each one found straddles a one-object resonance $2M_k=\varepsilon_q$. The search finds no body with $N\ge3$ levels and one $v$. So for $5\le L\le16$, as far as the numerical search is complete, one $v$ exists at order $\gamma^2$ only for two-level bodies at these isolated ratios.
- **(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), first order in $g_s$, under $\tau<d_s/2$, $N_f|g_s|\le d_{k,\sigma}/4$, $d_s\ge d_{k,\sigma}-2N_f|g_s|$: $a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}W'_{k,\sigma}(j)+O(\tau^2N_f^2g_s^2/d_s^2)+O(\tau^4N_f|g_s|/d_s^3)$, for a start with real envelope in the dressed band, $W'$ the central difference of the held weight ($\theta=0$). The fall from rest thus 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) the upper-branch isolation 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 (R1 fails); (5.2) and (5.3) still hold for the lower branch.
- **(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 solutions of (5.12) for (a) are determined numerically, for $5\le L\le16$ only, by a root search whose completeness is not proven (two grids differ by one point at $L=15$). For $L>16$ their existence and the absence of solutions with $N\ge3$ levels are not shown.
- 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 are bounds, not leading terms.
- $N_f\ge2$ 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/v2/output/`.

`code/v2/check_full_grip.py` (copied from v1, unchanged) 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/v2/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/v2/check_inertia_ac.py` → `check_inertia_ac.txt`:
- (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): all double points of the (a) chord curve for $5\le L\le16$ (root search in $(u_0+u_1,u_0u_1)$ on two grids, which agree except one point at $L=15$). 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/v2/check_small_hopping.py` → `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/v2/check_fall_full_grip.py` → `check_fall_full_grip.txt`: (5.14) from the joint description of body, branch and field object with a held source. The start is the dressed band state (direct rotation), and $a(0)$ comes from the double commutator and from the joint evolution $e^{-iC\lambda}$ (the two agree to all digits). Parameters and agreement are given in Step 7.

`code/v2/check_weight_next_order.py` → `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
- 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)

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