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

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

- **Subproject:** 07-ilang-weight
- **Package:** 05-inertia-of-mark
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-21

## 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 $\tau$.
- **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). Fourier sums on the ring, for example $\sum_{j=1}^{L-1}\sin^{-2}(\pi j/L)=(L^2-1)/3$, and 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. (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 it fails:
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 $n,\gamma$.** $\varepsilon_1\to0$ and the upper branch fails. 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 denominators stay away from zero by R1 ($|2R_k|<\varepsilon_1$) for FG, and by (6.4) at $\tau=0$ ($2M_k\neq\varepsilon_q$) for (a) and (c).

### 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.

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 $M_k^2$; the branch-odd term $b\gamma^2\sigma M_k/\tau^2$ cannot remove them.
- **$t_f\ll M_k$.** Both equal $\sigma\frac{n\gamma^2M_k^3}{2t_f^2\tau^2}\langle\frac{1+\cos q}{1-\cos q}\rangle_q(1+O(t_f/M_k))$. They are branch-odd but proportional to $M_k^3$, which $b\gamma^2\sigma M_k$ removes for at most one value of $M_k$.

So there is no single $v$ at order $\gamma^2$ in either limit. Away from the resonances $2M_k=\varepsilon_q$, $D$ is analytic in $t_f$, so the condition can hold at most at isolated values of $t_f$.

**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).** Control: 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 (a) and (c), $b$ removes the odd part of $D_\sigma$ and the even part is a constant.
### 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).
- **Leading order.** The product is the same for all levels and both branches.
- **First dependence.** At relative order $\gamma$, $\sigma\gamma M_k\Xi$, with one coefficient $\Xi$ for all three couplings. Since $\ell,\ell_0=O(L)$ and $\ell^{(2)}=O(L^3)$, $\lvert\gamma M_k\Xi\rvert=O(\varkappa_k)$.

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.** The premises of (7.7) are:
- a start with real envelope in an isolated band;
- the held weight as the place dependence at first order in $g_s$;
- the coefficient of $\tau^2$.

FG provides all three: by (5.1)–(5.4) the band at order $\tau^2$ has the same form, $E^0_\sigma+\text{const}-\frac{\tau^2}2J\cos2P$, and $W$ is that of (3.12) with $\theta=0$. So (7.7) applies to FG with $E''$ of (5.2), and the fall from rest inherits the level dependence of (5.10).

**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 denominators of (5.9) avoid zero when $\varepsilon_1>2M_k$ or $4t_f<2M_k$.
- **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.
- **$t_f$ small against $\tau$.**
  - The FG upper branch is not isolated, since $d_{k,\sigma_u}<\varepsilon_1\simeq4\pi^2t_f/L^2<2\tau$.
  - The lower branch is isolated for $t_f\ll\tau\ll|\gamma|M_k$. There $J=\langle K^\dagger\Omega_\sigma\vert R\vert K\Omega_\sigma\rangle=O(t_f^{2N_f})$: every object must move from beside the body on one side to the other side, by two hops of amplitude $t_f$. At $t_f=0$, $R$ is diagonal in places and $J=0$.
  - So $\mu\to\infty$ while $\mathrm{Cost}_{k,\sigma_l}\to-N_f|\gamma|M_k$: the inertia does not follow the cost.
- **$L\to\infty$ 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 $\gamma$.
  - Isolation fails as $\gamma\to0$ (both branches) and as $L\to\infty$ at fixed $n,\gamma$ (upper branch).
- **(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). $D_{k,\sigma}$ is given by (5.9), at leading order in $\tau$. In the limits $t_f\gg M_kL^2$ and $t_f\ll M_k$ no change of $v$ makes it level- and branch-independent, so there is no single $v$ at order $\gamma^2$. For other $t_f$ the condition can hold only at isolated values of $t_f$.
- **(5.10)** Item 3, at first order in $g_s$. $E''_{k,\sigma}(0)\Delta W_{k,\sigma}=\tau^2F\Lambda[1+\sigma\gamma M_k\Xi+O(\gamma^2)]$, with $\Lambda=1$ (FG), $n\gamma(1\mp\gamma n)$ ((a), (c)). It is the same for all levels and both branches at leading order. 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. (7.7) applies to FG with $E''$ of (5.2).
- **(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 $\varkappa_k$).
  - 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 $t_f\ll\tau$ the FG upper branch is not isolated. The FG lower branch is isolated for $\tau\ll|\gamma|M_k$, and there $\mu\to\infty$ as $t_f\to0$ at finite cost.
  - 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.

## 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

- The error term of (7.7) for FG is not re-derived.
- 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) is derived at $\tau\to0$ only; the finite-$\tau$ bracket of (6.12) is not evaluated, and (5.9) is not checked numerically.
- $N_f\ge2$ 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

`code/v1/check_full_grip.py` (check) 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/v1/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 $O(C^3)$.

## Methods used

- sector decomposition by joint translation; reflection symmetry
- Rayleigh–Schrödinger perturbation theory; Kato's isolation criterion
- spectral decomposition of resolvents; Franck–Condon overlaps of product states
- Perron–Frobenius; secular equation of a rank-one perturbation
- Fourier sums on the ring
- exact diagonalization; second-order perturbation in $P$ (numerical check)

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