Version 1 · earlier version; the current one is v3 · Review of v1

# The own mark of a body: dressed levels, clock spacings, inertia at second order

- **Subproject:** 06-ilang-source
- **Package:** 06-own-mark
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-19

## Setup and assumptions

**Field** (assumption, M8; the class of 04-profile). A ring of places $x\in\mathbb Z_L$, $L\ge3$; $N_f\ge1$ identical field objects with $c=+1$, no pair terms among them, own terms $C_1=-t_f\sum_x(\lvert x\rangle\langle x+1\rvert+\lvert x+1\rangle\langle x\rvert)$ with $t_f>0$ (assumption of the class); $\Phi_x$ is the number of field objects at $x$, $n:=N_f/L$, $C_f$ the sum of the own terms. One object has the eigenvectors $\lvert q\rangle=L^{-1/2}\sum_xe^{iqx}\lvert x\rangle$, $q\in\frac{2\pi}L\mathbb Z_L$, with eigenvalues $-2t_f\cos q$. In $\mathcal H_{\rm adm}$ (symmetric states) the field eigenstates are occupation states. The lowest one, $\lvert\Omega_f\rangle$ (all objects at $q=0$), is unique, with eigenvalue $-2t_fN_f$. An excited state has the label $Q=\sum_jq_j$ and the excitation energy $E_{\rm exc}=\sum_j\varepsilon_{q_j}>0$, where $\varepsilon_q:=2t_f(1-\cos q)\in(0,4t_f]$ for $q\ne0$.

**Body** (assumption). One Model D body on the same ring, with $\tau>0$, $m>0$, clock levels $\lvert f_k\rangle$, $M_k=m+d_k$ and $\delta_c:=M_{k+1}-M_k=\frac{2\pi}{N\lambda_0}$. The couplings (a) and (c), with constant $\gamma$, are those of the question. Records are taken at $\kappa\to0$. For $\kappa>0$ the record term $\kappa xX$ is not invariant under the joint translation of the ring, so the sectors below do not exist; this case is not treated.

**Inputs** (quoted). (5.5) and (5.6) of 04-ilang-time/05-moving-clock, with normalized plane waves $\lvert p\rangle\to L^{-1/2}\lvert p\rangle$:
$$C_k(p)=-\tau\sin p\,\sigma_1+\mu_k\sigma_3,\quad \omega_k(p)=\sqrt{\mu_k^2+\tau^2\sin^2p},\quad \beta_{k,+}=\begin{pmatrix}\cos\frac{\theta_k}2\\ \sin\frac{\theta_k}2\end{pmatrix},\ \beta_{k,-}=\begin{pmatrix}-\sin\frac{\theta_k}2\\ \cos\frac{\theta_k}2\end{pmatrix},\ \tan\theta_k=-\frac{\tau\sin p}{\mu_k},$$
with $\mu_k=M_k$. (6.7) of 04-ilang-time/06-inertia-internal-energy: $M_k=m+d_k$, gap $2M_k$, $\omega_k''(0)=\tau^2/M_k$. (4.2) of 04-profile: $E(g)=N_f\,\varepsilon_0(g)$, where $\varepsilon_0(g)$ is the lowest eigenvalue of the one-object contract $h(g)$ with the held sources.

**Notation.** $M_\sigma:=\sigma M_k$, $b_q:=\tau\lvert\sin q\rvert$, $K_\sigma(q):=\varepsilon_q(2M_\sigma-\varepsilon_q)+b_q^2$, and $S_0:=\sum_{q\ne0}\varepsilon_q^{-1}=\frac{L^2-1}{12t_f}$ (standard sum $\sum_{j=1}^{L-1}(1-\cos\frac{2\pi j}L)^{-1}=\frac{L^2-1}6$).

**Regime** (A13). Expansion in $\gamma$ at fixed $L,N_f,t_f,\tau,m,N,\lambda_0$; every $O(\gamma^3)$ is meant at fixed $L$. We use perturbation theory for an isolated eigenvalue, with its conditions stated in Step 3.

## Derivation

### Step 1. Sectors, continuation in $P$, invariance

Let $T$ shift the position label of the body and of every field object by $+1$, and let $Q_k=\lvert f_k\rangle\langle f_k\rvert$ act on the clock label. Both operators are built from places alone, commute with every swap of field objects, and commute with $C$. Every term of $C$ is invariant under a common shift on the ring, and the clock label enters $C$ only through $D$ (in $B$ and in the couplings), which is diagonal in $\lvert f_k\rangle$. The sector $(k,P)$ is $Q_k=1$, $T=e^{-iP}$, $P\in\frac{2\pi}L\mathbb Z_L$, inside $\mathcal H_{\rm adm}$; it contains the start $(p=P,\sigma,k)\otimes\lvert\Omega_f\rangle$.

Every matrix element of $C$ either keeps the body label $x_b$ or changes it by $\pm1$: the motion term does, and so does the bond factor of (c). For $L\ge3$ these three classes are distinct, so the split $C=C^{(0)}+C^{(+)}+C^{(-)}$ by the change of $x_b$ is fixed by $C$ alone. On the states $L^{-1/2}\sum_xe^{iPx}\lvert x,s,f_k\rangle\otimes\lvert x+\vec r\rangle$ (relative field positions $\vec r$, symmetrized), a step $x_b\to x_b+1$ at fixed field positions lowers every $r_j$ by one and gives a factor $e^{-iP}$. Hence
$$C_k(P)=\hat C^{(0)}+e^{-iP}\hat C^{(+)}+e^{iP}\hat C^{(+)\dagger},\tag{6.1}$$
where the $\hat C$ are independent of $P$ and act on (branch)$\otimes$(relative field configurations). For real $P$, (6.1) is a Hermitian family analytic in $P$, and at $P\in\frac{2\pi}L\mathbb Z_L$ it is $C$ on the sector $(k,P)$. The quantities $E_{k,\sigma}(P)$ and $E''_{k,\sigma}(0)$ refer to its eigenvalue branch.

**Invariance (M6, Law 5, (1.2)).**
- The sectors are fixed by places, types and families. The matrix (6.1) is fixed by the matrix elements of the total contract, and the dressed level by the family $C(\gamma)$ and its limit at $\gamma\to0$.
- Any replacement of listed terms that leaves $C(\gamma)$ unchanged therefore leaves unchanged: the eigenvalues of (6.1), their differences between levels, their $P$-derivatives, and their Taylor coefficients in $\gamma$.
- Relabelling the field objects maps each sector onto itself, and eigenvalues do not see the common phase.
- The splits used below (body, field and coupling; mean field and fluctuation) are only means of computation. For example, $\gamma M_k\sigma_3\Phi_{x_b}=\gamma nM_k\sigma_3+\gamma M_k\sigma_3(\Phi_{x_b}-n)$ moves a term diagonal in the body's places between a pair term and an own term (A5). This changes the body's "rest term" but not $C$. Only eigenvalue data of $C$ are reported.
- The parameters in the results are eigenvalue data of $C(0)$:
  - $M_k=\frac12[E_{k,+}(0)-E_{k,-}(0)]_{\gamma=0}$ and $\delta_c=M_{k+1}-M_k$;
  - $\tau^2=M_kE''_{k,+}(0)|_{\gamma=0}$;
  - $\varepsilon_q$ are the field's excitation energies;
  - $n$ counts objects and places.

### Step 2. The contract on a clock level; mean field and vertex

On $Q_k=1$, $D$ acts as $d_k$, so
$$\begin{aligned}\text{(a)}:\ C\big|_k&=A+M_k\sigma_3\textstyle\sum_x\lvert x\rangle\langle x\rvert(1+\gamma\Phi_x)+C_f,\\ \text{(c)}:\ C\big|_k&=\text{(a)}+\gamma\tfrac\tau2\textstyle\sum_x\bigl(\lvert x+1\rangle\langle x\rvert(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert(i\sigma_1)\bigr)\tfrac12(\Phi_x+\Phi_{x+1}).\end{aligned}\tag{6.2}$$
The level enters only through $M_k$. Hence, exactly and at every order in $\gamma$, $E_{k,\sigma}(P)=\mathcal E_\sigma(P;M_k)$ with one function $\mathcal E_\sigma$ for all levels. A clock level is dressed like a body without clock label whose $m$ is replaced by $M_k$.

**Mean field.** Write $\Phi_x=n+\delta\Phi_x$, and let $\gamma W$ be the part of (6.2) that contains $\delta\Phi_x$. The mean-field part is $A+(1+\gamma n)M_k\sigma_3+C_f$ for (a) and $(1+\gamma n)(A+M_k\sigma_3)+C_f$ for (c). By (5.5)–(5.6) of 04-ilang-time/05-moving-clock its body eigenvalues are $\sigma\tilde\omega_k(p)$, with $\tilde\omega_k^{(a)}=[(1+\gamma n)^2M_k^2+\tau^2\sin^2p]^{1/2}$ and $\tilde\omega_k^{(c)}=(1+\gamma n)\omega_k(p)$.

**Vertex.** $\Phi_x$ is a sum of one-object operators. It has $\langle\Omega_f\vert\delta\Phi_x\vert\Omega_f\rangle=0$ and $\delta\Phi_x\lvert\Omega_f\rangle=\frac{\sqrt{N_f}}L\sum_{q\ne0}e^{-iqx}\lvert q\rangle_f$, where $\lvert q\rangle_f$ is the normalized symmetric state with one object at $q$ (excitation $\varepsilon_q$). The sum over $x$ enforces $p'=P-q$. For (c), the bond factor gives $\frac\tau2\cdot\frac{1+e^{-iq}}2\cdot i(e^{iP}-e^{-i(P-q)})\sigma_1=-\frac\tau2(\sin P+\sin(P-q))\sigma_1$. With (5.5) this yields
$$\langle P-q,\sigma';q\vert\,\gamma W\,\vert P,\sigma;\Omega_f\rangle=\gamma\frac{\sqrt{N_f}}L\langle\beta_{\sigma'}(P-q)\vert\hat V_q\vert\beta_\sigma(P)\rangle,\qquad \hat V_q=\begin{cases}M_k\sigma_3&\text{(a)}\\ \tfrac12[C_k(P)+C_k(P-q)]&\text{(c)},\end{cases}\tag{6.3}$$
and all other matrix elements of $W$ from $\lvert P,\sigma;\Omega_f\rangle$ vanish.

### Step 3. Existence and isolation

At $\gamma=0$ the eigenvalues of the sector $(k,P)$ are $\sigma'\omega_k(P-Q)+E_{\rm exc}-2t_fN_f$. The dressed level starts at $\sigma\omega_k(P)-2t_fN_f$, and it is a simple eigenvalue iff
$$\sigma\omega_k(P)\neq\sigma'\omega_k(P-Q)+E_{\rm exc}\quad\text{for every excited field eigenstate }(Q,E_{\rm exc})\text{ and }\sigma'=\pm\tag{6.4}$$
(the case $\lvert\Omega_f\rangle$, $\sigma'=-\sigma$ is excluded by $\omega_k\ge M_k>0$). Under (6.4) the eigenvalue of the analytic Hermitian family (6.1), (6.2) is simple and isolated. By Rellich's theorem the dressed level exists, is analytic in $\gamma$ and $P$, and has the Rayleigh–Schrödinger expansion. This holds for $\lvert\gamma\rvert$ small against the distance of the level from the rest of the sector spectrum, divided by the vertex scale $M_k(n/L)^{1/2}$ of (6.3). That distance is at most $\varepsilon_{2\pi/L}+\omega_k(2\pi/L)-M_k=O(L^{-2})$: consider the state with the body at $-2\pi/L$ in the same branch and one field object at $2\pi/L$. The range therefore shrinks as $L$ grows. A level that is simple at $P=0$ stays simple for real $P$ near $0$, which is all that $E''(0)$ needs.

At $P=0$:
- **$\sigma=+$.** $\omega_k(Q)+E_{\rm exc}>M_k$ for every excited state, so only $\sigma'=-$ can coincide, at $E_{\rm exc}=M_k+\omega_k(Q)\ge2M_k$. Since $E_{\rm exc}\le4t_fN_f$, (6.4) holds if $2t_fN_f<M_k$. The one-object excitations, the only ones that enter the second order, stay below the gap if $2t_f<M_k$. When field excitations reach $M_k+\omega_k(Q)$, the level lies among states "lower branch plus excited field". On the finite ring it is then isolated only away from exact coincidences, with a range in $\gamma$ set by the nearest one. For $L\to\infty$ these states become dense (decay through the field); this case is not treated.
- **$\sigma=-$.** Only $\sigma'=-$ can coincide, at $E_{\rm exc}=\omega_k(Q)-M_k<(M_k^2+\tau^2)^{1/2}-M_k$. So (6.4) holds if $\varepsilon_{2\pi/L}>(M_k^2+\tau^2)^{1/2}-M_k$. For one-object excitations it holds if $\tau^2\le2t_fM_k$, since $\omega_k(q)-M_k<\tau^2\sin^2q/(2M_k)\le\varepsilon_q\tau^2/(2t_fM_k)$ when $\sin q\ne0$, and $\omega_k(q)-M_k=0<\varepsilon_q$ when $\sin q=0$.

For $N_f=1$ the one-object excitations are all the excitations.

### Step 4. The dressed level through second order

We take the mean-field contract as unperturbed and $\gamma W$ as the perturbation. The first order vanishes. The second order involves only $\lvert P-q,\sigma';q\rangle$, $q\ne0$, with denominators $\sigma\omega_k(P)-\sigma'\omega_k(P-q)-\varepsilon_q$; bare values suffice, since mean-field values change only $O(\gamma^3)$. The sum over $\sigma'$ uses $\sum_{\sigma'}\frac{\lvert\beta_{\sigma'}\rangle\langle\beta_{\sigma'}\rvert}{z-\sigma'\omega'}=(z-C_k(p'))^{-1}=\frac{z+C_k(p')}{z^2-\omega'^2}$, which holds because $C_k(p')^2=\omega_k(p')^2$. The result is
$$E_{k,\sigma}(P)=\sigma\tilde\omega_k(P)-2t_fN_f+\gamma^2\frac nL\sum_{q\ne0}G_{k,\sigma}(P,q)+O(\gamma^3),\qquad G_{k,\sigma}=\langle\beta_\sigma(P)\vert\hat V_q\,(z_q-C_k(P-q))^{-1}\hat V_q\vert\beta_\sigma(P)\rangle,\tag{6.5}$$
with $z_q=\sigma\omega_k(P)-\varepsilon_q$. To make $G$ explicit, set $\Omega:=\sigma\omega_k(P)$, $\omega'^2:=M_k^2+\tau^2\sin^2(P-q)$ and $s_\pm:=M_k^2\pm\tau^2\sin P\sin(P-q)$. From (5.6), $\langle\sigma_1\rangle=-\tau\sin P/\Omega$ and $\langle\sigma_3\rangle=M_k/\Omega$ in $\beta_\sigma(P)$. This gives $\langle\sigma_3C_k(p')\sigma_3\rangle=s_-/\Omega$ and $\langle C_k(p')\rangle=s_+/\Omega$. For (c), $\hat V_q\beta_\sigma(P)=\frac12(\Omega+C_k(p'))\beta_\sigma(P)$, and functions of $C_k(p')$ commute. Hence
$$G^{(a)}=\frac{M_k^2(z_q+s_-/\Omega)}{z_q^2-\omega'^2},\qquad G^{(c)}=\frac{z_q(\Omega^2+\omega'^2)+2\Omega\omega'^2+(s_+/\Omega)(\Omega^2+\omega'^2+2\Omega z_q)}{4(z_q^2-\omega'^2)}.$$
$G$ depends on $\sigma$ only through $\Omega$, and $\sigma\tilde\omega_k$ likewise. The $\sigma=-$ results therefore follow from the $\sigma=+$ ones by $M_k\to-M_k$, with $\omega_k(P)$ read as $M_k(1+\tau^2\sin^2P/M_k^2)^{1/2}$.

At $P=0$ we have $\Omega=M_\sigma$ and $z_q^2-\omega'^2=-K_\sigma(q)$:
$$G^{(a)}_{k,\sigma}(0,q)=-\frac{M_k^2(2M_\sigma-\varepsilon_q)}{K_\sigma(q)},\qquad G^{(c)}_{k,\sigma}(0,q)=-\frac{M_k^2(2M_\sigma-\varepsilon_q)+b_q^2(M_\sigma-\varepsilon_q/4)}{K_\sigma(q)}.\tag{6.6}$$
For both couplings $\sigma\tilde\omega_k(0)=(1+\gamma n)M_\sigma$ exactly. The $\gamma^2$ coefficient of $E_{k,\sigma}(0)$, the cost of the own mark, is therefore $\Delta_{k,\sigma}:=\gamma^2\frac nL\sum_{q\ne0}G_{k,\sigma}(0,q)$.

- **Sign, $\sigma=+$.** If $2t_f<M_k$, then $2M_k-\varepsilon_q>0$, so $K_+>0$ and both numerators are positive: $\Delta_{k,+}<0$, for every $\tau$.
- **Sign, $\sigma=-$.** If $\tau^2\le2t_fM_k$, then $K_-=-[(M_k+\varepsilon_q)^2-\omega_k(q)^2]<0$ (Step 3), while $M_k^2(2M_k+\varepsilon_q)$ and $M_k^2(2M_k+\varepsilon_q)+b_q^2(M_k+\varepsilon_q/4)$ are positive: $\Delta_{k,-}<0$.
- **Dependences.** $\Delta$ is linear in $N_f$ at fixed $L$ (the squared vertex is proportional to $N_f$, the denominators do not depend on it). It depends on $k$ only through $M_k$ and on $\sigma$ through $M_\sigma$; $t_f$ and $L$ enter through $\varepsilon_q$ and the set of labels.

**Held body, $\tau=0$.** The body label is conserved, and the field sees the held source $g=\sigma\gamma M_k$ at $x_b$. $\varepsilon_0$ is simple (Perron–Frobenius, $t_f>0$), hence analytic in $g$. By (4.2) of 04-profile,
$$E_{k,\sigma}(P)\big|_{\tau=0}=\sigma M_k+N_f\,\varepsilon_0(\sigma\gamma M_k)\ \ \text{for all }P\ \text{and all }\gamma,\qquad N_f\varepsilon_0(g)=-2t_fN_f+gn-g^2\tfrac nLS_0+O(g^3).\tag{6.7}$$
The second expression uses $\lvert\langle q\vert x_b\rangle\langle x_b\vert0\rangle\rvert^2=L^{-2}$. Hence $\Delta_{k,\sigma}|_{\tau=0}=-\gamma^2M_k^2N_f(L^2-1)/(12t_fL^2)$ for both branches and both couplings. It is proportional to $M_k^2$ and to $1/t_f$, and at fixed $n$ it grows like $L$.

### Step 5. Clock spacings

Let $G_+(M,q)$ denote $G_{k,+}(0,q)$ of (6.6) as a function of $M_k=M$ (by (6.2)). Then
$$E_{k+1,+}(0)-E_{k,+}(0)=(1+\gamma n)\,\delta_c+\gamma^2\frac nL\sum_{q\ne0}\bigl[G_+(M_{k+1},q)-G_+(M_k,q)\bigr]+O(\gamma^3).\tag{6.8}$$
The first order is a common rescaling by $1+\gamma n$ for both couplings. The second-order change is a common rescaling iff $F(M):=\sum_qG_+(M,q)$ is affine on $\{M_k\}$; this is a condition only for $N\ge3$, since $N=2$ has a single spacing.

At fixed $\varepsilon=\varepsilon_q$, $b=b_q$, set $K=\varepsilon(2M-\varepsilon)+b^2$. Then $G^{(a)}_+=-\frac{M^2}\varepsilon+\frac{M^2b^2}{\varepsilon^2(2M+a)}$ with $a:=\frac{b^2}\varepsilon-\varepsilon$, $\frac{M^2}{2M+a}=\frac M2-\frac a4+\frac{a^2}{4(2M+a)}$, and $G^{(c)}_+=G^{(a)}_+-\frac{b^2}{2\varepsilon}\bigl[1+\frac{\varepsilon^2/2-b^2}K\bigr]$. Differentiating twice in $M$:
$$\partial_M^2G^{(a)}_+=-\frac2\varepsilon\Bigl[1-\frac{b^2(b^2-\varepsilon^2)^2}{K^3}\Bigr],\qquad \partial_M^2G^{(c)}_+=-\frac2\varepsilon\Bigl[1-\frac{b^6}{K^3}\Bigr].\tag{6.9}$$
- **(c).** $K>b^2$ whenever $\varepsilon<2M$, so $\partial_M^2G^{(c)}_+<0$.
- **(a).** If $\varepsilon\le M$, then $K>\varepsilon^2$ and $K>b^2$. For $b\ge\varepsilon$, $b^2(b^2-\varepsilon^2)^2<b^6<K^3$; for $b<\varepsilon$, $b^2(\varepsilon^2-b^2)^2<b^2\varepsilon^4<K^3$. So $\partial_M^2G^{(a)}_+<0$.

Hence $F$ is strictly concave for $M\ge m$ if $\varepsilon_q<2m$ for all $q$ (c), sufficient $2t_f<m$, or if $\varepsilon_q\le m$ for all $q$ (a), sufficient $4t_f\le m$. The second-order change of the spacing, $\gamma^2[F(M_{k+1})-F(M_k)]$, then strictly decreases with $k$. It is not $\delta_c$ times one factor for all $k$. At $\tau=0$ the change is $-\gamma^2\frac nLS_0\,\delta_c(M_k+M_{k+1})$ by (6.7).

### Step 6. Curvature and inertia

$\sigma\tilde\omega''_k(0)$ is $\sigma\tau^2/((1+\gamma n)M_k)$ for (a) and $\sigma(1+\gamma n)\tau^2/M_k$ for (c). Differentiating (6.5) twice at $P=0$, where $G$ is analytic under (6.4), gives
$$E''_{k,\sigma}(0)=\sigma\frac{\tau^2}{M_k}\times\begin{cases}1-\gamma n+\gamma^2n^2&\text{(a)}\\ 1+\gamma n&\text{(c)}\end{cases}+\Delta''_{k,\sigma}+O(\gamma^3),\qquad \Delta''_{k,\sigma}:=\gamma^2\frac nL\sum_{q\ne0}\partial_P^2G_{k,\sigma}(P,q)\big|_{P=0}.\tag{6.10}$$

**Leading order in $\tau^2$.** Take (a), $\sigma=+$, and expand to first order in $\tau^2$ and second order in $P$:
- $z_q+s_-/\Omega=2M-\varepsilon+Z$, with $Z=\frac{\tau^2P^2}{2M}-\frac{\tau^2}M[\sin P\sin(P-q)+\frac{P^2}2]$;
- $z_q^2-\omega'^2=-K_0+Y$, with $K_0:=\varepsilon(2M-\varepsilon)$ and $Y=\frac{\tau^2P^2(M-\varepsilon)}M-\tau^2\sin^2(P-q)$.

So $G=-\frac{M^2}\varepsilon-\frac{M^2}{K_0}[Z+Y/\varepsilon]$, with $Z''=-2\tau^2c/M$, $Y''=2\tau^2\frac{M-\varepsilon}M-2\tau^2\cos2q$, $c:=\cos q$, and $4\sin^2q/\varepsilon_q=2(1+c)/t_f$. For (c), the same expansion turns the numerator of $G^{(c)}$ into $4M^2(2M-\varepsilon)+4Mu(3M-\varepsilon)+w(4M-\varepsilon)+2Mv(2M-\varepsilon)$, with $u=\frac{\tau^2P^2}{2M}$, $w=\tau^2\sin^2(P-q)$, $v=\frac{\tau^2}M[\sin P\sin(P-q)-\frac{P^2}2]$. With $\varepsilon=\varepsilon_q$:
$$\partial_P^2G^{(a)}_{k,+}\big|_0=\frac{2M_k\tau^2(1+c)(1-M_k/t_f)}{\varepsilon(2M_k-\varepsilon)},\qquad \partial_P^2G^{(c)}_{k,+}\big|_0=-\frac{\tau^2(1+c)\bigl[2M_k^2/t_f+2M_k(2c-1)-\varepsilon c\bigr]}{\varepsilon(2M_k-\varepsilon)}.\tag{6.11}$$
The relative corrections are $O(\tau^2/M_k^2)$ and $O(\tau^2/(M_kt_fq_1^2))$ with $q_1=2\pi/L$; the latter comes from the term $2\tau^2\sin q\cos q\,P$ of the denominator. For $\sigma=-$, replace $M_k\to-M_k$.

**Sign for $2t_f<M_k$.** Every term of (6.11) is $\le0$, and $<0$ for $c\ne-1$:
- (a): because $1-M_k/t_f<0$;
- (c): set $\mu:=M_k/t_f>2$. The bracket divided by $t_f$ is $2\mu^2+2\mu(2c-1)-2c(1-c)$. It increases with $c$ on $[-1,1]$ (its derivative is $\ge4\mu-6>0$) and equals $2(\mu-1)(\mu-2)>0$ at $c=-1$.

So $\Delta''_{k,+}<0$ at leading order in $\tau^2$.

**Inertia against cost.** $E_{k,+}(0)$ contains the field's eigenvalue $-2t_fN_f$, so the comparison is between changes. Write $\delta X:=X(\gamma)-X(0)$, a difference of eigenvalue data of two total contracts of the family, which M6 allows. The base value is $1/E''_{k,+}(0)|_{\gamma=0}=M_k/\tau^2$ ((6.7) of 04-ilang-time/06-inertia-internal-energy). Then $\delta E_{k,+}(0)=\gamma nM_k+\Delta_{k,+}+O(\gamma^3)$, while $\delta(1/E''_{k,+}(0))$ equals $\frac{\gamma nM_k}{\tau^2}-\frac{M_k^2}{\tau^4}\Delta''_{k,+}$ for (a) and $\frac{M_k}{\tau^2}(-\gamma n+\gamma^2n^2)-\frac{M_k^2}{\tau^4}\Delta''_{k,+}$ for (c), up to $O(\gamma^3)$. Hence
$$\delta\Bigl(\frac1{E''_{k,+}(0)}\Bigr)-\frac{\delta E_{k,+}(0)}{\tau^2}=\begin{cases}-\dfrac1{\tau^2}\Bigl[\dfrac{M_k^2}{\tau^2}\Delta''_{k,+}+\Delta_{k,+}\Bigr]&\text{(a)}\\[2mm] \dfrac{M_k}{\tau^2}(-2\gamma n+\gamma^2n^2)-\dfrac1{\tau^2}\Bigl[\dfrac{M_k^2}{\tau^2}\Delta''_{k,+}+\Delta_{k,+}\Bigr]&\text{(c)}\end{cases}\ +O(\gamma^3).\tag{6.12}$$
- **(a).** The first-order parts agree. The own-mark parts do not: for $2t_f<M_k$, $\Delta_{k,+}<0$ (Step 4), and $-\frac{M_k^2}{\tau^4}\Delta''_{k,+}>0$ at leading order in $\tau^2$. The own mark lowers the cost and raises the inertia, so the bracket is negative.
- **(c).** The two sides differ already at first order. The uniform part multiplies the motion and rest terms alike: the curvature grows by the factor $1+\gamma n$, while the cost grows by $\gamma nM_k$.
- **Heavy body.** For $M_k\gg t_f$, at leading order in $\tau^2$ and for both couplings: $-\frac{M_k^2}{\tau^4}\Delta''_{k,+}\approx\frac{\gamma^2nM_k^3(L-1)(L-2)}{6\tau^2t_f^2L}$, using $\sum_{q\ne0}\frac{1+\cos q}{\varepsilon_q}=\frac{(L-1)(L-2)}{6t_f}$. By contrast $\Delta_{k,+}/\tau^2\approx-\frac{\gamma^2nM_k^2(L^2-1)}{12\tau^2t_fL}$. The relative changes therefore grow like $+M_k^2$ for the inertia and $-M_k$ for the cost.

By (6.2), for (a) and (c), $1/E''_{k,\sigma}(0)$ and $\delta E_{k,\sigma}(0)$ of every level are those of a clockless body with rest parameter $M_k$: one function of $M_k$ for all levels.

### Step 7. Controls

**(i) $\tau\to0$.** (6.7) holds for both couplings: $E''_{k,\sigma}\equiv0$ at every $\gamma$, so the curvature inertia is not defined, with or without the mark. Item 3 is void, and the mark changes only the cost. The spacings are $\delta_c+N_f[\varepsilon_0(\gamma M_{k+1})-\varepsilon_0(\gamma M_k)]$. $\varepsilon_0$ is strictly concave: $\varepsilon_0''=2\sum_{j\ne0}\lvert\langle j\vert x_b\rangle\langle x_b\vert\varphi_0\rangle\rvert^2/(\varepsilon_0-\varepsilon_j)<0$, because the lowest eigenvector $\varphi_0$ of $h(g)$ has positive components and so is not $\lvert x_b\rangle$. Hence, at every order, the spacing changes strictly decrease with $k$: there is no common rescaling for $N\ge3$.

**(ii) Rest-only coupling.** Replace the (a) term by $\gamma\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes m\mathbb 1_N\otimes\Phi_x$ (a′), and add the bond term of (c) to get (c′). The level-$k$ contract now depends on $M_k$ and on $m$ separately, so the analogue of (6.2) fails. At $P=0$ the mean field gives $M_k+\gamma nm$, and the vertex is $m\sigma_3$ (a′) or $m\sigma_3+\frac\tau2\sin q\,\sigma_1$ (c′). Repeating Step 4 gives
$$G^{(a')}_{k,+}(0,q)=-\frac{m^2}{\varepsilon_q}+\frac{m^2b_q^2}{\varepsilon_qK_+(q)},\qquad G^{(c')}_{k,+}(0,q)=-\frac{m^2}{\varepsilon_q}+\frac{(m-\varepsilon_q/2)^2b_q^2}{\varepsilon_qK_+(q)}.\tag{6.13}$$
For $2t_f<m$, $1/K_+$ is positive, strictly decreasing and strictly convex in $M_k$.
- **Spacings.** They are unchanged at first order. At $\tau=0$ they are unchanged at every order: the shift $N_f\varepsilon_0(\gamma m)$ is the same for all levels. For $\tau>0$ their second-order changes are negative and strictly increasing in $k$, so there is no common factor for $N\ge3$.
- **Inertia, (a′).** $G^{(a')}=(m^2/M_k^2)G^{(a)}$ at every $P$, and the mean field gives $1/E''=(M_k+\gamma nm)/\tau^2$. So (6.12) holds with the bracket multiplied by $m^2/M_k^2$: equal at first order; unequal at second order for $2t_f<m$, at leading order in $\tau^2$.
- **Inertia, (c′).** $E''=(1+\gamma n)^2\tau^2/(M_k+\gamma nm)$ at first order, so $\delta(1/E'')=\gamma n(m-2M_k)/\tau^2\ne\delta E/\tau^2=\gamma nm/\tau^2$.
- **Dependence on $M_k$.** In neither control is level $k$ dressed like a clockless body with rest parameter $M_k$. At $\tau=0$ the own-mark cost $-\gamma^2m^2\frac nLS_0$ is the same for every level, whereas that body under (a) has $-\gamma^2M_k^2\frac nLS_0$.

## Result

Class: couplings (a), (c); every $N_f\ge1$, $L\ge3$, $t_f>0$, every level $k$, both branches; records at $\kappa=0$. Every $O(\gamma^3)$ is at fixed $L$.

- **(6.1), (6.2)** (definition and theorem, exact). The sectors $(k,P)$ and the continuation $C_k(P)$ are fixed by places and by the total contract, so the dressed levels, their differences, their $P$-curvature and their $\gamma$-coefficients are M6-invariant. For (a) and (c) the level-$k$ contract depends on $k$ only through $M_k$. At every order, therefore, $E_{k,\sigma}(P)=\mathcal E_\sigma(P;M_k)$, where $\mathcal E_\sigma(P;M)$ is the dressed level of a body without clock label and with rest parameter $M$ under the same coupling. This is one function for all levels, and it carries the cost, the spacings and the curvature inertia.
- **(6.4)** (criterion). The starting level is a simple eigenvalue of its sector at $\gamma=0$ iff (6.4) holds. Under (6.4) the dressed level exists, is isolated, and is analytic in $\gamma$ and $P$. Sufficient conditions at $P=0$:
  - $\sigma=+$: sufficient $2t_fN_f<M_k$; the one-object excitations stay off the gap if $2t_f<M_k$.
  - $\sigma=-$: sufficient $\varepsilon_{2\pi/L}>(M_k^2+\tau^2)^{1/2}-M_k$; for one-object excitations, $\tau^2\le2t_fM_k$.

  When field excitations reach $M_k+\omega_k(Q)\ge2M_k$, the upper level lies among lower-branch states with an excited field. It is then isolated on the finite ring only away from exact coincidences.
- **(6.5)** (formula, $O(\gamma^3)$, under (6.4)). $E_{k,\sigma}(P)=\sigma\tilde\omega_k(P)-2t_fN_f+\gamma^2\frac nL\sum_{q\ne0}G_{k,\sigma}(P,q)$ for real $P$ near $0$, with $G$ explicit for both couplings. The $\sigma=-$ result follows from $\sigma=+$ by $M_k\to-M_k$, with $\omega_k$ read as in Step 4.
- **(6.6)** (formula, $O(\gamma^3)$, under (6.4)). The own-mark cost at $P=0$ is $\Delta_{k,\sigma}=\gamma^2\frac nL\sum_qG_{k,\sigma}(0,q)$, with closed forms for (a) and (c).
  - Sign: $\Delta_{k,+}<0$ for $2t_f<M_k$ and every $\tau$; $\Delta_{k,-}<0$ for every $\tau$ with $\tau^2\le2t_fM_k$.
  - Dependences: linear in $N_f$ at fixed $L$; $k$ only through $M_k$; $\sigma$ through $M_\sigma$.
- **(6.7)** (formula, exact in $\gamma$, $\tau=0$). $E_{k,\sigma}=\sigma M_k+N_f\varepsilon_0(\sigma\gamma M_k)$, independent of $P$. Its second order is $-\gamma^2M_k^2N_f(L^2-1)/(12t_fL^2)$ for both branches and both couplings, growing like $L$ at fixed $n$.
- **(6.8), (6.9)** (negative, $O(\gamma^3)$, $N\ge3$). The first-order change of the spacings is the common factor $1+\gamma n$. The second-order change strictly decreases with $k$, so it is not a common rescaling of the clock. This holds for (c) if $2t_f<m$ and for (a) if $4t_f\le m$ (both sufficient conditions). At $\tau=0$ the change is $-\gamma^2\frac nLS_0\delta_c(M_k+M_{k+1})$.
- **(6.10), (6.11)** (formula, $O(\gamma^3)$, under (6.4)). (6.10) gives the curvature at all $\tau$. (6.11) gives its leading order in $\tau^2$, with relative corrections $O(\tau^2/M_k^2,\ \tau^2L^2/(M_kt_f))$. There, for $2t_f<M_k$, $\Delta''_{k,+}<0$ for both couplings.
- **(6.12)** (negative, $O(\gamma^3)$). The change of the curvature inertia is not the change of the cost divided by $\tau^2$:
  - (a): equal at first order; unequal at second order for $2t_f<M_k$ at leading order in $\tau^2$, where the own mark raises the inertia and lowers the cost;
  - (c): unequal already at first order.

  For (a) and (c), by (6.2), the curvature inertia and the cost of level $k$ are those of a clockless body with rest parameter $M_k$, for every level.
- **(6.13)** (control, $O(\gamma^3)$).
  - $\tau\to0$: $E''\equiv0$, so the inertia is undefined. The spacing changes strictly decrease with $k$ at every order, so there is no common rescaling for $N\ge3$.
  - Rest-only coupling, $2t_f<m$: the spacings are not changed at first order (nor at any order at $\tau=0$). For $\tau>0$ they change at second order without a common factor ($N\ge3$).
  - Rest-only coupling, inertia: the change of the inertia equals the change of the cost$/\tau^2$ at first order for (a′) but not for (c′). Through second order it equals it for neither: for (a′) at leading order in $\tau^2$ with $2t_f<m$; for (c′) already at first order.
  - Rest-only coupling, dependence on $M_k$: level $k$ is not dressed like a clockless body with rest parameter $M_k$.

## Consistency checks

1. **Dimensions.** $\gamma$ and $\Phi_x$ are numbers. $G$ has the dimension of a cost ($M_k^2/\varepsilon_q$), and so do $E''$ ($P$ is a number) and $\tau^2/M_k$. Both $1/E''$ and $M_k/\tau^2$ are inverse costs. Passed.
2. **Held body.** At $\tau=0$, (6.6) gives $G=-M_k^2/\varepsilon_q$ for both couplings and both branches, i.e. $\Delta_{k,\sigma}=-\gamma^2M_k^2\frac nLS_0$. This is the second order of $N_f\varepsilon_0(\sigma\gamma M_k)$ from (4.2) of 04-profile in (6.7). Passed.
3. **Branch sum rule.** $\sum_{\sigma'}\lvert\langle\beta_{\sigma'}\vert\hat V_q\vert\beta_\sigma\rangle\rvert^2=\langle\hat V_q^2\rangle$ requires $G\to-\langle\hat V_q^2\rangle/\varepsilon_q$ for large $\varepsilon_q$. (6.6) gives $-M_k^2/\varepsilon_q$ for (a) and $-(M_k^2+b_q^2/4)/\varepsilon_q$ for (c), which equal $\langle M_k^2\rangle$ and $\langle\frac14(C_k(0)+C_k(-q))^2\rangle$ at $P=0$. Passed.

## Open issues

- The sign of $\Delta''_{k,+}$ (the mark raises the inertia) is proven only at leading order in $\tau^2$. At $\tau=1$ and $\tau=0.25$ it is checked only numerically, for the parameters of the code.
- The concavity condition for (a), $4t_f\le m$, is sufficient but not necessary. The code finds decreasing spacing changes at $m=3$, $t_f=1$.
- The upper level when field excitations reach the gap (embedded level, decay through the field as $L\to\infty$) is not treated, and neither is the limit $L\to\infty$ at fixed $n$, where the own-mark cost grows like $L$.
- Records with $\kappa>0$, which break the joint translation, and orders above $\gamma^2$ are not treated.
- The code checks only $N_f=1$, so multi-object coincidences in (6.4) are not tested numerically.
- The fall of the dressed body in the field of another source is left to a later package.

## Code

- `code/v1/check_own_mark.py` (role: check).
  - **Method.** Exact diagonalization of the sector matrix (6.1) for $N_f=1$ on rings $L=6,10$, with couplings (a) and (c), levels $k=0,1,2$ ($t_f=1$, $m=3$, $\delta_c=0.5$), both branches, $\tau=1,0.25$ and $\gamma=0.01,0.02,0.04,0.08$. The exact curvature at $P=0$ comes from the second-order Hellmann–Feynman formula, cross-checked by a finite difference of (6.1). The second-order curvature is the exact $P$-derivative of (6.5), computed by a Cauchy integral.
  - **Outputs.** `code/v1/output/check_own_mark.log` (summary) and `code/v1/output/check_own_mark.csv` (all rows).
  - **Findings.**
    - (6.6) agrees with (6.5) at $P=0$ to $6.7\times10^{-16}$.
    - The deviation of the exact $E_{k,\sigma}(0)$, $E''_{k,\sigma}(0)$ and the spacings from (6.5), (6.10), (6.8) scales as $\gamma^{p}$ with $p=2.91$–$3.11$ in all cases.
    - The relative deviation of (6.11) from (6.10) decreases fourfold when $\tau$ is halved ($\le1.6\times10^{-2}$ at $\tau=0.2$).
    - The second-order spacing change per unit spacing is $-0.505$ ($k=0$) and $-0.585$ ($k=1$) for (a), $L=6$, $\tau=1$.
    - In all 24 cases of section D, $\Delta_{k,+}<0$ and $\Delta''_{k,+}<0$.

## Methods used

- Joint-translation sectors and relative coordinates (analytic continuation in the total label)
- Mean-field and fluctuation split; Rayleigh–Schrödinger perturbation theory to second order
- Analytic perturbation theory of a simple eigenvalue (Rellich); 2×2 resolvent identity
- Concavity of the lowest eigenvalue; Perron–Frobenius
- Exact diagonalization; Hellmann–Feynman second derivative; Cauchy-integral derivative

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