Version 2 · current

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

- **Subproject:** 07-ilang-weight
- **Package:** 03-weight-of-mark
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-20

## Changes from previous version

- Setup (Regime), Step 6(c), (3.15): for coupling (a), $\nu:=\gamma n$ is kept to all orders. $B^{(a)}$ has $2\ell_0/(1+\gamma n)$ in place of $2\ell_0$, and the criterion uses $\Theta^{(a)}_D$. The expansion needs $\lvert\nu\rvert\chi\ll\lvert1+\nu\rvert$, and the case $\gamma n=-1$ is stated (I1).
- Step 6(d), (3.16): the (a) column carries $\ell_0/(1+\gamma n)$. The growth with $L$ on the ring and the square torus has the factor $(2+\gamma n)/(1+\gamma n)$. The ring zeros of $[\Theta^{(a)}_D]$ depend on $\gamma n$ (I1). The errors are $O(\eta R^2\ln(L/R)/L^2)$ (square) and $O(\eta R^2/L^3)$ (cubic), for non-cancelling pairs (I2).
- Step 5: sizes of the $\ell^{(2)}$ terms corrected: $O(R^2\ln(L/R)/L^2)$ (square), $\lvert r\rvert\lvert r'\rvert/L^3$ (cubic) (I2).
- Step 6(e), (3.17): the cubic full-grip branch break gets the term $-\ell_0\lvert r\rvert\lvert r'\rvert/L^3$ and the error $O(L^{-2}+R^3/L^4)$ (I2). The (a) level and branch breaks keep $\gamma n$ (I1).
- Steps 4–6, Code: the weight checks use exact two-point eigenvalues on pairs whose $r'$ is not the antipode, and report relative errors of weight differences. New checks: a $\gamma n$ scan for (a) (T), distance scans with $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$ (K2), lattice sums (U). Code path `code/v2` (I3).

## Response to verification

- I1: Accepted and fixed in Setup (Regime), Step 6(c) (3.15), Step 6(d) (3.16), Step 6(e) and the Result. The $\gamma n$ contribution is kept exactly: $B^{(a)}=g_b[\Theta^{(a)}_D]/[\delta N]$. This is the verifier's expression, and it holds for $\lvert\gamma n\rvert\chi\ll\lvert1+\gamma n\rvert$. At $\gamma n=-1$ weight and cost are not proportional at leading order. Code check T.
- I2: Accepted and fixed in Step 5, Step 6(d) (errors after (3.16)) and Step 6(e). Square torus: $O(\eta R^2\ln(L/R)/L^2)$. Cubic branch break: the term $-\ell_0\lvert r\rvert\lvert r'\rvert/L^3$ is kept, with remainder $O(L^{-2}+R^3/L^4)$. Relative-error statements are restricted to non-cancelling pairs. Code check U.
- I3: Accepted and fixed in Steps 4–6 and Code. Checks A, E, I and J now use exact two-point eigenvalue differences (two levels, both branches, four $\gamma$, six graphs) and report relative errors of weight differences. T scans $\gamma n$. K2 scans distances with $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$. K is described as an antipode scan from the density.

## Setup and assumptions

- **Field** (assumption, M8). $N_f\ge1$ identical field objects ($c=+1$), no pair terms among them, on $X_f$: a ring ($D=1$), a square torus ($D=2$) or a cubic torus ($D=3$) of side $L\ge3$, $\lvert X_f\rvert=L^D$. Own term $C_1=-t_f\sum_{\{x,y\}}(\lvert x\rangle\langle y\rvert+\lvert y\rangle\langle x\rvert)$, $t_f>0$; $\Phi_x$ the number of field objects at $x$; $n=N_f/\lvert X_f\rvert$. $C_1+2Dt_f=t_f\Lambda$, $\Lambda$ the graph Laplacian, with eigenvalues $\omega(\mathbf k)$ of (4.4); $\omega_{\min}:=\min_{\mathbf k\ne0}\omega(\mathbf k)$.
- **State** (assumption, M9, A11). In each sector the lowest state of the field; the highest state for even $L$ through the mirror (3.18).
- **Body** (A15, M8). One held Model D body at $x$ ($\tau=0$), in the eigenvector $\lvert\sigma\rangle\lvert f_k\rangle$ of $S:=\sigma_3\otimes(m\mathbb 1_N+D)$, eigenvalue $\sigma M_k$, $M_k=m+2\pi k/(N\lambda_0)>0$. Coupling (a): $C_{\rm D}+\gamma\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes\Phi_x$; at $\tau=0$ the motion term vanishes, so (c) equals (a). Full grip: the body's own term is $A=0$, the pair term is the same. $\theta:=1$ for (a), $\theta:=0$ for the full grip; $g_b:=\sigma\gamma M_k$.
- **Source.** Held at $x_s$ in an eigenvector of its coupled operator, pair term $g_s\Phi_{x_s}$; its own eigenvalue $c_s$ does not depend on $x$. First order in $g_s$.
- **Cost, weight:** as defined in the question. $r:=x_s-x$, $r':=x_s-x'$ (displacements; $\lvert r\rvert$ the distance); $[f]:=f(r)-f(r')$.
- **Notation.** $\ell(y):=\Lambda^+(x,x+y)$, $\ell_0:=\ell(0)$, $\ell^{(j)}:=(\Lambda^+)^j$ in the same way ($\ell^{(2)}$ as in (4.8)). For $\delta/t_f<\omega_{\min}$:
$$\ell_\delta(y):=\frac1{\lvert X_f\rvert}\sum_{\mathbf k\ne0}\frac{e^{i\mathbf k\cdot y}}{\omega(\mathbf k)-\delta/t_f},\qquad \ell^{(2)}_\delta:=\sum_y\ell_\delta(y)^2=\frac1{\lvert X_f\rvert}\sum_{\mathbf k\ne0}\frac1{(\omega(\mathbf k)-\delta/t_f)^2},$$
and $\ell_\delta=\sum_{j\ge1}(\delta/t_f)^{j-1}\ell^{(j)}$ for $\lvert\delta\rvert/t_f<\omega_{\min}$.
- **Inputs** (as quoted): (4.2), (4.4), (4.8) of 06-ilang-source/04-profile; (6.7) of 06-ilang-source/06-own-mark; (7.1), (7.2), (7.6) of 06-ilang-source/07-dressed-fall, with $\delta N(x)=-\frac{2ng_s}{t_f}\ell(r)$.
- **Regime** (A13). Exact statements hold for all $\gamma$. Expansions in $\gamma$ need $\chi:=\lvert g_b\rvert\ell_0/t_f\ll1$. For coupling (a), $\nu:=\gamma n$ is kept to all orders ($n$ is independent of $\gamma$), and the expansions need in addition $\lvert\nu\rvert\chi\ll\lvert1+\nu\rvert$. Large-distance forms hold for $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$, with $R:=\max(\lvert r\rvert,\lvert r'\rvert)$ and $R_<:=\min(\lvert r\rvert,\lvert r'\rvert)$. Their errors relative to place differences hold for non-cancelling pairs ($\lvert r\rvert/\lvert r'\rvert$ bounded away from 1, so that the leading form of $[\ell]$ dominates). Standard mathematics: rank-one perturbations, Hellmann–Feynman for a nondegenerate eigenvalue, Perron–Frobenius, Fourier analysis on tori, asymptotics of lattice Green functions (constants evaluated by the code).

## Derivation

### Step 1. Reduction to one field object

At $\tau=0$ the body has no motion term, so its position label $x$ is conserved. Every term that contains body labels contains them through $S$, so the eigenbasis of $S$ is conserved too. In the sector $(x,k,\sigma)$, with the source in its eigenvector, the total contract is $\theta\sigma M_k+c_s+\sum_{j=1}^{N_f}h_x^{(j)}$ with $h_x=C_1+g_bP_x+g_sP_{x_s}$ (7.1). There are no pair terms among the field objects, so this is a sum of one-object operators on the symmetric subspace. $-h_x$ has nonnegative off-diagonal entries on a connected graph, so by Perron–Frobenius its lowest eigenvector is nondegenerate. The product of $N_f$ copies of it is symmetric, so (4.2) gives
$$E_{k,\sigma}(x)=\theta\sigma M_k+c_s+N_f\,\varepsilon_0(x),\qquad \varepsilon_0(x)=\min\operatorname{spec}h_x .\tag{3.1}$$
For (a) this is (7.2). The code (check R) diagonalizes the full contract with branch and clock labels and finds that its spectrum is the union of the sector spectra, to $4\times10^{-14}$, for both couplings.

### Step 2. Lowest eigenpair of one object with one point term

Let $h_g:=C_1+gP_x$, with lowest eigenvalue $-2Dt_f+\delta$. From $(E-C_1)\psi=g\psi(x)\lvert x\rangle$ we get $\psi\propto(E-C_1)^{-1}\lvert x\rangle$ and $1=g\langle x\vert(E-C_1)^{-1}\vert x\rangle$. Separating the $\mathbf k=0$ term gives $\lvert X_f\rvert\delta\,\langle x+y\vert(E-C_1)^{-1}\vert x\rangle=1-\eta\ell_\delta(y)$ with $\eta:=\lvert X_f\rvert\delta/t_f$. Hence
$$\psi(x+y)\propto1-\eta\,\ell_\delta(y),\qquad \frac g{t_f}=\frac{\eta}{1-\eta\,\ell_\delta(0)},\tag{3.2}$$
and, since $\sum_y\ell_\delta(y)=0$,
$$\rho_g(y):=\frac{\lvert\psi(x+y)\rvert^2}{\lVert\psi\rVert^2}=\frac{(1-\eta\ell_\delta(y))^2}{\lvert X_f\rvert+\eta^2\ell^{(2)}_\delta}.\tag{3.3}$$
The function $1/(\lvert X_f\rvert\delta)-\ell_\delta(0)/t_f$, which equals $1/g$, decreases strictly on $(-\infty,0)$, where it runs over $(-\infty,0)$, and on $(0,t_f\omega_{\min})$, where it runs over $\mathbb R$. So the lowest eigenvalue is the unique root on $(0,t_f\omega_{\min})$ for $g>0$ and on $(-\infty,0)$ for $g<0$: $\operatorname{sign}\delta=\operatorname{sign}g$ and $\delta/t_f<\omega_{\min}$. With $\delta/t_f=\eta/\lvert X_f\rvert$ and $\hat g:=g/t_f$, (3.2) gives $\eta=\hat g-\ell_0\hat g^2+(\ell_0^2-\ell^{(2)}_0/\lvert X_f\rvert)\hat g^3+O(\hat g^4)$. Code checks B and C: (3.3) agrees with the diagonalized density to $1.1\times10^{-13}/\lvert X_f\rvert$, and $N_f\delta$ agrees with the eigenvalue to $1.1\times10^{-14}$.

### Step 3. Cost (item 1)

Without the source, the description without the body (the field alone) has lowest eigenvalue $-2Dt_fN_f$. By (3.1) and (3.2):
$$\mathrm{Cost}_{k,\sigma}=\theta\sigma M_k+C^{\rm mark}_{k,\sigma},\qquad C^{\rm mark}_{k,\sigma}:=N_f\bigl[\varepsilon_0(g_b)-\varepsilon_0(0)\bigr]=n\,t_f\,\eta(g_b),\tag{3.4}$$
$$C^{\rm mark}_{k,\sigma}=n g_b-\frac{n\ell_0}{t_f}g_b^2+\frac n{t_f^2}\Bigl(\ell_0^2-\frac{\ell^{(2)}_0}{\lvert X_f\rvert}\Bigr)g_b^3+O(g_b^4),\qquad g_b=\sigma\gamma M_k .\tag{3.5}$$
$C^{\rm mark}$ does not depend on $x$ (translations). It increases strictly in $g_b$ ($\mathrm d\varepsilon_0/\mathrm dg=\rho_g(0)>0$) and has the sign of $g_b$. Values of $\ell_0$:
- ring: $\ell_0=(L^2-1)/(12L)$, which reproduces (6.7);
- square torus: $\ell_0=\frac1{2\pi}\ln L+c_\square+o(1)$ with $c_\square=0.04877$;
- cubic torus: $\ell_0=G_3-\xi_3/L+o(1/L)$ with $G_3=0.252731$ and $\xi_3=2.837297/(4\pi)$ (code check S).

Code check D: the remainder of (3.5) truncated at $g_b^2$ scales as $\gamma^{2.97\text{–}2.98}$.

What the total contract fixes (M6):
- (i) $E_{k,\sigma}-E_{k',\sigma'}=\theta(\sigma M_k-\sigma'M_{k'})+C^{\rm mark}_{k,\sigma}-C^{\rm mark}_{k',\sigma'}$ is fixed by the total contract of the description with the body alone.
- (ii) $C^{\rm mark}$ is the difference between the descriptions with and without the coupling, so it is fixed by those two total contracts.
- (iii) $\mathrm{Cost}$ itself needs the reference description without the body. It shifts by $c$ if $c\mathbb 1$ is added to the body's own term. That change alters the evolution of the description with the body only by a common phase (1.2), so it changes no reading. $\theta\sigma M_k$ is what this comparison adds to $C^{\rm mark}$.
- (iv) The costs of single contract terms are not fixed: listing $B$ inside the pair term (A5) changes them and leaves $C$ unchanged.

In the full grip, $\mathrm{Cost}=C^{\rm mark}\to0$ for $\gamma\to0$ at every level.

### Step 4. Weight, exact in γ (item 2)

$h_x=h_{g_b}+g_sP_{x_s}$, and the lowest eigenvalue of $h_{g_b}$ is nondegenerate (Step 1). By Hellmann–Feynman, $\varepsilon_0(x)=\varepsilon_0(g_b)+g_s\rho_{g_b}(x_s-x)+O(g_s^2)$, where $\varepsilon_0(g_b)$, $\theta\sigma M_k$ and $c_s$ do not depend on $x$. For both couplings, therefore,
$$W_{k,\sigma}(x)-W_{k,\sigma}(x')=N_fg_s\bigl[\rho_{g_b}(r)-\rho_{g_b}(r')\bigr]+O(g_s^2):\tag{3.6}$$
the source weighs the body by the field density that the body makes at the source's place. With (3.3):
$$W_{k,\sigma}(x)-W_{k,\sigma}(x')=\frac{n g_s\bigl[(1-\eta\ell_\delta(r))^2-(1-\eta\ell_\delta(r'))^2\bigr]}{1+\eta^2\ell^{(2)}_\delta/\lvert X_f\rvert}+O(g_s^2),\qquad \eta=\eta(g_b),\ \ \delta=\frac{t_f\eta}{\lvert X_f\rvert}.\tag{3.7}$$
Code check A takes $\Delta W$ from exact lowest eigenvalues of $h_x$ with both point terms. It uses symmetric differences in $g_s$ at $g_s=0.02\,t_f/\ell_0$ and half of it, Richardson-extrapolated to first order in $g_s$. The run covers four place pairs per graph, with $r'$ not the antipode, two levels, both branches and $\chi=0.01$–$0.08$ on all six graphs. The relative error of $\Delta W$ against (3.7) is $\le7.1\times10^{-7}$. Normalized by the maximal density instead, the error is $\le5.3\times10^{-9}$. On the cubic torus the error is set by the eigenvalue noise ($\sim10^{-14}$).

### Step 5. Weight through second order and at large distance (item 2)

With the expansions of Step 2, (3.3) gives $\lvert X_f\rvert\rho_g(y)=1-2\hat g\ell(y)+\hat g^2\bigl[\ell(y)^2+2\ell_0\ell(y)-\tfrac{2\ell^{(2)}(y)+\ell^{(2)}_0}{\lvert X_f\rvert}\bigr]+O(\hat g^3)$. In (3.6) this gives
$$\Delta W_{k,\sigma}=\sigma\gamma M_k[\delta N]+\gamma^2M_k^2[\Theta_D]+O(g_s^2)+O(g_s\gamma^3M_k^3),\qquad \Theta_D(x):=\frac{ng_s}{t_f^2}\Bigl[\ell(r)^2+2\ell_0\ell(r)-\frac{2\ell^{(2)}(r)}{\lvert X_f\rvert}\Bigr],\tag{3.8}$$
with $\Delta W_{k,\sigma}:=W_{k,\sigma}(x)-W_{k,\sigma}(x')$. Code check E (eigenvalue weights): the remainder scales as $\gamma^{2.95\text{–}3.00}$. Since $[\ell^2]=[\ell](\ell(r)+\ell(r'))$, (3.8) reads $\Delta W=[\delta N]\bigl\{\sigma\gamma M_k-\frac{\gamma^2M_k^2}{2t_f}\bigl(\ell(r)+\ell(r')+2\ell_0-T_2\bigr)\bigr\}$ with $T_2:=2[\ell^{(2)}]/(\lvert X_f\rvert[\ell])$.

**Ring.** $\ell=\frac{L^2-1}{12L}-\frac u{2L}$ and $\ell^{(2)}=\mathrm{const}-\frac{u^2+2u}{24L}$, with $u=r(L-r)$. The second follows from $\Lambda u=2$ and $\Lambda u^2=12u-2L^2-2$. Exactly in $r$:
$$\Delta W_{k,\sigma}\big|_{\rm ring}=\sigma\gamma M_k\frac{ng_s}{Lt_f}(u-u')+\gamma^2M_k^2\frac{ng_s}{3L^2t_f^2}\Bigl[u^2-u'^2-\frac{L^2-2}2(u-u')\Bigr].\tag{3.9}$$
For $1\ll r,r'\ll L$ this is $\simeq\frac{ng_s}{t_f}(r-r')\bigl[\sigma\gamma M_k-\gamma^2M_k^2\frac L{6t_f}\bigr]$, a uniform pull.

**Square torus.** $\ell(r)=\frac1{2\pi}\bigl[\ln\frac L{\lvert r\rvert}+\kappa_\square\bigr]+O(\lvert r\rvert^{-2})+O(\lvert r\rvert^2/L^2)$ with $\kappa_\square=2\pi c_\square-\gamma_E-\tfrac32\ln2=-1.3105$. For $\lvert y\rvert\ll L$, $\ell^{(2)}(0)-\ell^{(2)}(y)=\frac1{\lvert X_f\rvert}\sum_{\mathbf k\ne0}\frac{1-\cos\mathbf k\cdot y}{\omega(\mathbf k)^2}=\frac{\lvert y\rvert^2}{8\pi}\ln\frac L{\lvert y\rvert}+O(\lvert y\rvert^2)$: the small-$\mathbf k$ part of the sum, cut off at $2\pi/L$ and $1/\lvert y\rvert$. For non-cancelling pairs this gives $T_2=O(R^2\ln(L/R)/L^2)$ (code check U), which is dropped. Then
$$\Delta W_{k,\sigma}\big|_{\square}\simeq\frac{ng_s}{\pi t_f}\ln\frac{\lvert r\rvert}{\lvert r'\rvert}\Bigl[\sigma\gamma M_k-\frac{\gamma^2M_k^2}{2t_f}\bigl(\ell(r)+\ell(r')+2\ell_0\bigr)\Bigr].\tag{3.10}$$

**Cubic torus.** $\ell(r)=\frac1{4\pi\lvert r\rvert}-\frac{\xi_3}L+O(\lvert r\rvert^{-3})+O(\lvert r\rvert^2/L^3)$. In the same way, $\ell^{(2)}(0)-\ell^{(2)}(y)=\frac{\lvert y\rvert}{8\pi}+\mathrm{const}+O(\lvert y\rvert^{-1})+O(\lvert y\rvert^2/L)$ (the Fourier transform of $k^{-4}$ in three dimensions). Hence, for non-cancelling pairs, $T_2=\frac{\lvert r\rvert\lvert r'\rvert}{L^3}\bigl[1+O(R/L)+O(R_<^{-2})\bigr]$ (code check U), which is dropped. Then
$$\Delta W_{k,\sigma}\big|_{\rm cubic}\simeq-\frac{ng_s}{2\pi t_f}\Bigl(\frac1{\lvert r\rvert}-\frac1{\lvert r'\rvert}\Bigr)\Bigl[\sigma\gamma M_k-\frac{\gamma^2M_k^2}{2t_f}\bigl(\ell(r)+\ell(r')+2\ell_0\bigr)\Bigr].\tag{3.11}$$
Code check S confirms the asymptotic forms of $\ell$: the deviation falls as $\lvert r\rvert^{-2}$ on the square torus and as $\lvert r\rvert^{-3}$ on the cubic torus, and $\ell_0+\xi_3/L=0.252731$ at $L=32,64,96$. Code check U ($r=je_1$, $r'=2je_1$, $j=2,4,8$):
- square torus, $L=64$–$512$: $T_2L^2/(R^2\ln(L/R))$ stays in $0.38$–$0.47$, while $T_2L^2/R^2$ grows from $0.5$–$1.2$ to $1.6$–$2.3$;
- cubic torus, $L=128$: $T_2L^3/(\lvert r\rvert\lvert r'\rvert)=0.82$–$0.89$, approaching 1 as $L$ grows.

### Step 6. Weight against cost (item 3)

**(a) The weight is a function of the mark cost.** By (3.4), $\eta=C^{\rm mark}/(nt_f)$ and $\delta=t_f\eta/\lvert X_f\rvert$. So (3.7) depends on $k$, $\sigma$, $\gamma$ and the coupling only through $C^{\rm mark}$:
$$\Delta W_{k,\sigma}=\mathcal W_{x,x'}\bigl(C^{\rm mark}_{k,\sigma}\bigr)+O(g_s^2),\tag{3.12}$$
where $\mathcal W$ is (3.7) at $\eta=C/(nt_f)$: one function for all levels, both branches, every $\gamma$ and both couplings. Expanding (3.3) in $\eta$ instead of $\hat g$ gives
$$\mathcal W(C)=FC+F_2C^2+F_3C^3+O(C^4),\qquad F=-\frac{2g_s}{t_f}[\ell]=\frac{[\delta N]}n,\quad F_2=\frac{g_s}{nt_f^2}\Bigl[\ell^2-\frac{2\ell^{(2)}}{\lvert X_f\rvert}\Bigr],\quad F_3=\frac{2g_s}{n^2t_f^3\lvert X_f\rvert}\Bigl[\ell\,\ell^{(2)}+\ell^{(2)}_0\ell-\frac{\ell^{(3)}}{\lvert X_f\rvert}\Bigr].\tag{3.13}$$
Code check H (eigenvector density at all places): the remainder after $F_3$ scales as $\gamma^{3.87\text{–}4.00}$.

**(b) Full grip.** Here $\mathrm{Cost}=C^{\rm mark}$, so
$$\Delta W_{k,\sigma}=\mathrm{Cost}_{k,\sigma}F\bigl[1+B^{\rm fg}_{k,\sigma}\bigr],\qquad B^{\rm fg}_{k,\sigma}=-\frac{\eta}2\Bigl[\ell(r)+\ell(r')-\frac{2[\ell^{(2)}]}{\lvert X_f\rvert[\ell]}\Bigr]+O(\eta^2\ell^2),\quad \eta=\frac{\mathrm{Cost}_{k,\sigma}}{nt_f}=\frac{g_b}{t_f}+O(g_b^2).\tag{3.14}$$
Criterion, for a pair with $F\ne0$: a factor $G(x,x')$ independent of $(k,\sigma)$ with $\Delta W=\mathrm{Cost}\,G+O(\gamma^3)$ exists iff $F_2(x,x')=0$. Proof: the order $\gamma$ forces $G=F+\gamma G_1+O(\gamma^2)$. At order $\gamma^2$, $\Delta W-\mathrm{Cost}\,G$ is $-\sigma\gamma^2nM_kG_1+\gamma^2n^2M_k^2F_2$. The first part is odd in $\sigma$ and the second even, so both vanish. The first breaking term is $F_2\mathrm{Cost}^2$: even in $\sigma$, $\propto M_k^2$, with parameter $\eta\ell$.

**(c) Coupling (a).** At fixed $\gamma$, $n$ and places, the internal state enters only through $s:=\sigma M_k$, with $g_b=\gamma s$. By (3.4), (3.5) and (3.8), with $\nu=\gamma n$ kept to all orders,
$$\mathrm{Cost}=(1+\nu)s-\frac{\nu\gamma\ell_0}{t_f}s^2+O(\lvert\nu s\rvert\chi^2),\qquad \Delta W=\gamma[\delta N]s+\gamma^2[\Theta_D]s^2+O(g_s\gamma^3s^3).$$
So $\mathrm{Cost}/((1+\nu)s)=1-\frac{\nu}{1+\nu}\bigl(\frac{g_b\ell_0}{t_f}+O(\chi^2)\bigr)$ and, for non-cancelling pairs, $\Delta W/(\gamma[\delta N]s)=1+g_b[\Theta_D]/[\delta N]+O(\chi^2)$. Dividing, for $1+\nu\ne0$, $F\ne0$ and $\lvert\nu\rvert\chi\ll\lvert1+\nu\rvert$:
$$\Delta W_{k,\sigma}=\mathrm{Cost}_{k,\sigma}F^{(a)}\bigl[1+B^{(a)}_{k,\sigma}\bigr],\quad F^{(a)}:=\frac{\gamma[\delta N]}{1+\gamma n},\quad B^{(a)}_{k,\sigma}=\frac{g_b[\Theta^{(a)}_D]}{[\delta N]}=-\frac{g_b}{2t_f}\Bigl[\ell(r)+\ell(r')+\frac{2\ell_0}{1+\gamma n}-T_2\Bigr],\quad \Theta^{(a)}_D:=\Theta_D+\frac{\gamma n}{1+\gamma n}\,\frac{\ell_0}{t_f}\,\delta N,\tag{3.15}$$
with remainder $O\bigl(\chi^2(1+\lvert\nu/(1+\nu)\rvert)^2\bigr)$ in $B^{(a)}$.

Criterion ($\gamma\ne0$): a factor $G(x,x')$ independent of $(k,\sigma)$ with $\Delta W=\mathrm{Cost}\,G$ up to relative $O(\chi^2)$ exists iff $[\Theta^{(a)}_D]=0$. Proof: $\Delta W-\mathrm{Cost}\,G=(\gamma[\delta N]-(1+\nu)G)s+(\gamma^2[\Theta_D]+\nu\gamma\ell_0G/t_f)s^2+O(s^3)$. The first part is odd in $\sigma$ and the second even, so both vanish: $G=F^{(a)}$, and then $\gamma^2[\Theta^{(a)}_D]=0$. Otherwise the first breaking term is $\gamma^2M_k^2[\Theta^{(a)}_D]$.

In (a) only the mark is weighed, $\mathcal W(C^{\rm mark})$, while the cost also contains $\sigma M_k$. At leading order the mark's share of the cost is $\nu/(1+\nu)$. Only this share of the term $2\ell_0$, the mark's own second-order cost in (3.5), is compensated. The full grip compensates all of it, and (3.15) tends to (3.14) at first order as $\lvert\gamma n\rvert\to\infty$.

**Cancelling cost.** For $\gamma n=-1$ (possible for $\gamma<0$), $\mathrm{Cost}_{k,\sigma}=-\frac{\ell_0M_k^2}{nt_f}(1+O(\chi))$ is even in $\sigma$, while $\Delta W=\sigma\gamma M_k[\delta N](1+O(\chi))$ is odd. The two branches then have weight-to-cost ratios of opposite sign, and no common factor exists even at leading order. For $\lvert1+\gamma n\rvert\lesssim\lvert\gamma n\rvert\chi$ the expansion (3.15) does not apply.

Code check I (eigenvalue weights, both couplings, six graphs):
- $\lvert B\rvert\propto\gamma^{0.97\text{–}1.00}$;
- $\lvert B-B_{\rm 1st}\rvert\le0.31\chi^2$ (full grip) and $\le1.4\chi^2$ (a), where $B_{\rm 1st}$ is the first order of (3.14), (3.15);
- slopes: $\gamma^{1.95\text{–}2.00}$ for (a), and $\gamma^{1.90\text{–}1.94}$ for the full grip over the three largest $\gamma$ (at the smallest $\gamma$ the small full-grip remainder on the cubic torus reaches the eigenvalue noise).

Code check T applies the factor $N_f$ of (3.1) to the $N_f=1$ eigenvalues, for $\gamma n\in\{-0.5,1,4,20\}$:
- $\lvert B^{(a)}-B^{(a)}_{\rm 1st}\rvert\le1.31\,\chi^2(1+\lvert\nu/(1+\nu)\rvert)^2$, with slope $\gamma^{1.94\text{–}2.00}$;
- the form with $2\ell_0$ in place of $2\ell_0/(1+\gamma n)$ misses by factors $\ge2.4$ ($\gamma n=-0.5$) up to $\ge18$ ($\gamma n=20$);
- at $\gamma n=-1$ ($M_k=1.5$, $\chi=0.02$): $\mathrm{Cost}_{k,+}$ and $\mathrm{Cost}_{k,-}$ lie in $[-3.06,-2.94]\times10^{-2}$, against $-\ell_0M_k^2/(nt_f)=-3.00\times10^{-2}$. The two ratios $\Delta W/\mathrm{Cost}$ have opposite signs and agree in size to within $2\%$.

**(d) Dependence on the graph and the distance.** These forms are at first order, from (3.14), (3.15), (3.9)–(3.11):
$$\begin{aligned}
&\text{ring (all }r,r'):&&B^{\rm fg}=-\tfrac{\eta}{12L}\bigl[L^2-3-4(u+u')\bigr],&&B^{(a)}=-\tfrac{g_b}{12Lt_f}\bigl[L^2-3-4(u+u')+\tfrac{L^2-1}{1+\gamma n}\bigr];\\
&\text{square }(1\ll\lvert r\rvert,\lvert r'\rvert\ll L):&&B^{\rm fg}\simeq-\tfrac{\eta}{4\pi}\bigl[\ln\tfrac{L^2}{\lvert r\rvert\lvert r'\rvert}+2\kappa_\square\bigr],&&B^{(a)}\simeq-\tfrac{g_b}{4\pi t_f}\bigl[\ln\tfrac{L^2}{\lvert r\rvert\lvert r'\rvert}+2\kappa_\square\bigr]-\tfrac{g_b\ell_0}{(1+\gamma n)t_f};\\
&\text{cubic }(1\ll\lvert r\rvert,\lvert r'\rvert\ll L):&&B^{\rm fg}\simeq-\tfrac{\eta}{8\pi}\bigl(\tfrac1{\lvert r\rvert}+\tfrac1{\lvert r'\rvert}\bigr)+\tfrac{\eta\xi_3}L,&&B^{(a)}\simeq-\tfrac{g_b}{t_f}\bigl[\tfrac1{8\pi}\bigl(\tfrac1{\lvert r\rvert}+\tfrac1{\lvert r'\rvert}\bigr)-\tfrac{\xi_3}L+\tfrac{\ell_0}{1+\gamma n}\bigr],
\end{aligned}\tag{3.16}$$
with $\ell_0=\frac{\ln L}{2\pi}+c_\square+o(1)$ (square) and $\ell_0=G_3-\xi_3/L+o(1/L)$ (cubic). Errors, for non-cancelling pairs, are:
- square torus: $O(\eta R_<^{-2})+O(\eta R^2\ln(L/R)/L^2)$, the second from $T_2$;
- cubic torus: $O(\eta R_<^{-3})+O(\eta R^2/L^3)$;
- the second-order remainders of (3.14) and (3.15).

In the (a) column $\eta$ stands for $g_b/t_f$.
- Ring, $r,r'\ll L$: $B^{\rm fg}\simeq-\eta\bigl[\frac L{12}-\frac{r+r'}3\bigr]$ and $B^{(a)}\simeq-\frac{g_b}{t_f}\bigl[\frac L{12}\,\frac{2+\gamma n}{1+\gamma n}-\frac{r+r'}3\bigr]$, up to $O(\eta R^2/L)$. The distance-independent term, which grows with $L$, dominates, except in $B^{(a)}$ near $\gamma n=-2$. There it cancels, and $B^{(a)}=\frac{g_b}{6Lt_f}[1+2(u+u')]$ grows with the distance.
- Square torus: both breaks grow like $\ln L$, with coefficient $-\frac{\eta}{2\pi}$ (full grip) and $-\frac{g_b}{2\pi t_f}\frac{2+\gamma n}{1+\gamma n}$ (a). At $\gamma n=-2$ the growth cancels, and $B^{(a)}\simeq\frac{g_b}{4\pi t_f}\bigl[\ln(\lvert r\rvert\lvert r'\rvert)-2\kappa_\square+4\pi c_\square\bigr]$ grows with the distance.
- Cubic torus: $B^{\rm fg}$ decays like $1/\lvert r\rvert+1/\lvert r'\rvert$ down to $O(\eta/L)$. $B^{(a)}$ tends to $-g_b\ell_0/((1+\gamma n)t_f)$ at every large distance. This is nonzero for every $\gamma n$. It is suppressed by $1/\lvert1+\gamma n\rvert$ for $\lvert\gamma n\rvert\gg1$, and it falls to $O(\eta/L)$ only for $\lvert\gamma n\rvert$ of order $L$ or larger.
- The full grip has no ring pair with $F\ne0$ and $F_2=0$: $F_2\propto[\ell]\,[L^2-3-4(u+u')]$, and $L^2-3\not\equiv0\pmod 4$. For (a), $[\Theta^{(a)}_D]=0$ with $F\ne0$ on the ring pairs with $4(u+u')=L^2-3+(L^2-1)/(1+\gamma n)$. For a pair with $4(u+u')\ne L^2-3$ this fixes one value of $\gamma n$. As $\gamma n\to0$ the condition tends to $u+u'=L^2/2-1$ (even $L$: the antipode and its neighbour).

Code check K uses the eigenvector density along an axis, with $r'$ the antipode (outside the regime $\lvert r'\rvert\ll L$) and $\lvert g_b\rvert\ell_0/t_f=0.05$. It confirms the first-order breaks of (3.14) and (3.15) at all distances along the axis:
- cubic torus, $L=20$: $B^{\rm fg}=-6.99\times10^{-3}$ at $\lvert r\rvert=1$ and $8.6\times10^{-4}$ at $\lvert r\rvert=10$; $B^{(a)}$ lies between $-0.054$ and $-0.047$ ($\sigma=+1$), against $-g_b\ell_0/t_f=-0.05$;
- ring, $L=64$: $B^{\rm fg}$ rises from $2.5\times10^{-3}$ to $4.8\times10^{-2}$, and $B^{(a)}$ from $-4.6\times10^{-2}$ to $-1.9\times10^{-3}$.

Code check K2 uses exact eigenvalue weights with $r=je_1$, $r'=2je_1$, both levels and branches and $\chi\le0.03$: ring $L=128$, $j\le16$; square torus $L=48$, $j\le6$; cubic torus $L=32$, $j\le4$. It finds:
- $\lvert B_{\rm num}-B_{\rm 1st}\rvert\le0.46\chi^2$ (full grip) and $\le2.1\chi^2$ (a);
- $\lvert B_{\rm 1st}-(3.16)\rvert\le0.055\lvert\eta\rvert e$ (square) and $\le0.014\lvert\eta\rvert e$ (cubic), with $e$ the error scales above;
- on the ring, $B^{\rm fg}_{\rm 1st}+\eta L/12=(0.77\text{–}1.0)\,\eta(r+r')/3$;
- on the cubic torus ($M_k=1.5$), $\lvert B^{\rm fg}\rvert$ falls from $3.1\times10^{-3}$ ($j=2$) to $1.0\times10^{-3}$ ($j=4$), while $\lvert B^{(a)}\rvert$ stays in $3.0$–$3.4\times10^{-2}$, against $\lvert g_b\rvert\ell_0/t_f=0.03$.

**(e) Differences between levels and branches (M6).** For two internal states 1 and 2, with $C_i:=C^{\rm mark}_i$, (3.12)–(3.13) give
$$\frac{\Delta W_1-\Delta W_2}{\mathrm{Cost}_1-\mathrm{Cost}_2}=\frac{C_1-C_2}{\theta(\sigma_1M_{k_1}-\sigma_2M_{k_2})+C_1-C_2}\Bigl[F+F_2(C_1+C_2)+F_3(C_1^2+C_1C_2+C_2^2)+O(C^3)\Bigr].\tag{3.17}$$
- **Two levels, same branch.** The relative break is first order, as for one level:
  - full grip: $\sigma\gamma n(M_k+M_{k'})F_2/F+O(\gamma^2)$;
  - (a): $\sigma\gamma(M_k+M_{k'})[\Theta^{(a)}_D]/[\delta N]+O\bigl(\chi^2(1+\lvert\nu/(1+\nu)\rvert)^2\bigr)$. Here the prefactor of (3.17) is $\frac{\nu}{1+\nu}\bigl(1-\frac{\sigma\gamma(M_k+M_{k'})\ell_0}{(1+\nu)t_f}+\dots\bigr)$, and $\chi$ is taken at the larger level.
- **Two branches of one level.** $C_++C_-=-2n\ell_0\gamma^2M_k^2/t_f+O(\gamma^4)$ and $C_+^2+C_+C_-+C_-^2=n^2\gamma^2M_k^2+O(\gamma^4)$. For (a), $C_+-C_-=2\nu M_k\bigl(1+(\ell_0^2-\ell^{(2)}_0/\lvert X_f\rvert)\gamma^2M_k^2/t_f^2+O(\chi^4)\bigr)$. Exchanging the branches changes the sign of $g_b$ and leaves the ratio unchanged, so the ratio is even in $g_b$. The ratio (3.17) is
  - full grip: $F\bigl[1+\gamma^2M_k^2\,\tfrac{n^2F_3-2n\ell_0F_2/t_f}{F}+O(\gamma^4)\bigr]$;
  - (a): $F^{(a)}\bigl[1+\gamma^2M_k^2\bigl(\tfrac{\ell_0^2-\ell^{(2)}_0/\lvert X_f\rvert}{(1+\gamma n)t_f^2}+\tfrac{n^2F_3-2n\ell_0F_2/t_f}{F}\bigr)+O\bigl(\chi^4(1+\lvert\nu/(1+\nu)\rvert)^2\bigr)\bigr]$.

  Branch differences are universal through first order and break at second order wherever the coefficient is nonzero. Written out, $t_f^2\frac{n^2F_3-2n\ell_0F_2/t_f}{F}=\ell_0\bigl(\ell(r)+\ell(r')-T_2\bigr)-\frac{\ell^{(2)}_0}{\lvert X_f\rvert}-\frac{[\ell\,\ell^{(2)}]}{\lvert X_f\rvert[\ell]}+\frac{[\ell^{(3)}]}{\lvert X_f\rvert^2[\ell]}$. On the cubic torus, for non-cancelling pairs with $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$, the last three terms are $O(L^{-2})$, and $T_2$ is as in Step 5. So the full-grip break is
  $$\frac{\gamma^2M_k^2\ell_0}{t_f^2}\Bigl[\ell(r)+\ell(r')-\frac{\lvert r\rvert\lvert r'\rvert}{L^3}\Bigr]+O\Bigl(\frac{\gamma^2M_k^2}{t_f^2}\bigl(L^{-2}+R^3/L^4\bigr)\Bigr).$$
  It decays like $1/\lvert r\rvert+1/\lvert r'\rvert$ down to $O(\gamma^2M_k^2/(t_f^2L))$, since $\lvert r\rvert\lvert r'\rvert/L^3\le1/L$. The break of (a) tends to $\gamma^2M_k^2G_3^2/((1+\gamma n)t_f^2)$.

Code check U ($L=32$–$128$, $j=2$–$8$) computes the exact coefficient. With the term $-\ell_0\lvert r\rvert\lvert r'\rvert/L^3$ included, the remainder times $L^2$ is at most $0.15$ in size. Without that term, the remainder times $L^2$ grows with $\lvert r\rvert\lvert r'\rvert/L$, up to $0.34$.

Code check J (eigenvalue weights, six graphs) compares the predicted breaks with the computed ones:
- at the smallest $\gamma$ ($\chi=0.01$), the relative deviation is $\le1.6\times10^{-2}$ for branch pairs and $\le1.4\times10^{-2}$ for level pairs. The exception is one full-grip branch pair on the cubic torus $L=12$, at $9\times10^{-2}$, where the break is small and the eigenvalue noise dominates;
- at $\chi=0.08$, it is $\le7.8\times10^{-3}$ for branch pairs and up to $0.10$ for level pairs ((a)). The level-pair residuals grow with $\gamma$ as the next order.

**(f) Highest state (mirror).** For even $L$ the graph is bipartite. $U=(-1)^{\sum_\mu x_\mu}$ gives $UC_1U=-C_1$ and $UP_yU=P_y$, so $\max\operatorname{spec}h(g_b,g_s)=-\min\operatorname{spec}h(-g_b,-g_s)$; the reference value is $+2Dt_fN_f$. Since $\Delta W$ is linear in $g_s$, (3.12) gives
$$\Delta W^{\rm high}_{k,\sigma}(g_b)=\Delta W_{k,\sigma}(-g_b)=\sum_{j\ge1}(-1)^jF_j\bigl(C^{\rm mark,high}_{k,\sigma}\bigr)^j,\qquad C^{\rm mark,high}(g_b)=-C^{\rm mark}(-g_b)=ng_b+\frac{n\ell_0}{t_f}g_b^2+O(g_b^3).\tag{3.18}$$
So (3.14)–(3.17) carry over with $F\mapsto-F$ and $F^{(a)}\mapsto-F^{(a)}$. The first-order relative breaks change sign; the second-order ones (branch pairs, which are even in $g_b$) do not. Code check M: the eigenvalue identity holds to machine precision.

## Result

Class for all results (unless stated otherwise):
- the field class of the setting (ring, square torus or cubic torus, every $N_f\ge1$, $t_f>0$);
- one held Model D body at $\tau=0$, every level $k$ and both branches, with coupling (a) ($=$(c)) or the full grip;
- one held source, at first order in $g_s$ (remainder $O(g_s^2)$);
- the lowest field state, and the highest for even $L$ by (3.18).

- **(3.1)** In each sector $(x,k,\sigma)$ the eigenvalue is $\theta\sigma M_k+c_s+N_f\min\operatorname{spec}(C_1+g_bP_x+g_sP_{x_s})$, $g_b=\sigma\gamma M_k$, exactly.
- **(3.2), (3.3)** The lowest eigenpair of $C_1+gP_x$ is given exactly by $\psi\propto1-\eta\ell_\delta$, with $g/t_f=\eta/(1-\eta\ell_\delta(0))$, $\eta=\lvert X_f\rvert\delta/t_f$, $\operatorname{sign}\delta=\operatorname{sign}g$; its density is (3.3).
- **(3.4), (3.5)** Item 1 (cost):
  - $\mathrm{Cost}_{k,\sigma}=\theta\sigma M_k+C^{\rm mark}_{k,\sigma}$ with $C^{\rm mark}=N_f[\varepsilon_0(g_b)-\varepsilon_0(0)]=nt_f\eta(g_b)$, exact for every $\gamma$;
  - through $g_b^3$: $C^{\rm mark}=ng_b-n\ell_0g_b^2/t_f+n(\ell_0^2-\ell^{(2)}_0/\lvert X_f\rvert)g_b^3/t_f^2$, with remainder $O(g_b^4)$ for $\lvert g_b\rvert\ell_0/t_f\ll1$.
  - The total contract with the body fixes the level and branch differences of $E$. $C^{\rm mark}$ is fixed as a difference between the descriptions with and without the coupling. $\mathrm{Cost}$ needs the reference description and shifts with a constant in the body's own term, which changes no reading. The costs of single contract terms are not fixed.
- **(3.6), (3.7)** Item 2 (weight), exact in $\gamma$, both couplings: $W_{k,\sigma}(x)-W_{k,\sigma}(x')=N_fg_s[\rho_{g_b}(r)-\rho_{g_b}(r')]$, the field density made by the body read at the source, in the closed form (3.7).
- **(3.8)** Through $\gamma^2$: $\Delta W=\sigma\gamma M_k[\delta N]+\gamma^2M_k^2[\Theta_D]$, with $\Theta_D=\frac{ng_s}{t_f^2}[\ell^2+2\ell_0\ell-2\ell^{(2)}/\lvert X_f\rvert]$; remainder $O(g_s\gamma^3M_k^3)$; same for both couplings.
- **(3.9)–(3.11)** Large-distance forms:
  - ring: (3.9), exact in $r$, which is (7.6) (a uniform pull for $r\ll L$);
  - square torus: $\ln(\lvert r\rvert/\lvert r'\rvert)$ law (3.10);
  - cubic torus: $1/\lvert r\rvert$ law (3.11).

  On both tori the second-order coefficient is $-\frac{\gamma^2M_k^2}{2t_f}(\ell(r)+\ell(r')+2\ell_0)$ relative to $\sigma\gamma M_k$. (3.10) and (3.11) hold for $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$ and non-cancelling pairs, up to the errors stated in Step 5. The term $T_2$ dropped from the bracket is $O(R^2\ln(L/R)/L^2)$ on the square torus and $\frac{\lvert r\rvert\lvert r'\rvert}{L^3}[1+O(R/L)+O(R_<^{-2})]$ on the cubic torus.
- **(3.12), (3.13)** Exactly in $\gamma$, $\Delta W_{k,\sigma}=\mathcal W_{x,x'}(C^{\rm mark}_{k,\sigma})$, with one function $\mathcal W$ for all levels, branches, $\gamma$ and both couplings. $\mathcal W(C)=FC+F_2C^2+F_3C^3+O(C^4)$, with $F=[\delta N]/n$ and $F_2$, $F_3$ as in (3.13). $\mathcal W$ is not linear wherever $F_2(x,x')\ne0$: on the ring for every pair with $F\ne0$, and on the tori for generic pairs (3.16).
- **(3.14)** Item 3, full grip:
  - weight $=$ cost $\times F$ at leading order, with the universal $F=[\delta N]/n$ (remainder $O(\gamma^2)$);
  - a universal factor exists at order $\gamma^2$ iff $F_2(x,x')=0$; otherwise the first breaking term is $F_2\mathrm{Cost}^2$, with relative size $B^{\rm fg}=-\frac\eta2[\ell(r)+\ell(r')-2[\ell^{(2)}]/(\lvert X_f\rvert[\ell])]$, $\eta=\mathrm{Cost}/(nt_f)$.
- **(3.15)** Item 3, coupling (a):
  The first three statements keep $\gamma n$ to all orders. They hold for $\gamma\ne0$ and $1+\gamma n\ne0$ with $\lvert\gamma n\rvert\chi\ll\lvert1+\gamma n\rvert$, $\chi=\lvert g_b\rvert\ell_0/t_f$, and for non-cancelling pairs with $F\ne0$.
  - Weight $=$ cost $\times F^{(a)}$, $F^{(a)}=\gamma[\delta N]/(1+\gamma n)$, at leading order, with relative remainder $O(\chi(1+\lvert\gamma n/(1+\gamma n)\rvert))$.
  - A factor independent of level and branch exists up to relative $O(\chi^2)$ iff $[\Theta^{(a)}_D]=0$, with $\Theta^{(a)}_D=\Theta_D+\frac{\gamma n}{1+\gamma n}\frac{\ell_0}{t_f}\delta N$. Otherwise the first breaking term is $\gamma^2M_k^2[\Theta^{(a)}_D]$. Its relative size is $B^{(a)}=-\frac{g_b}{2t_f}[\ell(r)+\ell(r')+2\ell_0/(1+\gamma n)-2[\ell^{(2)}]/(\lvert X_f\rvert[\ell])]$, with remainder $O(\chi^2(1+\lvert\gamma n/(1+\gamma n)\rvert)^2)$.
  - Only the share $\gamma n/(1+\gamma n)$ of the mark's term $2\ell_0$ is compensated. As $\lvert\gamma n\rvert\to\infty$, $B^{(a)}$ tends to $B^{\rm fg}$ at first order.
  - At $\gamma n=-1$, the cost $-\ell_0M_k^2/(nt_f)(1+O(\chi))$ is even in $\sigma$ and the weight is odd. No common factor exists even at leading order.
- **(3.16)** Dependence on the graph (first order in $\gamma$):
  - ring ($r,r'\ll L$, up to $O(\eta R^2/L)$): $B^{\rm fg}\simeq-\eta[\frac L{12}-\frac{r+r'}3]$ and $B^{(a)}\simeq-\frac{g_b}{t_f}[\frac L{12}\frac{2+\gamma n}{1+\gamma n}-\frac{r+r'}3]$. The term growing with $L$ dominates, except in $B^{(a)}$ near $\gamma n=-2$. At $\gamma n=-2$ it cancels, and $B^{(a)}=g_b[1+2(u+u')]/(6Lt_f)$ grows with the distance. No ring pair with $F\ne0$ has $F_2=0$.
  - square torus: both breaks grow like $\ln L$, that of (a) with coefficient $-\frac{g_b}{2\pi t_f}\frac{2+\gamma n}{1+\gamma n}$. At $\gamma n=-2$ the growth cancels, and $B^{(a)}$ grows like $\ln(\lvert r\rvert\lvert r'\rvert)$.
  - cubic torus: the full-grip break decays like $1/\lvert r\rvert+1/\lvert r'\rvert$ down to $O(\eta/L)$. That of (a) tends to $-g_b\ell_0/((1+\gamma n)t_f)$ at every large distance, with $\ell_0\to G_3=0.252731$. This is nonzero for every $\gamma n$.
  - errors, for non-cancelling pairs: $O(\eta R_<^{-2})+O(\eta R^2\ln(L/R)/L^2)$ on the square torus, and $O(\eta R_<^{-3})+O(\eta R^2/L^3)$ on the cubic torus.

  Of the six combinations of coupling and graph, only the full grip on the cubic torus has a relative break that decays with distance down to $O(\eta/L)$. Its size is $O(\eta(1/\lvert r\rvert+1/\lvert r'\rvert))+O(\eta/L)+O(\eta^2)$, the same for all levels and branches. Coupling (a) on the cubic torus keeps the floor $\lvert g_b\rvert\ell_0/(\lvert1+\gamma n\rvert t_f)$. That floor reaches this accuracy only for $\lvert\gamma n\rvert$ of order $L$ or larger, where (a) approaches the full grip.
- **(3.17)** Differences, for every pair of internal states, exactly in $\gamma$ through the divided difference of $\mathcal W$:
  - between two levels of one branch, universality breaks at first order in $\gamma$ wherever $F_2\ne0$ (full grip) or $[\Theta^{(a)}_D]\ne0$ (a), as for one level;
  - between the two branches of one level, it holds through first order and breaks at order $\gamma^2M_k^2$ wherever the coefficient given in Step 6(e) is nonzero. For (a) that coefficient contains $(\ell_0^2-\ell^{(2)}_0/\lvert X_f\rvert)/((1+\gamma n)t_f^2)$, with the same conditions on $\gamma n$ as (3.15).

  On the cubic torus, for non-cancelling pairs with $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$, the full-grip branch break is $\frac{\gamma^2M_k^2\ell_0}{t_f^2}[\ell(r)+\ell(r')-\frac{\lvert r\rvert\lvert r'\rvert}{L^3}]+O(\frac{\gamma^2M_k^2}{t_f^2}(L^{-2}+R^3/L^4))$. It decays like $1/\lvert r\rvert$ down to $O(\gamma^2M_k^2/(t_f^2L))$. The branch break of (a) tends to $\gamma^2M_k^2G_3^2/((1+\gamma n)t_f^2)$.
- **(3.18)** In the highest state (even $L$): $\Delta W^{\rm high}(g_b)=\Delta W(-g_b)=\sum_j(-1)^jF_j(C^{\rm mark,high})^j$, with $C^{\rm mark,high}(g_b)=-C^{\rm mark}(-g_b)$. (3.14)–(3.17) carry over with $F\mapsto-F$ and $F^{(a)}\mapsto-F^{(a)}$. The first-order relative breaks change sign; the second-order branch-pair breaks do not.
- **Controls (M9).**
  - For the positive answer (full grip, cubic torus), the failing descriptions are coupling (a) on the same graph with $\lvert\gamma n\rvert\ll L$, and the full grip on the ring or square torus.
  - For the negative answers, the order-$\gamma^2$ criteria of (3.14) and (3.15) are exact, so the control of each is a pair on which its criterion holds. On the ring these are only the pairs with $[\ell]=0$ (where $F=0$) and, for (a), the pairs with $4(u+u')=L^2-3+(L^2-1)/(1+\gamma n)$. On the tori the zeros of $F_2$ and $[\Theta^{(a)}_D]$ are not classified.

## Consistency checks

1. **Dimensions.** $g_b$, $g_s$, $t_f$, $M_k$, Cost and $W$ have the dimension of $C$; $\gamma$, $n$, $\ell^{(j)}$, $\eta$ and $F$ are dimensionless; $F_2\propto1/[C]$ and $F_3\propto1/[C]^2$. Every term of (3.5), (3.8) and (3.13)–(3.17) is homogeneous.
2. **Special cases.** For $\gamma\to0$: $\Delta W\to0$ and $\mathrm{Cost}\to\theta\sigma M_k$. On the ring:
   - $\ell_0=(L^2-1)/(12L)$ turns (3.5) into (6.7);
   - the $g_b^2g_s$ coefficient of (3.8) equals the $g_1^2g_2$ term of (4.8);
   - $\Theta_D-\Theta$ of (7.6) is constant in $x$ (code check F: $2\times10^{-15}$).

   For $\lvert\gamma n\rvert\to\infty$, $B^{(a)}$ of (3.15) tends to $B^{\rm fg}$ of (3.14) at first order ($2\ell_0/(1+\gamma n)\to0$, $\eta\to g_b/t_f$).
3. **Sum rule.** $\sum_x\rho_{g_b}(x_s-x)=1$ for every $\gamma$, so the $\gamma$-dependent part of $\sum_xW(x)$ must vanish. At order $\hat g$: $\sum_y\ell=0$. At order $\hat g^2$: $\sum_y[\ell^2+2\ell_0\ell-(2\ell^{(2)}+\ell^{(2)}_0)/\lvert X_f\rvert]=\ell^{(2)}_0-\ell^{(2)}_0=0$. Passed.

## Open issues

- Results for $N_f>1$ rest on the reduction (4.2) and on the factor $N_f$ in (3.1). The code diagonalizes $N_f=1$ only; its $\gamma n$ scan (check T) applies the factor $N_f$ to those eigenvalues.
- For coupling (a), the window $\lvert1+\gamma n\rvert\lesssim\lvert\gamma n\rvert\chi$ around $\gamma n=-1$ is treated only at $\gamma n=-1$ itself, and there only at leading order.
- The highest state is covered for even $L$ only; for odd $L$ the top of the one-object band is degenerate, and it is not treated.
- Terms of order $g_s^2$ (the source's own mark and its nonlinearity) are not treated.
- The large-distance constants $c_\square$, $G_3$ and $\xi_3$ are evaluated numerically (code check S), not in closed form.
- On the square and cubic tori, the pairs where $F_2$ or $[\Theta^{(a)}_D]$ vanish are not classified.
- Whether a coupling other than the full grip removes the $G_3$ break of (a), and how special the full grip is, belongs to item 5. Moving bodies, inertia and bonds are out of scope.

## Code

- `code/v2/check_weight_of_mark.py` (role: check). It works at $N_f=1$, $t_f=1$, on the ring ($L=32,64$), the square torus ($L=12,20$) and the cubic torus ($L=8,12$). It uses two levels ($M_k=1,1.5$), both branches and both couplings, with $\gamma$ chosen so that $\chi=0.01$–$0.08$ at the larger level. Weight differences come from exact lowest eigenvalues of one field object with the point terms of the body and the source (dense or Lanczos). It takes symmetric differences in $g_s$ at $0.02\,t_f/\ell_0$ and half of it, Richardson-extrapolated to first order in $g_s$, on four place pairs per graph, with $r'$ not the antipode. The results are compared with (3.1)–(3.18). The checks are:
  - R: full contract against (3.1);
  - A: eigenvalue weight differences against (3.7), relative and density-normalized errors;
  - B, C: exact forms (3.2)–(3.4);
  - D, E: $\gamma$-scaling of the remainders of (3.5), (3.8);
  - F: (7.6);
  - H: (3.13), from the eigenvector density at all places;
  - I: single-level breaks (3.14), (3.15);
  - J: differences (3.17);
  - T: coupling (a) for $\gamma n\in\{-0.5,1,4,20\}$ and $\gamma n=-1$, via the factor $N_f$ of (3.1);
  - K: breaks along an axis with $r'$ the antipode, from the eigenvector density (ring $L=64$, square $L=20$, cubic $L=12,20$);
  - K2: breaks from eigenvalues for $r=je_1$, $r'=2je_1$ in the regime $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$ (ring $L=128$, square $L=48$, cubic $L=32$), against (3.14)–(3.16);
  - M: mirror (3.18);
  - S: asymptotics of $\ell$ by FFT up to $L=512$ (2D) and $L=96$ (3D);
  - U: the lattice sums $T_2$ (square $L\le512$, cubic $L\le128$) and the cubic branch-pair coefficient of (3.17).

  Outputs: `code/v2/output/check_log.txt` (all numbers cited above) and `code/v2/output/check_results.json` (the same, machine-readable).

## Methods used

- sector reduction by conserved labels; Perron–Frobenius; one-object reduction for bosons without mutual pair terms
- rank-one (point-term) perturbation, secular equation, resolvent with the zero mode separated
- Hellmann–Feynman theorem; Rayleigh–Schrödinger expansion
- Fourier analysis on tori; pseudo-inverse of the graph Laplacian; asymptotics of lattice Green functions
- odd/even decomposition in the branch label
- bipartite (staggered) mirror
- numerical check by exact diagonalization (dense and Lanczos) and FFT

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