Version 5 · earlier version; the current one is v6 · External review, round 3: minor issues

# Grips within pair terms on the field of identical objects, and their weighing

- **Subproject:** 08-ilang-propagation
- **Package:** 03-gas-grip
- **Version:** v5
- **Mode:** external regeneration
- **Date:** 2026-10-24

## Changes from previous version

- Step 4, "Criterion" and (3.8): (i) is now stated only for grips with $\mu_-\le\lvert\mu\rvert\le\mu_+$, constants fixed independently of $\chi$, $L$ and the places, with the constants of its two directions named; new paragraph "Small $\mu$" on $\mu\to0$, with the next-order coefficient $\nu_1$ and the ring $L=4$ as the control; the remainder of (ii) is now $O(\chi^2q)$ (verification I1, I2).
- Step 3: new cancellation (c7), small $\lvert\mu\rvert$ in the denominator of $w^{\rm self}$ (I1).
- Step 4, new paragraph "Uniform remainders": exact one-object form (3.16) of $B^{\rm fg}$ and of $\mu w^{\rm self}$, and bounds with absolute constants on the remainders of (3.5), (3.7), (3.8), (3.9), (3.13) and on their derivatives (I2).
- Step 4: $q$ now contains $\bar T_3=2s_3/(\mathcal X^2\lvert[\ell]\rvert)\ge\lvert T_3\rvert$, which (3.16) needs; the uniform bounds get the ring term $b_{1,3}$; (3.15) bounds $\bar T_3$ and has a non-degeneracy condition on the ring, so that $q=O(1)$ at the $c$-non-degenerate pairs of the three graphs (I2).
- Step 4, (3.7) and the paragraph after (3.13): the quoted remainder $O(\eta^2\ell^2)$ is replaced by $O(\chi^2q)$ from (3.16) (I2).
- Step 5: the remainder of (3.9) is uniform by (3.16) (I2).
- Result ((3.7), (3.8), (3.9), (3.13), (3.15), new (3.16)), Open issues (second item replaced, item on small $\mu$ added), Methods used and the twin follow.

## Response to verification

- I1: Accepted and fixed in Step 4 ("Criterion", (3.8), paragraph "Small $\mu$") and Step 3 (c7). The criterion (3.8)(i) is restricted to $\mu_-\le\lvert\mu\rvert\le\mu_+$; its "if" direction carries the factor $1/\mu_-$, its "only if" direction the factor $\mu_+$. For $\mu\to0$ the remainder of (3.13) is amplified by $1/\lvert\mu\rvert$: $w^{\rm self}$ grows as $(1+\zeta)/\mu$ where $\lvert1+\zeta\rvert\gg\chi q$, and for $\lvert\mu\rvert\lesssim\chi q$ it is fixed only by the next order, given on a fixed graph. The counterexample is confirmed from the order-$\chi$ terms of Step 4 ($\zeta=-1$, $\nu_1=26/25$) and is now the control for the range of (i).
- I2: Accepted and fixed by establishing uniform bounds: Step 4 (paragraph "Uniform remainders", (3.16), $q$ with $\bar T_3$, (3.15)) and Step 5. The closed form (3.16) of the one-object problem bounds the remainders and their derivatives with absolute constants for $\chi\le\frac14$, $\chi q\le1$, on every graph of the class and at every pair with $[\ell]\ne0$; (3.15) bounds $q$ at the $c$-non-degenerate pairs of the three graphs, so the size and distance statements there, the torus statements included, are uniform. Only the next-order form at small $\mu$ is stated for a fixed graph, with graph-dependent constants.

## Setup and assumptions

**Description** (assumptions of the package, M8, M9, A11, A15, as fixed in question.md).
- *Field.* $N_f\ge1$ identical field objects ($c=+1$) without pair terms among them, each with the places $X_f$ of a ring ($D=1$), a square torus ($D=2$) or a cubic torus ($D=3$) of side $L\ge3$, $\mathcal X=L^D$. Own term of each object $C_1=-t_f\sum_{\{x,y\}}(\lvert x\rangle\langle y\rvert+\lvert y\rangle\langle x\rvert)=t_f(\Lambda-2D\,\mathbb 1)$, $\Lambda$ the graph Laplacian with the nonzero eigenvalues $\lambda_k\ge\lambda_1>0$. $P^{(j)}_x$ is the projector of field object $j$ on place $x$, $\Phi_x=\sum_jP^{(j)}_x$, $n=N_f/\mathcal X$; $\Pi_f$ projects onto the symmetric subspace of the field objects.
- *Body.* Held Model D ($\tau=0$). $S=\sigma_3\otimes(m\mathbb 1_N+D)$ has the eigenvalues $\sigma M_k$, $\sigma=\pm1$, $M_k=m+2\pi k/(N\lambda_0)>0$, all distinct. The grip term is
$$G_F=\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes F(\Phi_x),$$
$F$ real on $\{0,\dots,N_f\}$, $x$ the field place of the body under the coupling map of the quoted packages; no other rest and clock term. Coupling (a) is $F=1+\gamma\Phi$, the full grip $F=\gamma\Phi$.
- *Source.* A held object at $x_s$, own eigenvalue $c_s$, pair term $g_s\Phi_{x_s}$; first order in $g_s$.
- *State.* The eigenvector of $C$ connected, as the couplings go to zero, to the bodies in eigenvectors of $S$ and every field object in its lowest one-object state.
- *Pair terms* (A5): operators acting on at most two objects, own terms included. "Sum of pair terms" means an operator identity on $\mathcal H$, or on $\mathcal H_{\rm adm}$ (both sides compressed by its projector $\Pi$).

**Inputs**, as quoted in question.md, with its notation ($h_x$, $\varepsilon_0$, $\ell(y)=\Lambda^+(x,x+y)$, $\ell_0$, $\ell^{(j)}$, $r$, $r'$, $[f]$, $\delta N$, $\eta$, $T_2$, $\chi$; for the bond $\phi$, $\mathbf R$, $d$, $\Delta_{\mathbf e}$, $\Delta_{\mathbf r\mathbf r'}$, $W_b$, $K$, $\kappa$, $\rho_0$). To avoid a clash with the grip function, the coefficients $F,F_2,F_3$ of (3.13)–(3.15) of 07-ilang-weight/03-weight-of-mark are written $\Omega,\Omega_2,\Omega_3$, and $F_i$ of (4.9) of 07-ilang-weight/04-weight-of-bond is written $\mathcal F_i$:
$$\Omega:=-\frac{2g_s}{t_f}[\ell]=\frac{[\delta N]}{n}=\phi(x)-\phi(x').$$
- 07-ilang-weight/03-weight-of-mark@v2: (3.1) $E_{k,\sigma}(x)=\theta\sigma M_k+c_s+N_f\varepsilon_0(x)$; (3.4) $\mathrm{Cost}_{k,\sigma}=\theta\sigma M_k+C^{\rm mark}_{k,\sigma}$, $C^{\rm mark}=N_f[\varepsilon_0(g_b)-\varepsilon_0(0)]=nt_f\eta(g_b)$; (3.12)–(3.13) $\Delta W_{k,\sigma}=\Omega C^{\rm mark}+\Omega_2(C^{\rm mark})^2+\Omega_3(C^{\rm mark})^3+O((C^{\rm mark})^4)+O(g_s^2)$, $\Omega_2=\frac{g_s}{nt_f^2}[\ell^2-\frac{2\ell^{(2)}}{\mathcal X}]$; (3.14) full grip: $\Delta W=\mathrm{Cost}\,\Omega\,[1+B^{\rm fg}]$, $B^{\rm fg}=-\frac\eta2[\ell(r)+\ell(r')-T_2]+O(\eta^2\ell^2)$, $\eta=\frac{\mathrm{Cost}}{nt_f}=\frac{g_b}{t_f}+O(g_b^2)$; (3.15) coupling (a): $\Delta W=\mathrm{Cost}\,F^{(a)}[1+B^{(a)}]$, $F^{(a)}=\frac{\gamma[\delta N]}{1+\gamma n}$, $B^{(a)}=-\frac{g_b}{2t_f}[\ell(r)+\ell(r')+\frac{2\ell_0}{1+\gamma n}-T_2]$. Regime $\chi=\lvert g_b\rvert\ell_0/t_f\ll1$.
- 07-ilang-weight/04-weight-of-bond@v3: (4.2) bond cost $B(\mathbf r)=-\frac{2ng_1g_2}{t_f}\ell(\mathbf r)$; (4.9) $\mathcal F_i=\phi$ (full grip), $\mathcal F_i=\frac{n\gamma_i}{1+n\gamma_i}\phi$ (coupling (a)); (4.10) $\rho_i=\frac{\Delta_{\mathbf e}\Delta_{\mathbf r\mathbf r'}W_b(x_1;\cdot)}{\Delta_{\mathbf r\mathbf r'}B\,\Delta_{\mathbf e}\mathcal F_i(x_1)}$, $\rho_i^{\rm fg}=\rho_0+O(\eta K)$, $\rho_0=1+\kappa$, for $\ell(\mathbf r)\ne\ell(\mathbf r')$, $\Delta_{\mathbf e}\ell(\mathbf R)\ne0$ and $\eta K\ll1$, including configurations with $\rho_0=0$ (note (a) of question.md); (4.12) $\kappa^{\rm mid}_{D=3}=O(\frac{\lvert\mathbf r\rvert\lvert\mathbf r'\rvert}{d^2})$, $\kappa^{\rm mid}_{D=2}=O(\frac{\lvert\mathbf r'\rvert^2}{d^2}\ln\frac{L^2}{\lvert\mathbf r'\rvert d})$, $\kappa^{\rm mid}_{\rm ring}=0$, with the conditions of the note of question.md: on the ring $\kappa^{\rm mid}=0$ at leading order in $\eta$ for $x_s$ outside the arc between the bodies; the ratio is 1 on the cubic torus for $\lvert\mathbf r\rvert,\lvert\mathbf r'\rvert\ll d\ll L$, on the square torus only for $d$ large against $\lvert\mathbf r\rvert$ times a logarithm of $L$; (4.13) $\rho_i^{(a)}=\frac{1+n\gamma_i}{n\gamma_i}\rho_i^{\rm fg}(1+O(\eta))$. Regime $\eta=\max_i\lvert g_i\rvert\ell_0/t_f<1/12$, $\gamma_i>0$. Item 2 asks the bond only for a source outside the pair: on the ring outside the arc between the bodies, on the tori at $d$ large against the separations, as in (4.12) (note (b) of question.md).

**Approximations.** First order in $g_s$; expansion in $\chi$ (mark) and $\eta$ (bond) at the orders stated with each result.

## Derivation

### Step 1. The grip as a sum of many-object terms

Newton's forward-difference formula on the finite set $\{0,\dots,N_f\}$ gives $F(\nu)=\sum_{k=0}^{N_f}\binom\nu k\Delta^kF(0)$ for every integer $0\le\nu\le N_f$, with $\Delta^kF(0)=\sum_{i=0}^k(-1)^{k-i}\binom kiF(i)$. The projectors $P^{(j)}_x$ commute and are diagonal in the joint places; on a joint place with $\nu$ objects at $x$, $\Phi_x=\nu$ and the elementary symmetric polynomial $e_k(P^{(1)}_x,\dots,P^{(N_f)}_x)=\sum_{j_1<\dots<j_k}P^{(j_1)}_x\cdots P^{(j_k)}_x$ equals $\binom\nu k$. Hence

$$G_F=\sum_{k=0}^{N_f}\Delta^kF(0)\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes e_k\bigl(P^{(1)}_x,\dots,P^{(N_f)}_x\bigr),\tag{3.1}$$

and its $k$-th term acts on the body and $k$ field objects together. If $\Delta^kF(0)=0$ for $2\le k\le N_f$, only the body term ($k=0$) and the pair terms ($k=1$) remain.

### Step 2. Only affine grips are sums of pair terms

Let $N_f\ge2$ and $\Pi G_F\Pi=\Pi R\Pi$, with $R$ a sum of pair terms between any objects of the description. On $\mathcal H_{\rm adm}$ the body is taken to be the only object of its type, so $\Pi=\mathbb 1_b\otimes\Pi_f\otimes\Pi_O$, $O$ the objects other than the body and the field; an identity on $\mathcal H$ implies this compressed one.

(a) Let $e_\pm$ be eigenvectors of $S$ with the eigenvalues $\pm M_0=\pm m$, and $\mathcal D(X):=\langle x,e_+\rvert X\lvert x,e_+\rangle-\langle x,e_-\rvert X\lvert x,e_-\rangle$ (partial matrix elements in the body places). $\mathcal D$ commutes with the compression by $\Pi$, and $\mathcal D(G_F)=2M_0F(\Phi_x)$. A term of $R$ that does not act on the body gives zero, a term on the body alone a number, a term of the body with field object $j$ an operator $c_j$ on object $j$, a term of the body with $o\in O$ an operator on $o$.
(b) The map $X\mapsto\operatorname{Tr}_O(\Pi_OX\Pi_O)/\operatorname{Tr}\Pi_O$ turns the last kind into numbers: $2M_0\,\Pi_fF(\Phi_x)\Pi_f=\Pi_f\bigl(c_0+\sum_jc_j\bigr)\Pi_f$, $c_0$ a number.
(c) Every permutation $U$ of the field objects has $U\Pi_f=\Pi_f$, so $\Pi_f\sum_jc_j\Pi_f=\Pi_f\sum_j\bar c^{(j)}\Pi_f$, with $\bar c=N_f^{-1}\sum_ic_i$ acting on object $j$.
(d) In the occupation basis $\lvert\{n_y\}\rangle$, $\sum_yn_y=N_f$, of the symmetric subspace the diagonal elements of $\sum_j\bar c^{(j)}$ are $\sum_yn_y\bar c_{yy}$ (standard), those of $F(\Phi_x)$ are $F(n_x)$. So $2M_0F(n_x)=c_0+\sum_yn_y\bar c_{yy}$ for every occupation.
(e) All objects at one place $y\ne x$: $N_f\bar c_{yy}=2M_0F(0)-c_0$, the same for every $y\ne x$. Then $\nu$ objects at $x$ and $N_f-\nu$ at one $y\ne x$: $2M_0[F(\nu)-F(0)]=\nu(\bar c_{xx}-\bar c_{yy})$ for $0\le\nu\le N_f$, so $F$ is affine.

With Step 1, for $N_f\ge2$:

$$G_F\ \text{is a sum of pair terms}\iff\Delta^kF(0)=0\ \ (2\le k\le N_f)\iff F(\Phi)=\alpha+\beta\Phi,\qquad G_{\alpha,\beta}=\sum_{j=1}^{N_f}\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes\Bigl(\frac{\alpha}{N_f}\mathbb 1+\beta P^{(j)}_x\Bigr),\tag{3.2}$$

with $\alpha=F(0)$, $\beta=F(1)-F(0)$. Since $R$ is arbitrary, adding, removing or moving listed pair terms (M6) does not make a grip with $\Delta^kF(0)\ne0$ for some $k\ge2$ a sum of pair terms. (3.2) is an exact characterization, so the class has no control for it. Illustration outside the class (not an admissible description under A5, hence not a control, M9): $F(\Phi)=\Phi^2$ has $\Delta^2F(0)=2$; its term $2\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes\sum_{i<j}P^{(i)}_xP^{(j)}_x$ is a three-object term.

For $N_f=1$, $\Phi_x=P_x$ has the spectrum $\{0,1\}$, and every $F$ is affine there. For every $N_f\ge1$, $G_{\alpha,\beta}$ is the full grip $\beta\Phi$ together with the own term $\alpha S$ of the body (A5), so the two lists are one description (M6); coupling (a) is the full grip with the own term $S$. Two pairs $(\alpha,\beta)\ne(\alpha',\beta')$ give different $C$ on $\mathcal H_{\rm adm}$, since $\Phi_x$ takes the values $0$ and $1$ there:

$$N_f=1:\ \ F(\Phi_x)=F(0)\,\mathbb 1+\bigl[F(1)-F(0)\bigr]\Phi_x\ \text{for every }F;\qquad N_f\ge1:\ \ G_{\alpha,\beta}=\alpha\sum_x\lvert x\rangle\langle x\rvert\otimes S+\beta\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes\Phi_x .\tag{3.3}$$

### Step 3. Eigenvalue, cost and first-order part for an affine grip

All terms of the held body are $S\otimes(\cdot)$, so each eigenspace $\sigma M_k$ of $S$ is invariant, and $G_{\alpha,\beta}$ acts on it as $\alpha\sigma M_k+g_b\Phi_x$, $g_b:=\sigma\beta M_k$. On this sector $C=\alpha\sigma M_k+c_s+\sum_jh_x^{(j)}$, $h_x=C_1+g_bP_x+g_sP_{x_s}$ acting on object $j$. The lowest eigenvalue of $h_x$ is simple (Perron–Frobenius: connected graph, off-diagonal entries $\le0$), and the symmetric product of its eigenvectors is the stated state. As in (3.1), (3.4) of 07-ilang-weight/03-weight-of-mark, with $\theta$ replaced by $\alpha$:

$$E_{k,\sigma}(x)=\alpha\sigma M_k+c_s+N_f\varepsilon_0(x),\qquad \mathrm{Cost}_{k,\sigma}=\alpha\sigma M_k+C^{\rm mark}_{k,\sigma},\qquad C^{\rm mark}_{k,\sigma}=N_f\bigl[\varepsilon_0(g_b)-\varepsilon_0(0)\bigr].\tag{3.4}$$

$\alpha$ enters only through a constant that does not depend on the places; $C^{\rm mark}$ and every place dependence of $E$ are those of the full grip with $\gamma=\beta$.

The lowest eigenvector of $C_1$ is $u=\mathcal X^{-1/2}\sum_y\lvert y\rangle$, and $C_1+2Dt_f=t_f\Lambda$. Since $\ell_0=\mathcal X^{-1}\sum_k\lambda_k^{-1}\ge(\mathcal X\lambda_1)^{-1}$, $\lvert g\rvert/\mathcal X\le\chi t_f\lambda_1$, so for $\chi\ll1$ the shift stays far below the gap and the perturbation series of the simple eigenvalue applies: first order $g\lvert u(x)\rvert^2=g/\mathcal X$, second order $-g^2\langle u\rvert P_x(t_f\Lambda)^+P_x\lvert u\rangle=-g^2\ell_0/(\mathcal Xt_f)$, third order $\frac{g^3}{\mathcal Xt_f^2}(\ell_0^2-\ell^{(2)}_0/\mathcal X)$ with $0\le\ell^{(2)}_0/\mathcal X\le\ell_0^2$. With $\chi_s:=g_b\ell_0/t_f$ ($\lvert\chi_s\rvert=\chi$), $C^{\rm mark}=ng_b[1-\chi_s+O(\chi^2)]$. The parts of $\mathrm{Cost}$ and of $\Delta W$ linear in $(\alpha,\beta)$ follow from (3.4) and from (3.12)–(3.13) of 07-ilang-weight/03-weight-of-mark ($\Omega C^{\rm mark}=\Omega ng_b+O(g_b^2)$):

$$\mathrm{Cost}^{(1)}_{k,\sigma}=\sigma M_k(\alpha+\beta n),\qquad \Delta W^{(1)}_{k,\sigma}=g_b[\delta N]=\mu\,\Omega\,\mathrm{Cost}^{(1)}_{k,\sigma},\qquad \mu:=\frac{\beta n}{\alpha+\beta n},\qquad C^{\rm self}_{k,\sigma}:=\mathrm{Cost}_{k,\sigma}-\mathrm{Cost}^{(1)}_{k,\sigma}=-ng_b\chi_s\bigl[1+O(\chi)\bigr].\tag{3.5}$$

$\mathrm{Cost}^{(1)}=\sigma M_kF(n)$ is $\sigma M_k\langle F(\Phi_x)\rangle$ in the uncoupled field. The *mark fraction* $\mu=ng_b/\mathrm{Cost}^{(1)}$ does not depend on $k$ or $\sigma$, and $\mu=1$ iff $\alpha=0$. **Cancellations.** Exact zeros of a cost, a first-order part or a denominator, excluded in every ratio that uses them:
- (c1) $g_s=0$: $\Omega=0$ and $\Delta W=0$ for every grip, level and branch;
- (c2) $\beta=0$: $g_b=0$, no mark, $\mu=0$, $\Delta W=0$ for every level;
- (c3) $\alpha+\beta n=0$: $\mathrm{Cost}^{(1)}=0$ for every level and both branches, $\mu$ undefined;
- (c4) $[\ell]=0$, i.e. $\ell(r)=\ell(r')$: $\Omega=0$; on the ring iff $\lvert r\rvert=\lvert r'\rvert$, on the tori for instance at pairs related by a symmetry of the graph that fixes $x_s$;
- (c5) for the bond, the excluded cases of (4.10): $\ell(\mathbf r)=\ell(\mathbf r')$ (then $\Delta_{\mathbf r\mathbf r'}B=0$) and $\Delta_{\mathbf e}\ell(\mathbf R)=0$ (then $\Delta_{\mathbf e}\mathcal F_i=0$); both occur by graph symmetry, e.g. $\mathbf r'$ the image of $\mathbf r$ under a symmetry of the lattice that fixes $0$;
- (c6) in (3.11)(i), $\rho^{\rm fg}_i=0$;
- (c7) in $w^{\rm self}$ of (3.13), the factor $\mu$ of the denominator $\mu\Omega$ is small for $\lvert\alpha\rvert\gg n\lvert\beta\rvert$; the remainder of (3.13) is then amplified by $1/\lvert\mu\rvert$, and (3.8)(i) is stated only for $\lvert\mu\rvert$ bounded away from $0$ (paragraph "Small $\mu$", Step 4).

No cancellation occurs in $C^{\rm self}$: for every $g_b\ne0$, $C^{\rm self}=N_f[\varepsilon_0(g_b)-\varepsilon_0(0)-g_b/\mathcal X]<0$ exactly, since $\varepsilon_0(0)+g_b/\mathcal X=\langle u\rvert C_1+g_bP_x\lvert u\rangle>\varepsilon_0(g_b)$ (variational principle; strict because $(C_1+g_bP_x)u=\varepsilon_0(0)u+g_b\mathcal X^{-1/2}\lvert x\rangle$ is not proportional to $u$). For $\chi\ll1$, $\mathrm{Cost}_{k,\sigma}=\mathrm{Cost}^{(1)}_{k,\sigma}[1-\mu\chi_s(1+O(\chi))]$ vanishes only where $\mu\chi_s=1+O(\chi)$, for instance at $\alpha=-N_f[\varepsilon_0(\sigma\beta M_k)-\varepsilon_0(0)]/(\sigma M_k)$, and a level gap $\mathrm{Cost}_a-\mathrm{Cost}_{a'}$ (Step 5) only where $\lvert\mu\rvert\chi\ge\frac12-O(\chi)$. These zeros and the near-cancellations around them are excluded by the regime $\lvert\mu\rvert\chi\ll1$ of (3.7)–(3.9).

### Step 4. Weight of the mark

**Quotients of place differences.** Relative expansions after division by $\Omega\propto[\ell]$ contain $T_2=2[\ell^{(2)}]/(\mathcal X[\ell])$ and $T_3:=2[\ell^{(3)}]/(\mathcal X^2[\ell])$. Excluding $[\ell]=0$ does not bound them: $[\ell]$ can be small at pairs of places where $[\ell^{(2)}]/\mathcal X$ is not. They are kept exact, and their size enters the remainders through $q:=1+\lvert T_2\rvert/\ell_0+\bar T_3/\ell_0^2\ge1$, with $\bar T_3:=2s_3/(\mathcal X^2\lvert[\ell]\rvert)$, $s_3:=2\ell^{(3)}(0)-\ell^{(3)}(r)-\ell^{(3)}(r')$. Since $\ell^{(3)}(0)-\ell^{(3)}(y)=\mathcal X^{-1}\sum_{k\ne0}(1-\cos k\cdot y)\lambda_k^{-3}\ge0$, $\lvert[\ell^{(3)}]\rvert\le s_3$ and $\lvert T_3\rvert\le\bar T_3$; $\bar T_3$ is what the uniform bound (3.16) needs. $q$ is bounded at the $c$-non-degenerate pairs of the three graphs (3.15).

By Step 3, $\Delta W_{k,\sigma}$ for $(\alpha,\beta)$ is that of the full grip with $\gamma=\beta$, which (3.14) of 07-ilang-weight/03-weight-of-mark gives as $\Omega C^{\rm mark}(1+B^{\rm fg})$, with $\eta=C^{\rm mark}/(nt_f)$; $1+B^{\rm fg}:=\Delta W/(\Omega C^{\rm mark})$. Inserting $C^{\rm mark}=ng_b+C^{\rm self}=\mu\,\mathrm{Cost}^{(1)}+C^{\rm self}$:

$$\Delta W_{k,\sigma}=\Omega\,\bigl(1+B^{\rm fg}_{k,\sigma}\bigr)\bigl[\mu\,\mathrm{Cost}^{(1)}_{k,\sigma}+C^{\rm self}_{k,\sigma}\bigr].\tag{3.6}$$

(3.6) is (3.14) of 07-ilang-weight/03-weight-of-mark rewritten with $C^{\rm mark}=\mu\,\mathrm{Cost}^{(1)}+C^{\rm self}$. At fixed $\beta$, $\Delta W$ does not depend on $\alpha$: the rest term $\alpha\sigma M_k$ is not weighed. The common factor of the two terms in (3.6) is an allocation, not the weight of the self-cost; that weight is derived in (3.13). Dividing by $\mathrm{Cost}=\mathrm{Cost}^{(1)}+C^{\rm self}$ with (3.5), $\frac{\mu\mathrm{Cost}^{(1)}+C^{\rm self}}{\mathrm{Cost}^{(1)}+C^{\rm self}}=\mu\frac{1-\chi_s+O(\chi^2)}{1-\mu\chi_s+O(\mu\chi^2)}$, so, with $\epsilon:=(1+\lvert\mu\rvert)\chi+\lvert B^{\rm fg}\rvert$, which is small for $\lvert\mu\rvert\chi\ll1$ and $\chi q\ll1$ ($\lvert B^{\rm fg}\rvert=O(\chi q)$, below),

$$\Delta W_{k,\sigma}=\mathrm{Cost}_{k,\sigma}\,\mu\,\Omega\,\bigl[1+B^F_{k,\sigma}\bigr],\qquad B^F_{k,\sigma}=B^{\rm fg}_{k,\sigma}-(1-\mu)\chi_s+O(\epsilon^2)=-\frac{g_b}{2t_f}\Bigl[\ell(r)+\ell(r')+2(1-\mu)\ell_0-T_2\Bigr]+O(\chi^2q)+O(\epsilon^2),\tag{3.7}$$

that is, the mark weighing ratio $w_{k,\sigma}:=\frac{\Delta W_{k,\sigma}/\mathrm{Cost}_{k,\sigma}}{\Delta W^{(1)}_{k,\sigma}/\mathrm{Cost}^{(1)}_{k,\sigma}}=1+B^F_{k,\sigma}$.

**Weight of the self-cost.** The weight of the first-order part is exactly $\Delta W^{(1)}=\Omega ng_b$ (3.5), so the weight that the self-cost carries is $\Delta W-\Delta W^{(1)}$. By (3.12)–(3.13) of 07-ilang-weight/03-weight-of-mark with $C^{\rm mark}=ng_b+C^{\rm self}$, $C^{\rm self}=-ng_b^2\frac{\ell_0}{t_f}[1+O(\chi)]$, it is $\Omega C^{\rm self}+\Omega_2n^2g_b^2$ up to terms with one more factor $g_b/t_f$ and the remainder $O((C^{\rm mark})^4)$ of $\mathcal W$, used as quoted. With $[\ell^2]=[\ell]\,(\ell(r)+\ell(r'))$, $\Omega_2n^2g_b^2/(\Omega C^{\rm self})=\frac{[\ell^2]-2[\ell^{(2)}]/\mathcal X}{2\ell_0[\ell]}[1+O(\chi)]=\zeta\,[1+O(\chi)]$, $\zeta$ of (3.13), the $O(\chi)$ from the series of $C^{\rm self}$ (coefficient $1-\ell^{(2)}_0/(\mathcal X\ell_0^2)\in[0,1]$). The terms with one more factor are, relative to $\Omega C^{\rm self}$, $2\Omega_2ng_b/\Omega=-2\chi_s\zeta$ and, with $[\ell\ell^{(2)}]=\ell^{(2)}(r)[\ell]+\ell(r')[\ell^{(2)}]$,
$$\frac{\Omega_3n^3g_b^3}{\Omega C^{\rm self}}=\frac{\chi_sQ_3}{\ell_0^2}\bigl[1+O(\chi)\bigr],\qquad Q_3:=\frac{\ell^{(2)}(r)+\ell^{(2)}_0}{\mathcal X}+\frac{\ell(r')\,T_2-T_3}{2}.$$
Since $\lvert\ell(y)\rvert\le\ell_0$ and $\lvert\ell^{(2)}(y)\rvert\le\ell^{(2)}_0\le\mathcal X\ell_0^2$, $\lvert\zeta\rvert\le q$ and $\lvert Q_3\rvert\le2q\ell_0^2$, so these terms are $O(\chi q)$. Dividing by the weighing factor $\mu\Omega$ of the first-order part:

$$w^{\rm self}_{k,\sigma}:=\frac{(\Delta W_{k,\sigma}-\Delta W^{(1)}_{k,\sigma})/C^{\rm self}_{k,\sigma}}{\Delta W^{(1)}_{k,\sigma}/\mathrm{Cost}^{(1)}_{k,\sigma}}=\frac{1+\zeta+O(\chi q)}{\mu},\qquad \zeta:=\frac{\ell(r)+\ell(r')-T_2}{2\ell_0}.\tag{3.13}$$

$\zeta$ depends on the graph and the two places, not on the grip, the level or the branch. By (3.12)–(3.13) of 07-ilang-weight/03-weight-of-mark, $B^{\rm fg}=\Omega_2C^{\rm mark}/\Omega+\Omega_3(C^{\rm mark})^2/\Omega+\dots=-\eta\ell_0\zeta-\chi_s^2Q_3/\ell_0^2+\dots$, so the part of order $\chi^2$ of the quoted remainder $O(\eta^2\ell^2)$ is $O(\chi^2q)$, and $\lvert B^{\rm fg}\rvert=O(\chi q)$; by (3.16) below, $B^{\rm fg}=-\chi_s\zeta+O(\chi^2q)$, and the constants in these $O$ and in (3.13) are absolute. By definition $B^F=(w^{\rm self}-1)\,C^{\rm self}/\mathrm{Cost}$ exactly; with $C^{\rm self}/\mathrm{Cost}=-\mu\chi_s[1+O(\epsilon)]$ this is (3.7).

**$\zeta$ on the three graphs** (standard lattice sums). $\ell^{(j)}(y)=\mathcal X^{-1}\sum_{k\ne0}\lambda_k^{-j}e^{ik\cdot y}$, $k\in\frac{2\pi}{L}\mathbb Z_L^D$, $\lambda_k=\sum_{i=1}^D(2-2\cos k_i)$; $\lvert y\rvert$ is the length of the shortest representative.
- Ring: $\ell(y)=\frac{L^2-1}{12L}-\frac{\varrho}{2L}$, $\varrho:=\lvert y\rvert(L-\lvert y\rvert)$ (standard). Since $\Lambda(\Lambda^+)^2=\Lambda^+$, $\ell^{(2)}$ solves $\Lambda\ell^{(2)}=\ell$; the second difference of $\varrho$ is $-2$ and that of $\varrho^2$ is $2L^2+2-12\varrho$, so $\ell^{(2)}=c-\frac{\varrho^2}{24L}-\frac{\varrho}{12L}$, which also satisfies the equation at $y=0$, $2[\ell^{(2)}(1)-\ell^{(2)}(0)]=-\ell_0$. Hence $T_2=\frac{\varrho+\varrho'+2}{6L}$ exactly.
- Square torus, large $L$: $\ell_0=\frac1{2\pi}\ln L+O(1)$ and $\ell_0-\ell(y)=\frac1{2\pi}\ln\lvert y\rvert+c_0+O(\lvert y\rvert^{-2}+\lvert y\rvert^2/L^2)$ for $\lvert y\rvert\ge1$, $c_0$ a constant (potential kernel of the square lattice, with the torus correction); $\ell^{(2)}(0)-\ell^{(2)}(y)=\frac{\lvert y\rvert^2}{8\pi}\ln\frac{L}{\lvert y\rvert}+O(\lvert y\rvert^2)$.
- Cubic torus: $\ell_0=\ell_0^\infty+O(1/L)$ with the finite $\ell_0^\infty=(2\pi)^{-3}\int_{[-\pi,\pi]^3}\mathrm d^3k/\lambda(k)$; $\ell(y)=\frac1{4\pi\lvert y\rvert}-\frac{c_1}{L}+O(\lvert y\rvert^{-3}+\lvert y\rvert^2/L^3)$, $c_1$ a constant; $\ell^{(2)}(0)-\ell^{(2)}(y)=\frac{\lvert y\rvert}{8\pi}+O(1)+O(\lvert y\rvert^2/L)$ for $1\le\lvert y\rvert\le\frac{\sqrt3}2L$, the last term from the smallest $k\ne0$ (finite size).
- Uniform bounds on the three graphs (Riemann sums over $k\ne0$ with $1-\cos(k\cdot y)\le\min(2,\lvert k\rvert^2\lvert y\rvert^2/2)$ and $\lambda_k\ge4\lvert k\rvert^2/\pi^2$; standard): $0\le\ell^{(j)}(0)-\ell^{(j)}(y)\le C\,b_{D,j}(y)$, $C$ depending only on $D$, with $b_{2,2}=\lvert y\rvert^2(1+\ln\frac L{\lvert y\rvert})\le L^2$, $b_{2,3}=\lvert y\rvert^2L^2$, $b_{3,2}=\lvert y\rvert+\lvert y\rvert^2/L\le2\lvert y\rvert$, $b_{3,3}=\lvert y\rvert^2L$, $b_{1,3}=\lvert y\rvert^2L^3$ ($b=0$ at $y=0$). Both $\ell^{(j)}(r)-\ell^{(j)}(0)$ and $\ell^{(j)}(r')-\ell^{(j)}(0)$ are $\le0$, so $\lvert[\ell^{(j)}]\rvert$ is at most the larger bound: $\lvert[\ell^{(2)}]\rvert\le CL^2$, $\lvert[\ell^{(3)}]\rvert\le CL^4$ ($D=2$); $\lvert[\ell^{(2)}]\rvert\le2Cm$, $\lvert[\ell^{(3)}]\rvert\le Cm^2L$, $m:=\max(\lvert r\rvert,\lvert r'\rvert)$ ($D=3$); $\lvert[\ell^{(3)}]\rvert\le Cm^2L^3$ (ring). In each case $s_3$ is at most twice the bound on $\lvert[\ell^{(3)}]\rvert$.
- Ring: $\ell^{(3)}$ solves $\Lambda\ell^{(3)}=\ell^{(2)}$ and is, by the same argument, a polynomial of degree 3 in $\varrho$ with leading term $-\varrho^3/(720L)$; so $[\ell^{(3)}]/[\ell]$ is a polynomial in $\varrho,\varrho'$, and $T_3=O(L^2)$ for every pair with $[\ell]\ne0$.

**Non-degeneracy.** The bounds control $T_2$, $\bar T_3$ only together with a lower bound on $\lvert[\ell]\rvert$. A pair of places $(x,x')$ of one of the three graphs is *$c$-non-degenerate* if the lower bound on $\lvert[\ell]\rvert$ in (3.15) holds with a $c>0$ fixed as $L$ grows. Dividing the bounds by $\mathcal X\lvert[\ell]\rvert$ and $\mathcal X^2\lvert[\ell]\rvert$, and on the ring using the exact forms ($\ell_0=\frac{L^2-1}{12L}$, $\varrho,\varrho'\le L^2/4$, $m\le L/2$):

$$\begin{aligned}\text{square torus, }\lvert[\ell]\rvert\ge c:&\quad\lvert T_2\rvert\le\frac{2C}{c},\quad\bar T_3\le\frac{4C}{c};\\ \text{cubic torus, }\lvert[\ell]\rvert\ge\frac{c\,m}{L^2}:&\quad\lvert T_2\rvert\le\frac{4C}{cL},\quad\bar T_3\le\frac{4C}{cL^2};\\ \text{ring, }\lvert[\ell]\rvert\ge c\,m:&\quad\frac{T_2}{\ell_0}=\frac{2(\varrho+\varrho'+2)}{L^2-1}\le2,\quad\bar T_3\le\frac{2CL^2}{c}=O\Bigl(\frac{\ell_0^2}{c}\Bigr);\qquad\text{so }q=O(1)\text{ in the three cases.}\end{aligned}\tag{3.15}$$

The condition is sufficient, not necessary. For $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$ it holds when the two distances differ by a fixed factor, since then $[\ell]=\frac1{2\pi}\ln\frac{\lvert r'\rvert}{\lvert r\rvert}+o(1)$ on the square torus and $[\ell]=\frac1{4\pi}(\lvert r\rvert^{-1}-\lvert r'\rvert^{-1})(1+o(1))$ on the cubic torus, by the forms above, and on the ring, where $[\ell]=-(\lvert r\rvert-\lvert r'\rvert)(L-\lvert r\rvert-\lvert r'\rvert)/(2L)$ exactly; at distances of order $L$ on the tori it requires the inverse-Laplacian kernel of the continuum torus to differ at $r/L$ and $r'/L$ by a fixed amount. It fails near pairs at which $[\ell]$ is small against these scales; there $T_2$ and $T_3$ are kept exact, and $q$ is not bounded by (3.15) (on the ring $T_2/\ell_0\le2$ still holds).

With (3.13) and (3.15):

$$\begin{aligned}\text{ring:}\quad&\zeta=1-\frac{2+4(\varrho+\varrho')}{L^2-1},\qquad\varrho=\lvert r\rvert(L-\lvert r\rvert),\ \ \varrho'=\lvert r'\rvert(L-\lvert r'\rvert)\qquad\text{(exact, }[\ell]\ne0);\\ \text{square torus, (3.15):}\quad&\zeta=1-\frac{\ln(\lvert r\rvert\lvert r'\rvert)+O(1)}{4\pi\ell_0},\qquad 4\pi\ell_0=\ln L^2+O(1),\qquad \lvert r\rvert,\lvert r'\rvert\ge1;\\ \text{cubic torus, (3.15):}\quad&\zeta=\frac{1}{8\pi\ell_0}\Bigl(\frac1{\lvert r\rvert}+\frac1{\lvert r'\rvert}\Bigr)+O\Bigl(\frac1{\lvert r\rvert^3}+\frac1{\lvert r'\rvert^3}+\frac1L\Bigr),\qquad \lvert r\rvert,\lvert r'\rvert\gg1.\end{aligned}\tag{3.14}$$

The $O(1)$ and the $O(1/L)$ of the torus lines depend on $c$. At pairs of the tori where (3.15) fails, the torus lines hold for $(\ell(r)+\ell(r'))/(2\ell_0)$, and $\zeta$ is that minus $T_2/(2\ell_0)$, with $T_2$ exact.

So, at $c$-non-degenerate pairs, $\zeta\to0$ on the cubic torus as $\min(\lvert r\rvert,\lvert r'\rvert)\to\infty$, and on the square torus $\zeta\ll1$ only where $\ln\frac{L^2}{\lvert r\rvert\lvert r'\rvert}\ll\ln L$ ($\zeta=O(1/\ln L)$ at distances of order $L$), and $\zeta\to1$ where $\ln(\lvert r\rvert\lvert r'\rvert)/\ln L\to0$. On the ring $\zeta\ne0$ at every pair of places ($L^2-3$ is not divisible by 4), $\lvert\zeta\rvert\ll1$ only near the curve $4(\varrho+\varrho')=L^2$, and $\zeta=1-O((\lvert r\rvert+\lvert r'\rvert)/L)$ for $\lvert r\rvert,\lvert r'\rvert\ll L$.

**Uniform remainders.** The series above fix the terms of order $\chi$ and bound them by $q$. Their remainders are bounded on every graph of the class by the closed form of the one-object problem. Place the body at $0$ (translation invariance) and put $e:=\eta/\mathcal X$, so that $\varepsilon_0(g_b)=\varepsilon_0(0)+t_fe$, $\ell_e(y):=\mathcal X^{-1}\sum_{k\ne0}\cos(k\cdot y)/(\lambda_k-e)$ and $\ell'_e:=\mathcal X^{-1}\sum_{k\ne0}(\lambda_k-e)^{-2}$. The eigenvector equation of $C_1+g_bP_0$ gives $\psi\propto(\varepsilon_0(g_b)-C_1)^{-1}\lvert0\rangle$, with the entries $(1-\eta\ell_e(y))/(t_f\eta)$; its entry at $0$ gives $\eta=(g_b/t_f)/(1+g_b\ell_e(0)/t_f)$, hence $C^{\rm self}=nt_f\eta-ng_b=-nt_f\eta^2\ell_e(0)/(1-\eta\ell_e(0))$. Normalized, $\lvert\psi(y)\rvert^2=(1-\eta\ell_e(y))^2/(\mathcal X+\eta^2\ell'_e)$, and at first order in $g_s$ the source shifts each field object by $g_s\lvert\psi(r)\rvert^2$. With $C^{\rm mark}=nt_f\eta$ and $\Omega=-2g_s[\ell]/t_f$, exactly at first order in $g_s$, on every graph of the class and at every pair with $[\ell]\ne0$:

$$1+B^{\rm fg}=\frac{\Delta W}{\Omega C^{\rm mark}}=\frac{[\ell_e]}{[\ell]}\;\frac{1-\frac\eta2\bigl(\ell_e(r)+\ell_e(r')\bigr)}{1+\eta^2\ell'_e/\mathcal X},\qquad \mu\,w^{\rm self}=1+\frac{C^{\rm mark}}{C^{\rm self}}\,B^{\rm fg}=1-\frac{1-\eta\ell_e(0)}{\eta\,\ell_e(0)}\,B^{\rm fg}.\tag{3.16}$$

Let $\chi\le\frac14$. For $g_b>0$, $0<\eta\le g_b/t_f$ (variational bound of Step 3); for $g_b<0$, $e<0$ gives $0<\ell_e(0)\le\ell_0$ and $\lvert\eta\rvert\le\lvert g_b\rvert/(t_f(1-\chi))$. So $\lvert\eta\rvert\ell_0\le\frac43\chi$, $\lvert e\rvert\le\lvert\eta\rvert\ell_0\lambda_1\le\frac{\lambda_1}3$ (as $\mathcal X\ell_0\lambda_1\ge1$) and $\lvert\lambda_k-e\rvert\ge\frac23\lambda_k$. With $\ell^{(2)}_0\le\mathcal X\ell_0^2$ this gives $\lvert\ell_e(y)-\ell(y)\rvert\le\frac32\lvert\eta\rvert\ell_0^2$, $\eta^2\ell'_e/\mathcal X\le\frac94\eta^2\ell_0^2$, $\lvert\eta\rvert\ell_0\ge\frac23\chi$, and, with $\lvert\cos k\cdot r-\cos k\cdot r'\rvert\le(1-\cos k\cdot r)+(1-\cos k\cdot r')$, $[\ell_e]/[\ell]=1+\eta T_2/2+\vartheta$, $\lvert\vartheta\rvert\le\frac34\eta^2\bar T_3$. Expanding (3.16) in these quantities, for $\chi q\le1$: $B^{\rm fg}=-\eta\ell_0\zeta+R_B$ with $\lvert R_B\rvert\le K\chi^2q$, $\mu w^{\rm self}=1+\zeta+R_w$ with $\lvert R_w\rvert\le K\chi q$, $\eta\ell_0=\chi_s[1+O(\chi)]$ and $C^{\rm self}=-ng_b\chi_s[1+O(\chi)]$, where $K$ and the constants of these $O$ are absolute: the same for every $D$, $L$ and pair. Each derivative in $e$ of $(\lambda_k-e)^{-1}$ gives a factor at most $\frac3{2\lambda_k}\le\frac1{2\lvert e\rvert}$, so the derivatives in $g_b$ of these remainders obey the same bounds divided by $\lvert g_b\rvert$, which carries them to differences between two internal states (Step 5). Hence the remainders of (3.5), (3.7), (3.8), (3.9) and (3.13) hold with absolute constants for $\chi\le\frac14$, $\chi q\le1$, and are small where $\chi q\ll1$; the size of $q$ is given by (3.15), so the statements at $c$-non-degenerate pairs are uniform in $L$ and in the distances.

**Criterion.** By (3.16), $\mu\,(w^{\rm self}-1)=1+\zeta-\mu+R_w$, $\lvert R_w\rvert\le K\chi q$, with $\mu$ a constant of the grip and $\zeta$ a function of the places; dividing by $\mu$ multiplies the remainder by $1/\lvert\mu\rvert$. Let $0<\mu_-\le1$ and $\mu_+\ge2$ be constants fixed independently of $\chi$, $L$ and the places. At places with $[\ell]\ne0$ and $\chi q\ll1$ (in particular at the $c$-non-degenerate pairs of (3.15)):

$$\text{(i)}\ \ \mu_-\le\lvert\mu\rvert\le\mu_+:\ \ w^{\rm self}_{k,\sigma}=1+O(\chi q)\iff\mu=1+\zeta+O(\chi q);\qquad\text{(ii)}\ \ \alpha=0:\ \ w^{\rm self}_{k,\sigma}=1+\zeta+O(\chi q),\quad \Delta W_{k,\sigma}=\mathrm{Cost}_{k,\sigma}\,\Omega\,\bigl[1-\eta\ell_0\zeta+O(\chi^2q)\bigr].\tag{3.8}$$

In (i), $\lvert\mu-1-\zeta\rvert\le A\chi q$ gives $\lvert w^{\rm self}-1\rvert\le(A+K)\chi q/\mu_-$, and $\lvert w^{\rm self}-1\rvert\le A\chi q$ gives $\lvert\mu-1-\zeta\rvert\le(\mu_+A+K)\chi q$. Where $\zeta\to0$, (i) is $\mu=1$, i.e. $\alpha=F(0)=0$; where $\zeta\to1$ (the ring for $\lvert r\rvert,\lvert r'\rvert\ll L$, the square torus for $\ln(\lvert r\rvert\lvert r'\rvert)\ll\ln L$ at $c$-non-degenerate pairs) it is $\mu=2$, i.e. $\alpha=-\beta n/2$. By (3.14) and (3.15), at $\alpha=0$ the self-cost is weighed like the first-order part up to $O(1/\min(\lvert r\rvert,\lvert r'\rvert))+O(\chi)$ on the cubic torus at $c$-non-degenerate pairs; on the square torus, at $c$-non-degenerate pairs, only where $\ln\frac{L^2}{\lvert r\rvert\lvert r'\rvert}\ll\ln L$, up to $O(1/\ln L)+O(\chi)$ at distances of order $L$; on the ring only near the curve $4(\varrho+\varrho')=L^2$. At the other pairs of the tori the condition is $\lvert\zeta\rvert\ll1$ with $\zeta$ of (3.13), $T_2$ exact, where $\chi q\ll1$; it is not reduced to distances there.

*Small $\mu$* ($\lvert\alpha\rvert\gg n\lvert\beta\rvert$, (c7)). For every $\mu\ne0$, $w^{\rm self}=(1+\zeta+R_w)/\mu$ by (3.16). The weight of the self-cost does not depend on $\alpha$ at fixed $\beta$ (3.6), while that of the first-order part per cost is $\mu\Omega$; so, as $\mu\to0$, $w^{\rm self}$ grows as $(1+\zeta)/\mu$ where $\lvert1+\zeta\rvert\gg\chi q$, and for $\lvert\mu\rvert\lesssim\chi q$ the order of (3.13) leaves an error of order one or larger. There the next order decides. The terms of order $\chi$ found above give $\mu w^{\rm self}=1+\zeta+\chi_s\nu_1+O(\chi^2)$, $\nu_1:=\zeta(a-2)+Q_3/\ell_0^2$, $a:=1-\ell^{(2)}_0/(\mathcal X\ell_0^2)\in[0,1]$, where the constant of $O(\chi^2)$ depends on the graph and the pair (it contains $[\ell^{(4)}]/[\ell]$, not bounded here); so on a fixed graph $w^{\rm self}=(1+\zeta+\chi_s\nu_1)/\mu+O(\chi^2/\lvert\mu\rvert)$. Control for the range of (i): on the ring $1+\zeta=0$ exactly at the pairs with $2(\varrho+\varrho')=L^2-2$ (for even $L$: $\lvert r\rvert=L/2$, $\lvert r'\rvert=L/2-1$). For $L=4$, $\lvert r\rvert=2$, $\lvert r'\rvert=1$: $\ell=(\frac5{16},-\frac1{16},-\frac3{16})$, $\ell^{(2)}=(\frac9{64},-\frac1{64},-\frac7{64})$, $\ell^{(3)}=(\frac{17}{256},-\frac1{256},-\frac{15}{256})$ at $y=0,1,2$, so $\zeta=-1$, $a=\frac{16}{25}$, $Q_3/\ell_0^2=-\frac8{25}$, $\nu_1=\frac{26}{25}$; the grips with $\sigma=+1$, $\beta>0$, $\alpha+\beta n=nt_f/(M_k\ell_0)$ have $\mu=\chi_s=1+\zeta+O(\chi q)$, but $w^{\rm self}=\frac{26}{25}+O(\chi)$.

The leading factor $\mu\Omega$ of (3.7) is the same for all levels and both branches for every affine grip. Two bodies with grips $(\alpha_i,\beta_i)$ have the same leading factor iff $\mu_1=\mu_2$, i.e. $\alpha_1\beta_2=\alpha_2\beta_1$ (a condition, M7), which holds for all bodies with $\alpha_i=0$.

### Step 5. Level costs of one body

For two internal states $a\ne a'$ of one body, $g_a\ne g_{a'}$ (the $M_k$ are distinct and positive). The coefficients of $\mathcal W$ in (3.13) of 07-ilang-weight/03-weight-of-mark and of the series of $C^{\rm mark}(g)$ do not depend on the state, and $\Omega_2n^2/(\Omega n)=-\frac1{2t_f}[\ell(r)+\ell(r')-T_2]$. So $\Delta W_a=\Omega n\,g_a[1-\frac{g_a}{t_f}(\ell_0+\frac12[\ell(r)+\ell(r')-T_2])]+O(\Omega ng_a\chi^2q)$ (the terms of order $\chi^2$ carry $\zeta$ and $Q_3/\ell_0^2$, Step 4; by (3.16) the constant is absolute, also for the difference of the two states) and $\mathrm{Cost}_a=\frac{g_a}\beta(\alpha+\beta n)[1-\mu\frac{g_a\ell_0}{t_f}]+O(ng_a\chi^2)$. The differences are proportional to $g_a-g_{a'}$, and their quotient gives

$$\Delta W_a-\Delta W_{a'}=(\mathrm{Cost}_a-\mathrm{Cost}_{a'})\,\mu\,\Omega\Bigl[1-\frac{g_a+g_{a'}}{2t_f}\Bigl(\ell(r)+\ell(r')+2(1-\mu)\ell_0-T_2\Bigr)+O\bigl((1+\lvert\mu\rvert)^2\chi^2q\bigr)\Bigr],\tag{3.9}$$

with $\chi$ the larger of the two states. With $\Omega=\phi(x)-\phi(x')$ this is (4.9) of 07-ilang-weight/04-weight-of-bond with $\mathcal F_i=\mu_i\phi$, together with its relative remainder at first order. It involves only differences of costs of levels, which the total contract fixes (M6).

### Step 6. Weight of a bond

As in Step 3, the eigenvalue of the pair sector is $\alpha_1\sigma_1M^{(1)}_{k_1}+\alpha_2\sigma_2M^{(2)}_{k_2}+c_s+N_f\varepsilon_0$, $\varepsilon_0$ the lowest eigenvalue of $C_1+g_1P_{x_1}+g_2P_{x_2}+g_sP_{x_s}$, $g_i=\sigma_i\beta_iM^{(i)}_{k_i}$. The $\alpha_i$ add a constant that depends neither on $x_1$, nor on $\mathbf r$, nor on the source; so $B(\mathbf r)$ of (4.2), $W_b$ and the numerator of (4.10) are those of the full grips at the same $g_1,g_2$. The sector depends on the grips only through the $g_i$ and these constants, and every real $g_i$ is reached with $\gamma_i>0$ by the choice of $\sigma_i$; so (4.2), (4.10)–(4.12) hold for $\beta_i$ of either sign. With $\mathcal F_i=\mu_i\phi$ from (3.9) in the definition (4.10), under the conditions of (4.10) ($\ell(\mathbf r)\ne\ell(\mathbf r')$, $\Delta_{\mathbf e}\ell(\mathbf R)\ne0$, $\eta K\ll1$; cancellations (c1), (c2), (c5) excluded) and for a source outside the pair:

$$\rho^F_i=\frac{\rho^{\rm fg}_i}{\mu_i}=\frac{\alpha_i+n\beta_i}{n\beta_i}\bigl[\rho_0+O(\eta K)\bigr],\qquad \rho^{{\rm mid},F}_i=\frac{\alpha_i+n\beta_i}{n\beta_i}\bigl[1+\kappa^{\rm mid}+O(\eta K)\bigr],\tag{3.10}$$

with $\kappa^{\rm mid}$ of (4.12) on the three graphs. If the denominator uses the level factor of (3.9) with its remainder, the quotient changes by $\bigl(1+\bar B^{\rm fg}_i\bigr)/\bigl(1+\bar B^F_i\bigr)=1+O\bigl((1+\lvert\mu_i\rvert)\eta\bigr)$, the bracket of (3.9) at $\mu_i=1$ over that at $\mu_i$; this is the form of (4.13). The bond cost is weighed with $\phi$ times $\rho^{\rm fg}_i$, the first-order level costs of body $i$ with $\mu_i\phi$; for $\rho^{\rm fg}_i\ne0$:

$$\text{(i)}\ \ \rho^F_i=\rho^{\rm fg}_i\iff\mu_i=1\iff\alpha_i=0;\qquad\text{(ii)}\ \ \alpha_i=0:\ \ \rho^{{\rm mid},F}_i-1=\kappa^{\rm mid}+O(\eta K).\tag{3.11}$$

By (i), where $\rho^{\rm fg}_i\to1$ ($\lvert\kappa\rvert$, $\eta K\ll1$) the bond is weighed like the first-order level costs iff $\alpha_i=0$; for $\alpha_i\ne0$ it carries the extra factor $1/\mu_i=1+\alpha_i/(n\beta_i)$, the same at every separation, level and branch. By (4.12) with the conditions stated with it (note of question.md), the term $\kappa^{\rm mid}$ of (ii) is zero at leading order in $\eta$ on the ring for $x_s$ outside the arc between the bodies (a source on that arc, between the bodies, is outside the scope of the question, note (b)); it is $O(\lvert\mathbf r\rvert\lvert\mathbf r'\rvert/d^2)$ on the cubic torus, small for $\lvert\mathbf r\rvert,\lvert\mathbf r'\rvert\ll d\ll L$; and $O(\frac{\lvert\mathbf r'\rvert^2}{d^2}\ln\frac{L^2}{\lvert\mathbf r'\rvert d})$ on the square torus, small only for $d$ large against $\lvert\mathbf r\rvert$ times a logarithm of $L$.

**Mark and bond together.** For a source outside the pair, under the conditions of (4.10) and where $\kappa^{\rm mid}$ and $\eta K$ are small, the bond requires $\mu_i=1$ by (i), and then (3.8)(ii) leaves the self-cost of the mark with the factor $1+\zeta$, at pairs of places $(x,x')$ with $[\ell]\ne0$ and $\chi q\ll1$. So both are weighed like the first-order part iff $\alpha=0$ and $\lvert\zeta\rvert\ll1$, with $\zeta$ of (3.13), $T_2$ exact. By (3.14) and (3.15): on the cubic torus, at $c$-non-degenerate pairs with $1\ll\min(\lvert r\rvert,\lvert r'\rvert)$ and for $\lvert\mathbf r\rvert,\lvert\mathbf r'\rvert\ll d\ll L$, by the grips $F(\Phi)=\beta\Phi$, $\beta\ne0$ of either sign; on the square torus, at $c$-non-degenerate pairs, by the same grips only where $\ln\frac{L^2}{\lvert r\rvert\lvert r'\rvert}\ll\ln L$ (up to $O(1/\ln L)$ at source distances of order $L$) and for $d$ large against $\lvert\mathbf r\rvert$ times a logarithm of $L$; on the ring, for $x_s$ outside the arc between the bodies, by no affine grip, except for pairs of places near the curve $4(\varrho+\varrho')=L^2$. At pairs of the tori where (3.15) fails, the criterion stays $\alpha=0$ and $\lvert\zeta\rvert\ll1$ with $T_2$ exact, and is not reduced to distances.

### Step 7. Illustration outside the admissible class

$F(\Phi)=\Phi^2$ has $F(0)=0$ and, for $N_f\ge2$, is not a sum of pair terms (3.2), so it is not an admissible description under A5: the computation below is a mathematical illustration, not a control (M9). At first order in the grip the eigenvalue shift is $\sigma M_k\langle\Phi_x^2\rangle$ in the field state with the source, the product of $N_f$ copies of one vector $\psi$ with $N_f\lvert\psi(y)\rvert^2=n+\delta N(y)$. $\Phi_x$ is then binomial, $\langle\Phi_x^2\rangle=N_fp+N_f(N_f-1)p^2$, $p=\lvert\psi(x)\rvert^2$. With $n':=(1-1/N_f)n$:

$$\mathrm{Cost}^{(1)}=\sigma M_k\,n(1+n'),\qquad \Delta W^{(1)}=\sigma M_k[\delta N](1+2n')+O(g_s^2),\qquad \frac{\Delta W^{(1)}}{\mathrm{Cost}^{(1)}}=\Omega\,\frac{1+2n'}{1+n'} .\tag{3.12}$$

For $N_f=1$, $n'=0$ and the factor is 1, as it must be: $\Phi_x^2=\Phi_x$ is then the affine grip $(0,1)$ of (3.3), with $\mu=1$.

## Result

- **(3.1)** For every real $F$ on $\{0,\dots,N_f\}$ and every $N_f\ge1$, the grip term is the sum over $k$ of $\Delta^kF(0)$ times a term acting on the body and $k$ field objects together. Exact.
- **(3.2)** For $N_f\ge2$, $G_F$ is a sum of pair terms (on $\mathcal H$; on $\mathcal H_{\rm adm}$ when the body is the only object of its type) iff $\Delta^kF(0)=0$ for $2\le k\le N_f$, iff $F(\Phi)=\alpha+\beta\Phi$ on $\{0,\dots,N_f\}$; then $G_{\alpha,\beta}$ is a sum of $N_f$ pair terms between the body and single field objects. Adding or moving listed pair terms (M6) does not change this. Exact; a characterization, no control in the class. Illustration outside the class: $F=\Phi^2$, a three-object term.
- **(3.3)** For $N_f=1$ every $F$ gives the pair term of the affine grip with $\alpha=F(0)$, $\beta=F(1)-F(0)$. For every $N_f\ge1$ the grips within pair terms are the affine grips $(\alpha,\beta)$, each the same description as the full grip $\beta\Phi$ with the own term $\alpha S$; different $(\alpha,\beta)$ give different $C$. Exact; a characterization, no control in the class.
- **(3.4)** For an affine grip, a held body and the stated state: $E_{k,\sigma}=\alpha\sigma M_k+c_s+N_f\varepsilon_0(x)$, $\mathrm{Cost}_{k,\sigma}=\alpha\sigma M_k+C^{\rm mark}_{k,\sigma}$, $C^{\rm mark}$ that of the full grip with $\gamma=\beta$, $g_b=\sigma\beta M_k$. Exact.
- **(3.5)** First-order part $\mathrm{Cost}^{(1)}=\sigma M_k(\alpha+\beta n)$, its weight $\Delta W^{(1)}=\mu\Omega\,\mathrm{Cost}^{(1)}$ (first order in $g_s$), mark fraction $\mu=\beta n/(\alpha+\beta n)$, the same for all levels and both branches, and self-cost $C^{\rm self}=-ng_b\chi_s[1+O(\chi)]$ for $\chi\ll1$, with $C^{\rm self}<0$ exactly for every $\beta\ne0$. Excluded (cancellations of Step 3): $g_s=0$, $\beta=0$, $\alpha+\beta n=0$, $[\ell]=0$; for $\chi\ll1$, $\mathrm{Cost}_{k,\sigma}$ vanishes only where $\mu\chi_s=1+O(\chi)$, and a level gap of costs only where $\lvert\mu\rvert\chi\ge\frac12-O(\chi)$, both excluded by $\lvert\mu\rvert\chi\ll1$.
- **(3.6)** For every affine grip with $\beta\ne0$, $\alpha+\beta n\ne0$, every level and both branches, first order in $g_s$, regime $\chi\ll1$ of (3.14) of 07-ilang-weight/03-weight-of-mark: $\Delta W=\Omega(1+B^{\rm fg})[\mu\,\mathrm{Cost}^{(1)}+C^{\rm self}]$, a rewriting of that equation; at fixed $\beta$, $\Delta W$ does not depend on $\alpha$, so the rest term is not weighed. The common factor in front of $C^{\rm self}$ is an allocation, not the weight of the self-cost, which is (3.13). Excluded: (c1)–(c4) of Step 3. It holds for every affine grip, so the class has no control for it.
- **(3.7)** Under the conditions of (3.6), $\lvert\mu\rvert\chi\ll1$ and $\chi q\ll1$ ($q$ bounded at the $c$-non-degenerate pairs of (3.15)), on the three graphs: $\Delta W=\mathrm{Cost}\,\mu\Omega[1+B^F]$, and the mark weighing ratio is $w=1+B^F$, with the first breaking term $B^F=-\frac{g_b}{2t_f}[\ell(r)+\ell(r')+2(1-\mu)\ell_0-T_2]$, $T_2$ exact, and remainder $O(\chi^2q)+O(\epsilon^2)$ with absolute constants (3.16), $\epsilon=(1+\lvert\mu\rvert)\chi+\lvert B^{\rm fg}\rvert$.
- **(3.13)** Under the conditions of (3.6), at places with $[\ell]\ne0$ and $\chi q\ll1$, $q=1+\lvert T_2\rvert/\ell_0+\bar T_3/\ell_0^2$, $\bar T_3=2s_3/(\mathcal X^2\lvert[\ell]\rvert)\ge\lvert T_3\rvert$, $s_3=2\ell^{(3)}(0)-\ell^{(3)}(r)-\ell^{(3)}(r')$: the self-cost carries the weight $\Delta W-\Delta W^{(1)}$, and its weighing ratio against the first-order part is $w^{\rm self}=\mu^{-1}[1+\zeta+O(\chi q)]$, with an absolute constant in $O(\chi q)$ (3.16), so that its error is $O(\chi q/\lvert\mu\rvert)$; $\zeta=(\ell(r)+\ell(r')-T_2)/(2\ell_0)$, $T_2$ exact; $\zeta$ is the same for every affine grip, level and branch, and $B^F=(w^{\rm self}-1)C^{\rm self}/\mathrm{Cost}$ exactly.
- **(3.15)** Non-degeneracy. $T_2$ and $\bar T_3$ are not bounded by $[\ell]\ne0$ alone. A pair of places is $c$-non-degenerate, $c>0$ fixed as $L$ grows, if $\lvert[\ell]\rvert\ge c$ (square torus), $\lvert[\ell]\rvert\ge c\,m/L^2$ (cubic torus) or $\lvert[\ell]\rvert\ge c\,m$ (ring), $m=\max(\lvert r\rvert,\lvert r'\rvert)$. Then $\lvert T_2\rvert\le2C/c$, $\bar T_3\le4C/c$ (square torus), $\lvert T_2\rvert\le4C/(cL)$, $\bar T_3\le4C/(cL^2)$ (cubic torus), $\bar T_3\le2CL^2/c=O(\ell_0^2/c)$ (ring), $C$ depending only on $D$; on the ring $T_2/\ell_0=2(\varrho+\varrho'+2)/(L^2-1)\le2$ at every pair with $[\ell]\ne0$. So $q=O(1)$ at the $c$-non-degenerate pairs of the three graphs. The condition is sufficient, not necessary; for $1\ll\lvert r\rvert,\lvert r'\rvert\ll L$ it holds when the two distances differ by a fixed factor; at pairs where it fails, $T_2$ and $T_3$ are kept exact and $q$ is not bounded by it.
- **(3.14)** On the ring, exactly, for every pair with $[\ell]\ne0$, $\zeta=1-\frac{2+4(\varrho+\varrho')}{L^2-1}$, $\varrho=\lvert r\rvert(L-\lvert r\rvert)$. At $c$-non-degenerate pairs (3.15): on the square torus, large $L$, $\zeta=1-\frac{\ln(\lvert r\rvert\lvert r'\rvert)+O(1)}{4\pi\ell_0}$, $4\pi\ell_0=\ln L^2+O(1)$, for $\lvert r\rvert,\lvert r'\rvert\ge1$; on the cubic torus $\zeta=\frac{1}{8\pi\ell_0}(\frac1{\lvert r\rvert}+\frac1{\lvert r'\rvert})+O(\lvert r\rvert^{-3}+\lvert r'\rvert^{-3}+L^{-1})$ for $\lvert r\rvert,\lvert r'\rvert\gg1$; the $O(1)$ and $O(L^{-1})$ depend on $c$. Thus, at these pairs, $\zeta\to0$ on the cubic torus as $\min(\lvert r\rvert,\lvert r'\rvert)\to\infty$, and on the square torus $\zeta\ll1$ only where $\ln\frac{L^2}{\lvert r\rvert\lvert r'\rvert}\ll\ln L$, and $\zeta\to1$ where $\ln(\lvert r\rvert\lvert r'\rvert)/\ln L\to0$. At the other pairs of the tori the torus forms hold for $(\ell(r)+\ell(r'))/(2\ell_0)$, and $\zeta$ is that minus $T_2/(2\ell_0)$, $T_2$ exact. On the ring $\zeta\ne0$ at every pair of places, $\lvert\zeta\rvert\ll1$ only near the curve $4(\varrho+\varrho')=L^2$, and $\zeta=1-O((\lvert r\rvert+\lvert r'\rvert)/L)$ for $\lvert r\rvert,\lvert r'\rvert\ll L$.
- **(3.16)** On every graph of the class and at every pair with $[\ell]\ne0$, at first order in $g_s$, exactly: $1+B^{\rm fg}=\frac{[\ell_e]}{[\ell]}\,\frac{1-\frac\eta2(\ell_e(r)+\ell_e(r'))}{1+\eta^2\ell'_e/\mathcal X}$ and $\mu w^{\rm self}=1-\frac{1-\eta\ell_e(0)}{\eta\ell_e(0)}B^{\rm fg}$, with $e=\eta/\mathcal X$, $\ell_e(y)=\mathcal X^{-1}\sum_{k\ne0}\cos(k\cdot y)/(\lambda_k-e)$, $\ell'_e=\mathcal X^{-1}\sum_{k\ne0}(\lambda_k-e)^{-2}$. For $\chi\le\frac14$ and $\chi q\le1$: $B^{\rm fg}=-\eta\ell_0\zeta+R_B$, $\lvert R_B\rvert\le K\chi^2q$, and $\mu w^{\rm self}=1+\zeta+R_w$, $\lvert R_w\rvert\le K\chi q$, with $K$ absolute; the derivatives in $g_b$ of the remainders obey the same bounds divided by $\lvert g_b\rvert$. So the remainders of (3.5), (3.7), (3.8), (3.9) and (3.13) are uniform in $D$, $L$ and the places, and small where $\chi q\ll1$.
- **(3.8)** Under the conditions of (3.13), with constants $0<\mu_-\le1$ and $\mu_+\ge2$ fixed independently of $\chi$, $L$ and the places: (i) for grips with $\mu_-\le\lvert\mu\rvert\le\mu_+$, $w^{\rm self}=1+O(\chi q)$ iff $\mu=1+\zeta+O(\chi q)$, the constants of the two directions related by the factors $1/\mu_-$ and $\mu_+$ (Step 4); where $\zeta\to0$ this is $\mu=1$, i.e. $\alpha=F(0)=0$, and where $\zeta\to1$ (the ring for $\lvert r\rvert,\lvert r'\rvert\ll L$, the square torus for $\ln(\lvert r\rvert\lvert r'\rvert)\ll\ln L$ at $c$-non-degenerate pairs) it is $\mu=2$, i.e. $\alpha=-\beta n/2$. (ii) For $\alpha=0$, every level and both branches: $w^{\rm self}=1+\zeta+O(\chi q)$ and $\Delta W=\mathrm{Cost}\,\Omega[1-\eta\ell_0\zeta+O(\chi^2q)]$; the self-cost is weighed like the first-order part up to $O(1/\min(\lvert r\rvert,\lvert r'\rvert))+O(\chi)$ on the cubic torus at $c$-non-degenerate pairs, on the square torus at $c$-non-degenerate pairs only where $\ln\frac{L^2}{\lvert r\rvert\lvert r'\rvert}\ll\ln L$, up to $O(1/\ln L)+O(\chi)$ at distances of order $L$, on the ring only near the curve $4(\varrho+\varrho')=L^2$, and at the other pairs of the tori where $\lvert\zeta\rvert\ll1$ with $T_2$ exact. Small $\mu$ ($\lvert\alpha\rvert\gg n\lvert\beta\rvert$): $w^{\rm self}=(1+\zeta+O(\chi q))/\mu$ grows as $(1+\zeta)/\mu$ as $\mu\to0$ where $\lvert1+\zeta\rvert\gg\chi q$; for $\lvert\mu\rvert\lesssim\chi q$ it is not fixed at the order of (3.13), and on a fixed graph it is $(1+\zeta+\chi_s\nu_1)/\mu+O(\chi^2/\lvert\mu\rvert)$, $\nu_1$ of Step 4, with a constant that depends on the graph and the pair. Two bodies have the same leading factor $\mu\Omega$ iff $\alpha_1\beta_2=\alpha_2\beta_1$ (a condition, M7). Control: coupling (a) for (i); the ring for (ii); for the range of (i), the ring with $L=4$, $\lvert r\rvert=2$, $\lvert r'\rvert=1$ ($\zeta=-1$, $\nu_1=\frac{26}{25}$), where the grips with $\mu=\chi_s$ satisfy $\mu=1+\zeta+O(\chi q)$ but have $w^{\rm self}=\frac{26}{25}+O(\chi)$.
- **(3.9)** For two internal states of one body with an affine grip, at first order in $g_s$ and $\chi\ll1$, $\lvert\mu\rvert\chi\ll1$, $\chi q\ll1$, excluding (c1)–(c4): the difference of weights is the difference of costs times $\mu\Omega[1-\frac{g_a+g_{a'}}{2t_f}(\ell(r)+\ell(r')+2(1-\mu)\ell_0-T_2)+O((1+\lvert\mu\rvert)^2\chi^2q)]$ with an absolute constant (3.16), $T_2$ exact; this is (4.9) of 07-ilang-weight/04-weight-of-bond with $\mathcal F_i=\mu_i\phi$.
- **(3.10)** For two held bodies with affine grips $(\alpha_i,\beta_i)$, $\beta_i\ne0$, $\alpha_i+n\beta_i\ne0$, either sign, every level and both branches, in the regime and on the three graphs of (4.10)–(4.12), under the conditions of (4.10) ($\ell(\mathbf r)\ne\ell(\mathbf r')$, $\Delta_{\mathbf e}\ell(\mathbf R)\ne0$, $\eta K\ll1$), $g_s\ne0$, and for a source outside the pair: $\rho^F_i=\rho^{\rm fg}_i/\mu_i=\frac{\alpha_i+n\beta_i}{n\beta_i}[\rho_0+O(\eta K)]$, and with the midpoint as the place of the pair $\frac{\alpha_i+n\beta_i}{n\beta_i}[1+\kappa^{\rm mid}+O(\eta K)]$, $\kappa^{\rm mid}$ of (4.12).
- **(3.11)** Under the conditions of (3.10) and $\rho^{\rm fg}_i\ne0$: (i) $\rho^F_i=\rho^{\rm fg}_i$ iff $\alpha_i=0$; where $\lvert\kappa\rvert,\eta K\ll1$ the bond is weighed like the first-order level costs iff $\alpha_i=0$, and for $\alpha_i\ne0$ it carries the extra factor $1+\alpha_i/(n\beta_i)$. (ii) For $\alpha_i=0$, $\rho^{{\rm mid},F}_i-1=\kappa^{\rm mid}+O(\eta K)$, with $\kappa^{\rm mid}$ of (4.12) under its conditions: zero at leading order in $\eta$ on the ring for $x_s$ outside the arc between the bodies (a source on the arc is outside the scope); $O(\lvert\mathbf r\rvert\lvert\mathbf r'\rvert/d^2)$ on the cubic torus, small for $\lvert\mathbf r\rvert,\lvert\mathbf r'\rvert\ll d\ll L$; $O(\frac{\lvert\mathbf r'\rvert^2}{d^2}\ln\frac{L^2}{\lvert\mathbf r'\rvert d})$ on the square torus, small only for $d$ large against $\lvert\mathbf r\rvert$ times a logarithm of $L$. Mark and bond together, for a source outside the pair, where $\kappa^{\rm mid}$ and $\eta K$ are small and at pairs $(x,x')$ with $[\ell]\ne0$, $\chi q\ll1$: both are weighed like the first-order part iff $\alpha=0$ and $\lvert\zeta\rvert\ll1$ ($T_2$ exact), i.e. by $F(\Phi)=\beta\Phi$, $\beta\ne0$ of either sign, on the cubic torus at $c$-non-degenerate pairs with $1\ll\min(\lvert r\rvert,\lvert r'\rvert)$ and for $\lvert\mathbf r\rvert,\lvert\mathbf r'\rvert\ll d\ll L$; on the square torus, at $c$-non-degenerate pairs, by the same grips only where $\ln\frac{L^2}{\lvert r\rvert\lvert r'\rvert}\ll\ln L$ (up to $O(1/\ln L)$ at source distances of order $L$) and for $d$ large against $\lvert\mathbf r\rvert$ times a logarithm of $L$; on the ring, for $x_s$ outside the arc between the bodies, by no affine grip, except for pairs of places near the curve $4(\varrho+\varrho')=L^2$; at the other pairs of the tori the criterion is not reduced to distances. Control: coupling (a); the ring.
- **(3.12)** Illustration outside the admissible class (not a control, M9; for $N_f\ge2$, $F=\Phi^2$ is not a sum of pair terms by (3.2)): for $F=\Phi^2$ and $N_f\ge2$, first order in the grip and in $g_s$, the first-order part is weighed with $\Omega(1+2n')/(1+n')$, $n'=(1-1/N_f)n$, whose factor $(1+2n')/(1+n')$ lies strictly between 1 and 2 although $F(0)=0$.

## Consistency checks

1. Special grips: at $\alpha=0$, $\mu=1$ and (3.7), (3.9), (3.10) reduce to (3.14) of 07-ilang-weight/03-weight-of-mark and to (4.9), (4.10) of 07-ilang-weight/04-weight-of-bond for the full grip; at $\alpha=1$, $\beta=\gamma$, $\mu\Omega=F^{(a)}$, $2(1-\mu)\ell_0=2\ell_0/(1+\gamma n)$ gives $B^F=B^{(a)}$ of (3.15) of 07-ilang-weight/03-weight-of-mark exactly, $\mu\phi$ is (4.9) for coupling (a), and $1/\mu$ with the remainder factor of Step 6 is (4.13). Passed.
2. Dimensions: $\alpha,\beta,\mu,\Omega,\chi,\zeta,T_2,B^F,w^{\rm self},\rho^F_i$ are numbers ($\ell^{(j)}$ are entries of powers of the inverse of a graph Laplacian); $g_b$, $\mathrm{Cost}$, $C^{\rm self}$, $\Delta W$ and $B(\mathbf r)$ are costs ($g_b^2\ell_0/t_f$ and $ng_1g_2\ell/t_f$ included). Passed.
3. Ring $L=3$: $\Lambda=3\cdot\mathbb 1-J$ ($J$ the all-ones matrix) gives $\Lambda^+=\frac13(\mathbb 1-J/3)$, $(\Lambda^+)^2=\frac19(\mathbb 1-J/3)$; for $\lvert r\rvert=0$, $\lvert r'\rvert=1$: $\ell_0=\frac29$, $\ell(1)=-\frac19$, $[\ell]=\frac13$, $[\ell^{(2)}]=\frac19$, $T_2=\frac29$, $\zeta=-\frac14$, which (3.14) gives with $\varrho=0$, $\varrho'=2$. Passed.

## Open issues

- The size of $O(\eta K)$ of (4.10) on each graph is that of 07-ilang-weight/04-weight-of-bond; the constants $C$ of (3.15), the $O(1)$ terms of (3.14) on the two tori and the absolute constant $K$ of (3.16) are not evaluated.
- At pairs that are not $c$-non-degenerate, in particular on the ring at two places at nearly equal distance from the source, $\bar T_3$ and hence $q$ may grow with $L$ (on the ring $T_2/\ell_0\le2$ and the signed $T_3=O(\ell_0^2)$ stay bounded); the remainders there are bounded through $q$ by (3.16), and sharper bounds would need the signed quotients $[\ell^{(k)}]/[\ell]$ of all orders.
- For $\lvert\mu\rvert\lesssim\chi q$ the weight of the self-cost is given only on a fixed graph, at the next order ($\nu_1$), with a remainder whose constant depends on the graph and the pair.
- (3.15) is a sufficient condition; at pairs where it fails, $\zeta$ and the criteria (3.8), (3.11) are stated with $T_2$ exact and are not reduced to distances.
- The bond criterion uses the midpoint as the place of the pair; with body 1 as the place its first breaking term is $\kappa$ of (4.10), on the ring $\rho_0^{\rm ring}-1$ of (4.12).
- Grips outside pair terms ($N_f\ge2$, $F$ not affine) are not admissible descriptions under A5; only the illustration (3.12) is computed for them.
- On $\mathcal H_{\rm adm}$, (3.2) assumes the body is the only object of its type; several identical bodies are treated only through the identity on $\mathcal H$.
- Grips with a different internal operator for each value of $\Phi_x$, field states other than the stated one, and fields of cells are outside the scope.

## Methods used

- Newton forward differences on a finite set; elementary symmetric polynomials of commuting projectors
- partial matrix elements and partial traces; symmetrization on the symmetric subspace, occupation basis
- Perron–Frobenius for the lowest one-object eigenvector; product eigenvectors of non-interacting identical objects
- Rayleigh–Schrödinger perturbation theory of a simple isolated eigenvalue; pseudo-inverse of the graph Laplacian
- lattice Green functions: exact sums on the ring (second-difference equation), large-distance forms on the square and cubic torus
- uniform Riemann-sum bounds on lattice sums; lower bounds on denominators of quotients of differences (non-degeneracy)
- variational principle for the lowest eigenvalue
- closed-form resolvent of a rank-one perturbation; uniform bounds on remainders and their derivatives

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