Version 1 · earlier version; the current one is v6 · Review of v1

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

- **Subproject:** 08-ilang-propagation
- **Package:** 03-gas-grip
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-24

## 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) $\rho_i=\frac{\Delta_{\mathbf e}\Delta_{\mathbf r\mathbf r'}W_b(x_1;\cdot)}{\Delta_{\mathbf r\mathbf r'}B\,\Delta_{\mathbf e}\mathcal F_i(x_1)}$, $\rho_i^{\rm fg}=\rho_0+O(\eta K)$, $\rho_0=1+\kappa$; (4.12) $\kappa^{\rm mid}_{D=3}=O(\frac{\lvert\mathbf r\rvert\lvert\mathbf r'\rvert}{d^2})$, $\kappa^{\rm mid}_{D=2}=O(\frac{\lvert\mathbf r'\rvert^2}{d^2}\ln\frac{L^2}{\lvert\mathbf r'\rvert d})$, $\kappa^{\rm mid}_{\rm ring}=0$; (4.13) $\rho_i^{(a)}=\frac{1+n\gamma_i}{n\gamma_i}\rho_i^{\rm fg}(1+O(\eta))$. Regime $\eta=\max_i\lvert g_i\rvert\ell_0/t_f<1/12$, $\gamma_i>0$.

**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. Control: $F(\Phi)=\Phi^2$ 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) ($\Omega C^{\rm mark}=\Omega ng_b+O(g_b^2)$):

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

$\mathrm{Cost}^{(1)}=\sigma M_kF(n)$ 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 $\alpha=0$. Excluded: $\beta=0$ (then $g_b=0$, no mark, $\Delta W=0$ for every level), $\alpha+\beta n=0$ ($\mathrm{Cost}^{(1)}=0$ for every level and both branches), and places with $[\ell]=0$ ($\Omega=0$).

### Step 4. Weight of the mark

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 $C^{\rm mark}=ng_b+C^{\rm self}=\mu\,\mathrm{Cost}^{(1)}+C^{\rm self}$:

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

The rest term $\alpha\sigma M_k$ is not weighed; the first-order mark cost $ng_b$ and the self-cost $C^{\rm self}$ are weighed with the same factor $\Omega(1+B^{\rm fg})$. 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 $\epsilon:=(1+\lvert\mu\rvert)\chi+\lvert B^{\rm fg}\rvert$,

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

that is, the mark weighing ratio $w_{k,\sigma}:=\frac{\Delta W_{k,\sigma}/\mathrm{Cost}_{k,\sigma}}{\Delta W^{(1)}_{k,\sigma}/\mathrm{Cost}^{(1)}_{k,\sigma}}=1+B^F_{k,\sigma}$. In (3.6) the self-cost is weighed with $\Omega(1+B^{\rm fg})$ and the first-order part with $\mu\Omega(1+B^{\rm fg})$; the two agree iff $\mu=1$. Equivalently, the term $B^F-B^{\rm fg}=-\frac{\alpha}{\alpha+\beta n}\frac{g_b\ell_0}{t_f}$ of (3.7), which does not depend on the places, vanishes iff $\alpha=0$:

$$\text{the mark is weighed like the first-order part of the cost}\iff\mu=1\iff\alpha=F(0)=0;\qquad\text{then}\ \ \Delta W_{k,\sigma}=\mathrm{Cost}_{k,\sigma}\,\Omega\,\bigl[1+B^{\rm fg}_{k,\sigma}\bigr].\tag{3.8}$$

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) 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 $\Delta W_a=\Omega n\,g_a[1-\frac{g_a}{t_f}(\ell_0+\frac12[\ell(r)+\ell(r')-T_2])]+O(\Omega ng_a\chi^2)$ 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

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

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):

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

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 $\rho^{\rm fg}_i\ne0$:

$$\rho^F_i=\rho^{\rm fg}_i\iff\mu_i=1\iff\alpha_i=0;\qquad\text{mark and bond weighed like the first-order part of the cost}\iff F(\Phi)=\beta\Phi,\ \beta\ne0 .\tag{3.11}$$

For $\alpha_i\ne0$ the bond is weighed against the first-order level costs of body $i$ with the extra factor $1/\mu_i=1+\alpha_i/(n\beta_i)$, the same at every separation, level and branch.

### Step 7. Control outside pair terms

$F(\Phi)=\Phi^2$ has $F(0)=0$ and, for $N_f\ge2$, is not a sum of pair terms (3.2). 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 $n':=(1-1/N_f)n$:

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

## 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. Control: $F=\Phi^2$.
- **(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 $\chi\ll1$. Excluded: $\beta=0$, $\alpha+\beta n=0$, $[\ell]=0$.
- **(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: $\Delta W=\Omega(1+B^{\rm fg})[\mu\,\mathrm{Cost}^{(1)}+C^{\rm self}]$; the rest term is not weighed, the first-order mark cost and the self-cost are weighed with $\Omega(1+B^{\rm fg})$. Control: (3.12).
- **(3.7)** Under the conditions of (3.6) and $\lvert\mu\rvert\chi\ll1$, 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 $B^F=-\frac{g_b}{2t_f}[\ell(r)+\ell(r')+2(1-\mu)\ell_0-T_2]$ and remainder $O(\eta^2\ell^2)+O(\epsilon^2)$, $\epsilon=(1+\lvert\mu\rvert)\chi+\lvert B^{\rm fg}\rvert$.
- **(3.8)** Among the affine grips with $\beta\ne0$, $\alpha+\beta n\ne0$, at places with $[\ell]\ne0$: the self-cost of the mark is weighed with the factor of the first-order part iff $\mu=1$, iff $\alpha=F(0)=0$; then $\Delta W=\mathrm{Cost}\,\Omega[1+B^{\rm fg}]$ for every level and both branches, at the order of (3.14) of 07-ilang-weight/03-weight-of-mark. Two bodies have the same leading factor $\mu\Omega$ iff $\alpha_1\beta_2=\alpha_2\beta_1$ (a condition, M7). Control: coupling (a).
- **(3.9)** For two internal states of one body with an affine grip, at first order in $g_s$ and $\chi\ll1$, $\lvert\mu\rvert\chi\ll1$: the difference of weights is the difference of costs times $\mu\Omega[1-\frac{g_a+g_{a'}}{2t_f}(\ell(r)+\ell(r')+2(1-\mu)\ell_0-T_2)+O((1+\lvert\mu\rvert)^2\chi^2)]$; 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): $\rho^F_i=\rho^{\rm fg}_i/\mu_i=\frac{\alpha_i+n\beta_i}{n\beta_i}[\rho_0+O(\eta K)]$, 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 $\rho^{\rm fg}_i\ne0$: $\rho^F_i=\rho^{\rm fg}_i$ iff $\alpha_i=0$. Under the conditions of (3.8) and (3.10), the grips of (3.2)–(3.3) for which the mark and the bond are weighed like the first-order part of the cost are exactly $F(\Phi)=\beta\Phi$, $\beta\ne0$ of either sign, at the orders of (3.8) and (3.10). For $\alpha_i\ne0$ the bond carries the extra factor $1+\alpha_i/(n\beta_i)$. Control: coupling (a).
- **(3.12)** Control outside pair terms: 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) 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: $\alpha,\beta,\mu,\Omega,\chi,B^F,\rho^F_i$ are numbers; $g_b$, $\mathrm{Cost}$, $C^{\rm self}$, $\Delta W$ and $B(\mathbf r)$ are costs ($g_b^2\ell_0/t_f$ and $ng_1g_2\ell/t_f$ included). Passed.
3. $N_f=1$: $n'=0$ in (3.12) gives the factor $\Omega$, as it must, since $\Phi_x^2=\Phi_x$ is then the affine grip $(0,1)$ of (3.3), with $\mu=1$. Passed.

## Open issues

- The size of the distance-dependent breaking $B^{\rm fg}$ and of $O(\eta K)$ on each graph is that of the quoted packages; it is not re-derived here.
- Grips outside pair terms ($N_f\ge2$, $F$ not affine) are characterized only by the first-order control (3.12); their marks and bonds are not studied.
- 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

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