Version 2 · earlier version; the current one is v4 · External review, round 1: major errors

# Profile of the mutual cost of two held sources on a hopping field

- **Subproject:** 06-ilang-source
- **Package:** 04-profile
- **Version:** v2
- **Mode:** internal regeneration
- **Date:** 2026-10-19

## Changes from previous version

- Setup, "Mutual cost": the claim that $M$ is computed from the total contract and the state alone is withdrawn; $M$ refers to the split of the setting (review I1).
- Step 1: new paragraph "What the total contract fixes" and new result (4.17): the stated states are characterized by $C$ alone; $M$ changes under $G_1\to G_1+c$, $C_1\to C_1-c\lvert x_1\rangle\langle x_1\rvert$; the stated split is fixed only by a convention; in it, $M=\mathcal E-\overline{\mathcal E}+\overline M$ with $\overline M$ free of first-order terms in $g_1$ and in $g_2$ (I1).
- Steps 2–5, 7, 8 and Result (4.1), (4.4)–(4.7), (4.9), (4.13)–(4.15): every value and sign statement on $M$ is restated through (4.17) as a statement on the total cost $\mathcal E$ relative to its mean $\overline{\mathcal E}$, or on differences of $\mathcal E$, or marked as a statement in the stated split; formulas unchanged (I1).
- Step 7, control field: its value and decay are read as differences of $\mathcal E$ against far sources, since the open chain is not vertex-transitive (I1).
- Code: `check_exact_ring.py` also computes $\overline M$ and $\mathcal E-\overline{\mathcal E}$ on rings (check of (4.17)); all scripts re-run in `code/v2/`.
- Open issues: one item on what remains split-dependent; Methods: Hellmann–Feynman.

## Setup and assumptions

- **Field (assumption, M8).** $N_f\ge1$ identical field objects, family $c=+1$, no pair terms among them. Each has the places $X_f$ of a finite connected graph $\Gamma$ with adjacency matrix $A_\Gamma$, own term $C_1=-t_fA_\Gamma$, $t_f>0$, and $\Phi_x=\sum_{j}\lvert x\rangle\langle x\rvert_j$. Graphs: ring of $L\ge3$ places, square torus $L\times L$, cubic torus $L^3$ ($L\ge3$), nearest-neighbour edges. They are regular of degree $z=2D$ ($D=1,2,3$) and vertex-transitive. $\Lambda:=z\mathbb 1-A_\Gamma$ is the graph Laplacian, so $C_1=t_f(\Lambda-z\mathbb 1)$; $\Lambda^+$ is its pseudo-inverse ($\Lambda^+\mathbf 1=0$); $n:=N_f/\lvert X_f\rvert$.
- **Sources (A15, M8).** Held bodies at $x_1\neq x_2$ in eigenvectors of their $G$ with eigenvalues $g_1,g_2$. A held body has no motion term and its state is an eigenvector of every operator by which it enters $C$, so its state is conserved and $C$ acts on the field as $\sum_j h_j(g)$ with $h(g):=C_1+V$, $V:=g_1\lvert x_1\rangle\langle x_1\rvert+g_2\lvert x_2\rangle\langle x_2\rvert$. Two bodies of one type in the swap-symmetrized state give the same $V$ in both terms; nothing changes.
- **State (assumption, M9).** Lowest state: the eigenvector of the field contract with sources connected to the lowest one without them. Control: the highest state.
- **Distance.** $d$ is the graph distance; on the tori $\mathbf d=x_1-x_2$ with components in $(-L/2,L/2]$, and $\lvert\mathbf d\rvert$ is its Euclidean length. On the ring $w:=d(L-d)/L^2\in[0,\tfrac14]$.
- **Mutual cost** (as in 03-held-sources). With $E(g_1,g_2)$ the field cost (1.8) in the stated state, the part depending on both $g_1$ and $g_2$ is
$$M(g_1,g_2):=E(g_1,g_2)-E(g_1,0)-E(0,g_2)+E(0,0).\tag{4.1}$$
It refers to the split of $C$ given by the setting (field's own term $C_1$, couplings $g_i\Phi_{x_i}$), which the total contract does not fix (A5, M6); Step 1 states what the total contract and the state do fix, (4.17). $M^{(k)}$ denotes its part of order $k$ in $(g_1,g_2)$.
- **Input** (item 4), (17.15) of 03-ilang-space/17-motion: for the chain $h_m$ with $K_{h_m}=mX$, $d_0(h_m,h_n)=\lvert m-n\rvert\,\sigma_X$.
- **Tools.** Perron–Frobenius; Rayleigh–Schrödinger perturbation theory for a simple isolated eigenvalue (convergent at least while $\lVert V\rVert$ is below half the gap, Kato); Fourier diagonalization on tori; standard lattice and continuum Green-function asymptotics. $\Lambda$ is the explicit matrix of $C_1$; no field equation is used (M4, M5). $g_i,t_f,J,\Delta$ are costs.

## Derivation

### Step 1. Exact reduction to one field object; what the total contract fixes

$h(g)$ has off-diagonal entries $-t_f\le0$ on the edges of a connected graph. By Perron–Frobenius its lowest eigenvalue $\varepsilon_0(g)$ is simple for every real $(g_1,g_2)$, with a positive eigenvector $\varphi_0(g)$. Since $\sum_jh_j\ge N_f\varepsilon_0(g)$, with equality only on $\varphi_0(g)^{\otimes N_f}$, and this vector is symmetric (admissible for $c=+1$) and continuous in $g$, it is the lowest state for every real $g$ (no level crossing):
$$E(g)=N_f\,\varepsilon_0(g),\qquad M=N_f\bigl[\varepsilon_0(g_1,g_2)-\varepsilon_0(g_1,0)-\varepsilon_0(0,g_2)+\varepsilon_0(0,0)\bigr].\tag{4.2}$$
For $c=-1$ and $N_f\ge2$ the lowest admissible state fills $N_f$ distinct orbitals, so $E\neq N_f\varepsilon_0$, and (4.2) fails (control).

**What the total contract fixes.** The argument above holds for $C_1+V$ with every real diagonal $V$. So the lowest state is the lowest admissible eigenvector of the total contract $C$ among the states in which the bodies are in their stated states, a characterization that does not refer to a split of $C$; so is the highest state of Step 7. Let $\mathcal E(x_1,x_2):=\langle\Psi\vert C\vert\Psi\rangle$ be the total cost in the stated state at fixed bodies, as a function of the places of the sources ($x_1=x_2$ allowed: both on one place). It is computed from the total contracts and the state, so it is invariant under every replacement of the listed terms that leaves $C$ unchanged (M6). $M$ is not. The replacement $G_1\to G_1+c$, $C_1\to C_1-c\lvert x_1\rangle\langle x_1\rvert$ leaves $C$ unchanged (A5) and gives $E'(a,b)=E(a-c,b)$, hence $M'=M(g_1,g_2)-M(-c,g_2)$. At second order in $(g_1,g_2,c)$ this is $M'=-(2n(g_1+c)g_2/t_f)\Lambda^+(x_1,x_2)$ (with (4.4) below): it vanishes for $c=-g_1$ and has the opposite sign for $c=-2g_1$, while $\mathcal E$ is unchanged. Among splits into a field's own term and couplings $g_i'\Phi_{x_i}$, the split of the setting is the only one in which the field's own term and the $g_i$ do not depend on the places, up to a constant that cancels in (4.1): if $C_f+g_1\Phi_{x_1}+g_2\Phi_{x_2}=C_f'+g_1'\Phi_{x_1}+g_2'\Phi_{x_2}+a$ for all $x_1\neq x_2$, two places $x_2,x_2'\neq x_1$ give $(g_2'-g_2)(\Phi_{x_2}-\Phi_{x_2'})=0$, so $g_2'=g_2$, and likewise $g_1'=g_1$. This requirement is a convention (M6 admits place-dependent replacements), so the $g_i$, their signs included, are parameters of the description in the stated split.

In this split, on a vertex-transitive $\Gamma$, $E(g_1,0)$ and $E(0,g_2)$ do not depend on the places. By Hellmann–Feynman, $\partial_{g_2}\varepsilon_0(g_1,g_2)\vert_{g_2=0}=\varphi_0(g_1,0)(x_2)^2$, whose sum over $x_2\in X_f$ is $1$ for every $g_1$. So $\sum_{x_2}M(x_1,x_2)$ has no term of first order in $g_2$; it does not depend on $x_1$, and summing over $x_1$ removes the terms of first order in $g_1$ in the same way. The same holds in the highest state. Hence (exact; stated split, vertex-transitive $\Gamma$, lowest state, or highest state for even $L$)
$$M(x_1,x_2)=\mathcal E(x_1,x_2)-\overline{\mathcal E}(x_1)+\overline M,\qquad \overline{\mathcal E}(x_1):=\frac1{\lvert X_f\rvert}\sum_{y\in X_f}\mathcal E(x_1,y),\qquad \overline M:=\frac1{\lvert X_f\rvert}\sum_{y\in X_f}M(x_1,y),\tag{4.17}$$
where $\overline M$ does not depend on the places and has no term of first order in $g_1$ or in $g_2$. Thus the dependence of $M$ on the places is that of $\mathcal E$, exactly, and through third order in $(g_1,g_2)$ $M$ equals $\mathcal E-\overline{\mathcal E}$, the total cost relative to its mean over the places of the second source. The values and signs of $M^{(2)}$ and $M^{(3)}$ on the ring and the tori below are statements on this invariant; the $g_i$ in their conditions are those of the stated split. Code (ii), rings $L=64,256$, $g_1=g_2=g$, $\gamma=gL/t_f\in\{0.1,\pm0.3,\pm1\}$: $\overline ML^2/(t_f\gamma^4)$ lies between $1.19\times10^{-4}$ and $1.47\times10^{-4}$ for $\lvert\gamma\rvert\le0.3$; $\lvert\overline M\rvert\le1.11\times10^{-3}\max_d\lvert M^{(2)}\rvert$ throughout; $\mathcal E-\overline{\mathcal E}$ agrees with $M^{(2)}+M^{(3)}$ of (4.5), (4.8) to $2.3\times10^{-4}$, $2.1\times10^{-3}$, $2.4\times10^{-2}$ of $\max_d\lvert M^{(2)}\rvert$ at $\lvert\gamma\rvert=0.1,0.3,1$ (`code/v2/output/exact_ring.txt`).

### Step 2. Second order in terms of the single-object contract

Let $\varphi_0$ be the simple isolated lowest eigenvector of a single-object contract, and $R:=\sum_{k\neq0}\lvert\varphi_k\rangle\langle\varphi_k\rvert/(\varepsilon_k-\varepsilon_0)$ its reduced resolvent. Rayleigh–Schrödinger gives $\varepsilon_0(g)=\varepsilon_0+\langle V\rangle-\langle VRV\rangle+O(V^3)$. The first-order term is additive. With $V=g_1\Phi^{(1)}_{x_1}+g_2\Phi^{(1)}_{x_2}$ (single-object coupled operators), the $g_1g_2$ part of the second-order term is, by (4.2),
$$M^{(2)}=-2N_fg_1g_2\,\operatorname{Re}\langle\varphi_0\vert\Phi^{(1)}_{x_1}R\,\Phi^{(1)}_{x_2}\vert\varphi_0\rangle=-2N_fg_1g_2\,\varphi_0(x_1)\varphi_0(x_2)R(x_1,x_2),\tag{4.3}$$
the second form for $\Phi^{(1)}_x=\lvert x\rangle\langle x\rvert$ and real $C_1$. On the three graph families, $\varphi_0=\lvert X_f\rvert^{-1/2}\mathbf 1$, $\varepsilon_0=-zt_f$, and $C_1-\varepsilon_0=t_f\Lambda$, so $R=\Lambda^+/t_f$. Fourier diagonalization then gives
$$M^{(2)}(x_1,x_2)=-\frac{2n\,g_1g_2}{t_f}\,\Lambda^+(x_1,x_2),\qquad \Lambda^+(x,y)=\frac1{\lvert X_f\rvert}\sum_{\mathbf k\neq0}\frac{e^{i\mathbf k\cdot(x-y)}}{\omega(\mathbf k)},\quad \omega(\mathbf k)=\sum_{\mu=1}^{D}4\sin^2\frac{k_\mu}2,\qquad \sum_{x_2\in X_f}M^{(2)}(x_1,x_2)=0,\tag{4.4}$$
with $\mathbf k\in(2\pi/L)\mathbb Z_L^D$. The sum rule follows from $\Lambda^+\mathbf 1=0$; it is the second-order part of $\overline M$ in (4.17), which vanishes. Since $\Lambda^+(x,x)>0$, for $g_1g_2>0$ the value at $x_2=x_1$ is negative, so $M^{(2)}(x_1,x_2)>0$ for some $x_2\neq x_1$, on every graph of the class. By (4.17), for $g_1g_2>0$ and at second order: two sources on one place have a total cost below the mean $\overline{\mathcal E}$, and for some $x_2\neq x_1$ the total cost lies above it.

### Step 3. Ring: closed form

Let $f(d):=\frac{L^2-1}{12L}-\frac{d(L-d)}{2L}$, $f(L)=f(0)$. Since $2p(d)-p(d+1)-p(d-1)=2$ for $p(d)=d(L-d)$, we get $(\Lambda f)(d)=-1/L$ for $1\le d\le L-1$ and $(\Lambda f)(0)=2f(0)-2f(1)=1-1/L$. Also $\sum_{d=0}^{L-1}p(d)=L(L^2-1)/6$ gives $\sum_df(d)=0$. Hence $f=\Lambda^+\delta_0$, i.e. $\Lambda^+(x_1,x_2)=f(d)$, and
$$M^{(2)}_{\rm ring}(d)=\frac{N_f\,g_1g_2}{t_f\,L^2}\Bigl[d(L-d)-\frac{L^2-1}6\Bigr]=\frac{n\,g_1g_2}{t_f}\Bigl[d\Bigl(1-\frac dL\Bigr)-\frac{L^2-1}{6L}\Bigr].\tag{4.5}$$
For $g_1g_2\neq0$ it is strictly monotone on $0\le d\le L/2$, and increasing for $g_1g_2>0$. By (4.17), (4.5) is the second-order part of $\mathcal E(d)-\overline{\mathcal E}$: its monotonicity is that of the total cost, and its sign compares $\mathcal E(d)$ with the mean $\overline{\mathcal E}$. At fixed $N_f$ it equals $(N_fg_1g_2/t_f)(w-\tfrac16+\tfrac1{6L^2})$, a function of $d/L$. At fixed $n$ its $d$-dependent part tends to $(ng_1g_2/t_f)\,d$ for $d\ll L$, while its constant part $-ng_1g_2L/(6t_f)$ diverges. Code (i): (4.3) evaluated numerically from $C_1$ agrees with (4.5) for $L\in\{3,4,16,101,400\}$, $t_f\in\{1,2\}$ and all $d$, to $\le1.5\times10^{-12}$ relative (`code/v2/output/second_order.txt`).

### Step 4. Tori: leading forms for $1\ll\lvert\mathbf d\rvert\ll L$

For $\lvert\mathbf d\rvert\gg1$, (4.4) is fixed by $\lvert\mathbf k\rvert\ll1$, where $\omega(\mathbf k)=\lvert\mathbf k\rvert^2(1+O(\lvert\mathbf k\rvert^2))$. By standard lattice–continuum matching, $\Lambda^+(\mathbf d)=G_L(\mathbf d)+O(\lvert\mathbf d\rvert^{-D})$, where $G_L$ is the zero-mean Green function of $-\nabla^2$ on the continuum torus of side $L$ ($-\nabla^2G_L=\delta-L^{-D}$). There is no constant offset: both functions have zero mean and the same small-$\mathbf k$ coefficients. Scaling gives $G_L(\mathbf r)=L^{2-D}G_1(\mathbf r/L)$. The square and cubic symmetry excludes quadratic harmonic terms, so near $0$:
- $D=2$ (Kronecker's second limit formula, $G_1=-\frac1{2\pi}\ln\lvert\theta_1(z\vert i)/\eta(i)\rvert+\frac{v^2}2$, $z=u+iv$, $\theta_1'(0)=2\pi\eta^3$): $G_1(r)=-\frac1{2\pi}(\ln r+\kappa_2)+\frac{r^2}4+O(r^4)$, with $\kappa_2=\ln(2\pi\eta(i)^2)=\ln\frac{\Gamma(1/4)^2}{2\sqrt\pi}=1.31053$;
- $D=3$ (Ewald sum of the simple cubic lattice): $G_1(r)=\frac1{4\pi r}-\frac{c_3}{4\pi}+\frac{r^2}6+O(r^4)$, with $c_3=2.837297$.

With (4.4):
$$M^{(2)}_{\rm sq}=-\frac{n\,g_1g_2}{\pi t_f}\Bigl[\ln\frac L{\lvert\mathbf d\rvert}-\kappa_2\Bigr]-\frac{n\,g_1g_2}{t_f}\Bigl[\frac{\lvert\mathbf d\rvert^2}{2L^2}+O(\lvert\mathbf d\rvert^{-2})+O\Bigl(\frac{\lvert\mathbf d\rvert^4}{L^4}\Bigr)\Bigr],\tag{4.6}$$
$$M^{(2)}_{\rm cub}=-\frac{n\,g_1g_2}{2\pi t_f}\Bigl[\frac1{\lvert\mathbf d\rvert}-\frac{c_3}L\Bigr]-\frac{n\,g_1g_2}{t_f}\Bigl[\frac{\lvert\mathbf d\rvert^2}{3L^3}+O(\lvert\mathbf d\rvert^{-3})+O\Bigl(\frac{\lvert\mathbf d\rvert^4}{L^5}\Bigr)\Bigr].\tag{4.7}$$
In the regime $1\ll\lvert\mathbf d\rvert\ll L$ and for $g_1g_2>0$, both are negative and increasing in $\lvert\mathbf d\rvert$; by (4.17), at second order, $\mathcal E(\mathbf d)$ lies below $\overline{\mathcal E}$ there and increases with $\lvert\mathbf d\rvert$. Their leading brackets are positive for $\lvert\mathbf d\rvert<e^{-\kappa_2}L=0.270L$ (square) and for $\lvert\mathbf d\rvert<L/c_3=0.352L$ (cubic). At fixed $n$, the $d$-dependent part of (4.6), $(ng_1g_2/\pi t_f)\ln\lvert\mathbf d\rvert$, does not depend on $L$, and its constant part diverges like $\ln L$. (4.7) tends to $-ng_1g_2/(2\pi t_f\lvert\mathbf d\rvert)$ as $L\to\infty$. At fixed $N_f$ both carry the factor $N_f/L^D$. Code (i), with $m:=M^{(2)}t_f/(ng_1g_2)$ computed from $C_1$ by conjugate gradients, compared with the bracket terms of (4.6) and (4.7):
- square, $L=64,128,256$: $\lvert m-m_{\rm lead}\rvert\le8.3\times10^{-3}$ on $4\le\lvert\mathbf d\rvert\le L/8$ (at $L=256$, $m$ runs there from $-0.91$ to $-0.25$); the largest deviation is at $\lvert\mathbf d\rvert=L/8$ and is close to $\lvert\mathbf d\rvert^2/2L^2=7.8\times10^{-3}$;
- cubic, $L=32,48,64$: $\lvert m-m_{\rm lead}\rvert\le9.6\times10^{-4}$ on $4\le\lvert\mathbf d\rvert\le L/8$, where $\lvert m\rvert\ge0.013$.

All three families give a dependence on $d$ that is not exponentially small (linear, logarithmic, $1/\lvert\mathbf d\rvert$). The reason is that in (4.4) the states a source excites reach down to the lowest one as $L\to\infty$: $t_f\,\omega_{\min}=4t_f\sin^2(\pi/L)\to0$.

### Step 5. Range of the second-order form

The third-order term $\langle VRVRV\rangle-\langle V\rangle\langle VR^2V\rangle$ has the $g_1g_2$ part (vertex-transitive $\Gamma$, $\ell_d:=\Lambda^+(x_1,x_2)$, $\ell_0:=\Lambda^+(x,x)$, $\ell^{(2)}$ the entries of $(\Lambda^+)^2$)
$$M^{(3)}=\frac{N_f\,g_1g_2(g_1+g_2)}{\lvert X_f\rvert\,t_f^2}\Bigl[2\ell_0\ell_d+\ell_d^2-\frac{\ell^{(2)}_0+2\ell^{(2)}_d}{\lvert X_f\rvert}\Bigr].\tag{4.8}$$
- **Ring.** $\ell_d=L(\tfrac1{12}-\tfrac w2)-\tfrac1{12L}$ by Step 3. Further, $\ell^{(2)}_d=L^3c+O(L)$, where $c$ is the zero-mean periodic solution of $-\mathrm d^2c/\mathrm du^2=\tfrac1{12}-\tfrac w2$ with $u=d/L$, i.e. $c=(\tfrac1{30}-w^2)/24$. The bracket is $\tfrac{L^2}{60}(1-10w+20w^2)$, and $M^{(3)}/M^{(2)}=-\frac{(g_1+g_2)L}{10t_f}\frac{1-10w+20w^2}{1-6w}$.
- **Square torus.** $2\ell_0+\ell_d=\frac1{2\pi}(3\ln L-\ln\lvert\mathbf d\rvert)+O(1)$, and the last term of (4.8) is $O(1)\ell_d$.
- **Cubic torus.** $\ell_0=W_3+O(1/L)$, with Watson's integral $W_3=0.252731$; $\ell_d=1/(4\pi\lvert\mathbf d\rvert)+O(1/L)$; the last term is $O(\lvert\mathbf d\rvert/L^2)\,\ell_d$.

On the ring $\lvert1-10w+20w^2\rvert\le1$ and $\max_d\lvert M^{(2)}\rvert=N_f\lvert g_1g_2\rvert(L^2-1)/(6t_fL^2)$. To leading order, the second-order form therefore has relative accuracy $\epsilon\ll1$ under
$$\text{ring: }\frac{(\lvert g_1\rvert+\lvert g_2\rvert)L}{10\,t_f}\le\epsilon\ \ (\text{relative to }\max_d\lvert M^{(2)}\rvert);\quad \text{square: }\frac{(\lvert g_1\rvert+\lvert g_2\rvert)(3\ln L-\ln\lvert\mathbf d\rvert)}{4\pi t_f}\lesssim\epsilon;\quad \text{cubic: }\frac{(\lvert g_1\rvert+\lvert g_2\rvert)\bigl(W_3+\frac1{8\pi\lvert\mathbf d\rvert}\bigr)}{t_f}\lesssim\epsilon.\tag{4.9}$$
By (4.2) these conditions do not involve $N_f$ or $n$. By (4.17), $M^{(2)}+M^{(3)}$ is also the expansion of the invariant $\mathcal E-\overline{\mathcal E}$ through third order, so (4.9) holds for it as well. At fixed $n$ and $g$ they bound $L=(N_f/n)^{1/D}$: the second-order form fails for $L\gtrsim t_f/\lvert g\rvert$ on the ring and for $\ln L\gtrsim t_f/\lvert g\rvert$ on the square torus; the cubic condition does not involve $L$. On the ring, near the zero $w=\tfrac16$ of $M^{(2)}$ only the condition relative to $\max_d\lvert M^{(2)}\rvert$ holds: the zero itself moves at third order.

### Step 6. Exact lowest eigenvalue on the ring (one field object)

For $\varepsilon\notin\operatorname{spec}C_1$, $(C_1+V)\psi=\varepsilon\psi$ is equivalent to $\psi=\mathcal G(\varepsilon)V\psi$ with $\mathcal G=(\varepsilon-C_1)^{-1}$. Taking components at $x_1,x_2$:
$$\bigl(1-g_1\mathcal G_0(\varepsilon)\bigr)\bigl(1-g_2\mathcal G_0(\varepsilon)\bigr)=g_1g_2\,\mathcal G_d(\varepsilon)^2,\qquad \mathcal G_d:=\mathcal G(x,x+d).\tag{4.10}$$
On the ring, for $\varepsilon=-2t_f\cos q$,
$$\mathcal G_d(\varepsilon)=\frac{\cos\bigl(q(L/2-d)\bigr)}{2t_f\sin q\,\sin(qL/2)},\qquad 0\le d\le L,\tag{4.11}$$
because $\varepsilon\mathcal G_d+t_f(\mathcal G_{d+1}+\mathcal G_{d-1})=\delta_{d,0}$: for $d\neq0$ by $\cos(a+q)+\cos(a-q)=2\cos q\cos a$, and at $d=0$ by $\cos(qL/2-q)-\cos q\cos(qL/2)=\sin q\sin(qL/2)$. Here $q\in(0,2\pi/L)$ above $-2t_f$ and $q=i\theta$, $\theta>0$, below. A rank-2 perturbation keeps $\varepsilon_0\le-2t_f\cos(2\pi/L)$ (Weyl), so $\varepsilon_0$ is given by the largest root $\theta$ if one exists, and otherwise by the smallest root $q$. For $g_1=g_2=g$ the unique positive lowest eigenvector is even under the reflection exchanging $x_1,x_2$, so $1=g(\mathcal G_0+\mathcal G_d)$:
$$t_f\sin q\,\sin\frac{qL}2=g\cos\frac{qd}2\cos\frac{q(L-d)}2,\qquad\text{one source: }\ 2t_f\sin q\tan\frac{qL}2=g,\qquad\varepsilon_0=-2t_f\cos q.\tag{4.12}$$
For $g<0$, $q=i\theta$: $t_f\sinh\theta\sinh\frac{\theta L}2=\lvert g\rvert\cosh\frac{\theta d}2\cosh\frac{\theta(L-d)}2$ and $\varepsilon_0=-2t_f\cosh\theta$. For bounded $\gamma:=gL/t_f$ one has $q=O(1/L)$. With $q=y/L$ and $u=d/L$, (4.12) becomes $Y\sin\frac Y2=\gamma\cos\frac{Yu}2\cos\frac{Y(1-u)}2$ and $y\tan\frac y2=\frac\gamma2$, up to relative $O(L^{-2})$, and $M=\frac{t_f}{L^2}\bigl(Y^2-2y^2\bigr)$. Expanding gives $Y^2-2y^2=\gamma^2(w-\tfrac16)+\tfrac{\gamma^3}{30}(1-10w+20w^2)+O(\gamma^4)$, which is (4.5) and (4.8) at $N_f=1$. So $ML^2/t_f$ depends only on $\gamma$ and $d/L$, and $\gamma$ is the expansion parameter.

Code (ii), $L=64,256$, $d\in\{1,L/8,L/4,L/2\}$, $\gamma\in\{0.03,0.1,0.3,1,3,-0.3,-1\}$:
- the roots of (4.10)–(4.12) agree with exact diagonalization to $\le4.4\times10^{-16}$; this includes $g_2=2g_1$;
- the residual $\lvert(M-M^{(2)}-M^{(3)})/M^{(2)}\rvert$ is $\le3.1\times10^{-4}$ for $\lvert\gamma\rvert\le0.1$, $\le2.9\times10^{-3}$ for $\lvert\gamma\rvert\le0.3$ and $\le3.3\times10^{-2}$ for $\lvert\gamma\rvert\le1$ (it is $O(\gamma^2)$);
- at $\gamma=3$ the second-order form is off by $42$–$43\%$ at $d=1$ and by $59\%$ at $d=L/4$ (`code/v2/output/exact_ring.txt`).

### Step 7. Control state and control field

**Highest state.** Let $L$ be even, so that $\Gamma$ is bipartite. With $S:=\operatorname{diag}((-1)^{\sum_\mu x_\mu})$, $SA_\Gamma S=-A_\Gamma$ and $SVS=V$, hence $Sh(g)S=-h(-g)$. The highest eigenvalue $\varepsilon_{\rm top}(g)=-\varepsilon_0(-g)$ is therefore simple for all real $g$, and Step 1 with the inequality reversed gives
$$M_{\rm top}(g_1,g_2)=-M_0(-g_1,-g_2),\qquad M^{(2)}_{\rm top}=+\frac{2n\,g_1g_2}{t_f}\Lambda^+(x_1,x_2).\tag{4.13}$$
(4.13) is an identity in the stated split: $SC_1S=-C_1$ and the map $g\to-g$ refer to it. Its second-order part is a statement on $\mathcal E-\overline{\mathcal E}$ by (4.17) for the highest state: at second order the dependence of the total cost on the places reverses, and so does its sign relative to $\overline{\mathcal E}$. For odd $L$ the highest eigenvalue of $C_1$ is degenerate (ring: $k=\pi(L\pm1)/L$), and the highest state is not fixed by connection. Code (i): $\lvert M^{(2)}_0+M^{(2)}_{\rm top}\rvert\le2.8\times10^{-15}$ on rings $L=4,16,64$; degeneracy at $L=15$.

**Control field (assumption, M8).** $N_f$ objects ($c=+1$, no pair terms among them) with the places $\{\varnothing\}\cup X_f$, $X_f$ a chain. The own term is $C^{c}_1=\Delta\sum_x\lvert x\rangle\langle x\rvert-J\sum_{\langle xy\rangle}(\lvert x\rangle\langle y\rvert+{\rm h.c.})$ with $\Delta>2J>0$, and the coupled operator is $\Phi^c_x=\sum_j(\lvert x\rangle\langle\varnothing\rvert+\lvert\varnothing\rangle\langle x\rvert)_j$. The lowest eigenvector is $\lvert\varnothing\rangle$ (eigenvalue $0$). Its gap is $\ge\Delta-2J$, since the block $K:=C^c_1\vert_{X_f}$ has spectrum in $[\Delta-2J,\Delta+2J]$. It is not sharp, since $\Phi^c_x\lvert\varnothing\rangle=\lvert x\rangle$. The reduction of Step 1 holds near $g=0$, and (4.3) gives $M^{(2)}_c=-2N_fg_1g_2K^{-1}(x_1,x_2)$. On an infinite chain, $f(d)=e^{-\theta\lvert d\rvert}/(2J\sinh\theta)$ with $\cosh\theta=\Delta/2J$ solves $Kf=\delta_0$: for $d\ne0$ $\Delta-2J\cosh\theta=0$, and at $d=0$ $\Delta-2Je^{-\theta}=2J\sinh\theta$. It is the bounded solution, so
$$M^{(2)}_c(d)=-\frac{N_f\,g_1g_2}{J\sinh(1/\xi)}\,e^{-d/\xi},\qquad \xi=\frac1{\operatorname{arccosh}(\Delta/2J)},\tag{4.14}$$
valid for $\lvert g_1\rvert+\lvert g_2\rvert\ll\Delta-2J$; $\xi\to\infty$ as $\Delta\to2J^+$. On a finite chain the ends add relative corrections $O(e^{-2\delta/\xi})$, with $\delta$ the distance of the sources from the ends. In this range the lowest state is the lowest eigenvector of $C$ (simple, isolated), so $\mathcal E$ is invariant as in Step 1. The chain is not vertex-transitive, but away from the ends the single-source terms do not depend on the places up to the same corrections. Since $M^{(2)}_c(d')\to0$, (4.14) is then the second-order part of $\mathcal E(d)-\mathcal E(d')$ for $d'\gg\xi$, up to $O(e^{-d'/\xi})$: its value, its sign and its decay are statements on the total cost; the factor $g_1g_2$ refers to the stated split. Code (iii): open chain of $400$ places, $x_1=100$, $d=1,\dots,160$, $\Delta/J\in\{2.02,2.1,2.5,3\}$. Numerical (4.3) from $C^c_1$ equals (4.14) to $\le5.7\times10^{-13}$ relative wherever $\lvert M\rvert>10^{-12}\lvert M(1)\rvert$. The fitted decay lengths $7.07695,3.17536,1.44270,1.03904$ equal $\xi$ to all digits shown (`code/v2/output/control_decay.txt`).

### Step 8. Field places and body positions

Body $b$ couples by $\sum_m\lvert h_m\rangle\langle h_m\rvert\otimes G\otimes\Phi_{\pi(m)}$, with $\pi$ a map from chain labels to ring places (assumption, M8). A held body at $h_{m_i}$ acts on the field as $g_i\Phi_{\pi(m_i)}$. By the rotation and reflection symmetry of the ring, $M(m_1,m_2)=F_L\bigl(d_\Gamma(\pi(m_1),\pi(m_2))\bigr)$ exactly, with $F_L$ the ring mutual cost; at second order $F_L$ is (4.5), which for $g_1g_2\neq0$ is strictly monotone on $\{0,\dots,\lfloor L/2\rfloor\}$. Let the positions used be $S=\{m_0,\dots,m_0+K\}$, $K\ge1$. Requiring $M=F_L(d_0/a)$ on $S$ for a constant $a>0$, with $d_0=\lvert m_1-m_2\rvert\sigma_X$ (17.15), is (by monotonicity) the same as requiring $d_\Gamma(\pi(m),\pi(m'))=\lvert m-m'\rvert\sigma_X/a$. For adjacent $m$ this makes $k:=\sigma_X/a$ a positive integer. Hence
$$d_\Gamma\bigl(\pi(m),\pi(m')\bigr)=k\lvert m-m'\rvert\ \ \forall m,m'\in S\iff \pi(m)=\pi(m_0)+s\,k\,(m-m_0)\bmod L,\ s=\pm1,\ kK\le\lfloor L/2\rfloor,\tag{4.15}$$
and then $M=F_L(k\,d_0/\sigma_X)$. Proof. ($\Leftarrow$) The ring distance of a difference $r\le\lfloor L/2\rfloor$ is $r$. ($\Rightarrow$) $kK=d_\Gamma(\pi(m_0),\pi(m_0+K))\le\lfloor L/2\rfloor$. Then $\pi(m_0+1)=\pi(m_0)+sk$, and inductively, of the two places at distance $k$ from $\pi(m)$, only $\pi(m)+sk$ is at distance $k(m+1-m_0)$ from $\pi(m_0)$; the other is at $k(m-1-m_0)$. Sufficiency holds at all orders, and necessity at second order. A map that fails:
$$\pi(m)=m\bmod L\ \text{on}\ S=\{0,\dots,K\},\ \lfloor L/2\rfloor<K\le L-1:\quad d_\Gamma(\pi(0),\pi(K))=L-K<K,\ \text{e.g. } M(0,L-1)=M(0,1)\ \text{while}\ d_0=(L-1)\sigma_X.\tag{4.16}$$
(4.15) and (4.16) use only the dependence of $M$ on the places, which by (4.17) is that of the total cost $\mathcal E$; they are statements on $\mathcal E$. The tie between field places and body positions is the coupling map $\pi$ alone, a choice of the description; whether anything enforces (4.15) is not asked here (M10).

## Result

- **(4.1)** Definition: the mutual cost is the double difference of the field cost $E(g_1,g_2)$ in the stated state, in the split of the setting. It depends on the split: $G_1\to G_1+c$, $C_1\to C_1-c\lvert x_1\rangle\langle x_1\rvert$ leaves $C$ unchanged and gives $M'=M(g_1,g_2)-M(-c,g_2)$.
- **(4.2)** Theorem (exact, all real $g_1,g_2$; every finite connected $\Gamma$, $N_f\ge1$, $c=+1$, no pair terms among field objects): the lowest state is $\varphi_0(g)^{\otimes N_f}$, connected to the lowest state without sources without level crossing, and $E=N_f\varepsilon_0(g)$, so $M$ is exactly $N_f$ times the one-object mutual cost. Control: $c=-1$, $N_f\ge2$.
- **(4.17)** Theorem (exact; stated split, vertex-transitive $\Gamma$, lowest state, or highest state for even $L$): the stated states are eigenvectors of $C$ (at the stated body states) characterized by $C$ alone, so the total cost $\mathcal E(x_1,x_2)$ is invariant under every replacement of the listed terms that leaves $C$ unchanged; $M=\mathcal E-\overline{\mathcal E}+\overline M$, where $\overline M$ does not depend on the places and has no term of first order in $g_1$ or in $g_2$. Hence the dependence of $M$ on the places is that of $\mathcal E$ at all orders, and $M^{(2)}$, $M^{(3)}$ are the second- and third-order parts of $\mathcal E-\overline{\mathcal E}$. Among splits into a field's own term and couplings $g_i\Phi_{x_i}$, the stated split is the only one in which both do not depend on the places (a convention); the $g_i$ and their signs are parameters in this split.
- **(4.3)** Formula (second order, remainder $O(N_f\lvert g_1g_2\rvert(\lvert g_1\rvert+\lvert g_2\rvert))$; simple isolated lowest eigenvector of the single-object contract): $M^{(2)}=-2N_fg_1g_2\operatorname{Re}\langle\varphi_0\vert\Phi^{(1)}_{x_1}R\Phi^{(1)}_{x_2}\vert\varphi_0\rangle$, which for the class is $-2N_fg_1g_2\varphi_0(x_1)\varphi_0(x_2)R(x_1,x_2)$.
- **(4.4)** Formula (second order; ring and tori, lowest state, stated split): $M^{(2)}=-(2ng_1g_2/t_f)\Lambda^+(x_1,x_2)$, and $\sum_{x_2\in X_f}M^{(2)}(x_1,x_2)=0$. Hence, for $g_1g_2>0$, $M^{(2)}(x_1,x_2)>0$ for some $x_2\neq x_1$. By (4.17) these are statements on $\mathcal E-\overline{\mathcal E}$ at second order.
- **(4.5)** Formula (second order; ring, every $L\ge3$, every $d$; stated split): $M^{(2)}=\frac{N_fg_1g_2}{t_fL^2}[d(L-d)-\frac{L^2-1}6]$. For $g_1g_2>0$ it is strictly increasing on $0\le d\le L/2$ and negative iff $d(L-d)<(L^2-1)/6$; by (4.17), at second order, $\mathcal E(d)$ is strictly increasing there and lies below $\overline{\mathcal E}$ iff $d(L-d)<(L^2-1)/6$. At fixed $N_f$ it is a function of $d/L$. At fixed $n$ it is linear in $d$ for $d\ll L$, with a constant part $\propto L$.
- **(4.6)** Formula (second order; square torus, $1\ll\lvert\mathbf d\rvert\ll L$, corrections $\lvert\mathbf d\rvert^2/2L^2$ and $O(\lvert\mathbf d\rvert^{-2})$; stated split): $M^{(2)}=-\frac{ng_1g_2}{\pi t_f}[\ln\frac L{\lvert\mathbf d\rvert}-\kappa_2]$, $\kappa_2=1.31053$. For $g_1g_2>0$ it is negative and increasing in $\lvert\mathbf d\rvert$, i.e. by (4.17), at second order, $\mathcal E(\mathbf d)$ lies below $\overline{\mathcal E}$ and increases with $\lvert\mathbf d\rvert$. At fixed $n$ the $d$-dependence is $\frac{ng_1g_2}{\pi t_f}\ln\lvert\mathbf d\rvert$. Checked to $8.3\times10^{-3}$ in $m$ ($L=64,128,256$; $4\le\lvert\mathbf d\rvert\le L/8$).
- **(4.7)** Formula (second order; cubic torus, $1\ll\lvert\mathbf d\rvert\ll L$, corrections $\lvert\mathbf d\rvert^2/3L^3$ and $O(\lvert\mathbf d\rvert^{-3})$; stated split): $M^{(2)}=-\frac{ng_1g_2}{2\pi t_f}[\frac1{\lvert\mathbf d\rvert}-\frac{c_3}L]$, $c_3=2.837297$. For $g_1g_2>0$ it is negative and increasing in $\lvert\mathbf d\rvert$, i.e. by (4.17), at second order, $\mathcal E(\mathbf d)$ lies below $\overline{\mathcal E}$ and increases with $\lvert\mathbf d\rvert$. At fixed $n$ it tends to $-ng_1g_2/(2\pi t_f\lvert\mathbf d\rvert)$. Checked to $9.6\times10^{-4}$ in $m$ ($L=32,48,64$; $4\le\lvert\mathbf d\rvert\le L/8$). At fixed $L$ and $g_i$, all of (4.4)–(4.7) scale as $N_f/t_f$.
- **(4.8)** Formula (third order; vertex-transitive $\Gamma$, lowest state): the $g_1g_2$ part of third order, $\propto g_1g_2(g_1+g_2)$.
- **(4.9)** Criterion (leading order in the small parameter; lowest state): the second-order form holds to relative accuracy $\epsilon$ under $(\lvert g_1\rvert+\lvert g_2\rvert)L/10t_f\le\epsilon$ on the ring (relative to $\max_d\lvert M^{(2)}\rvert$), under $(\lvert g_1\rvert+\lvert g_2\rvert)(3\ln L-\ln\lvert\mathbf d\rvert)/4\pi t_f\lesssim\epsilon$ on the square torus, and under $(\lvert g_1\rvert+\lvert g_2\rvert)(W_3+1/8\pi\lvert\mathbf d\rvert)/t_f\lesssim\epsilon$ on the cubic torus. These conditions do not involve $N_f$ or $n$. By (4.17) they hold for $\mathcal E-\overline{\mathcal E}$ as well.
- **(4.10)–(4.12)** Formula (exact; one field object on the ring, lowest state): the eigenvalue equation (4.10) with the ring Green function (4.11), and for $g_1=g_2=g$ the closed equation (4.12). For $g_1=g_2$ and $L\gg1$, $ML^2/t_f$ is a function of $\gamma=gL/t_f$ and $d/L$ alone, up to relative $O(L^{-2})$. Its expansion in $\gamma$ reproduces (4.5) and (4.8).
- **(4.13)** Theorem (exact, all orders, as an identity in the stated split; ring and tori with even $L$, highest state): $M_{\rm top}(g_1,g_2)=-M_0(-g_1,-g_2)$, so $M^{(2)}_{\rm top}=-M^{(2)}_0$. The sign and monotonicity statements of (4.5)–(4.7) reverse in the highest state; by (4.17), at second order, the dependence of the total cost on the places and its sign relative to $\overline{\mathcal E}$ reverse.
- **(4.14)** Example (second order, $\lvert g_1\rvert+\lvert g_2\rvert\ll\Delta-2J$; gapped control field on a chain, sources away from the ends, lowest state $\lvert\varnothing\rangle^{\otimes N_f}$): $M^{(2)}_c=-\frac{N_fg_1g_2}{J\sinh(1/\xi)}e^{-d/\xi}$ with $\xi=1/\operatorname{arccosh}(\Delta/2J)$. It is the second-order part of $\mathcal E(d)-\mathcal E(d')$ for $d'\gg\xi$, up to $O(e^{-d'/\xi})$, and it falls off faster than any power of $d$.
- **(4.15)** Criterion (bodies recorded on a chain, ring field, coupling map $\pi$; sufficiency at all orders, necessity at second order; a statement on $\mathcal E$ by (4.17)): the mutual cost is the same function of $d_0$ as of the graph distance, up to a constant factor, iff $\pi$ is the scaled embedding $\pi(m)=\pi(m_0)\pm k(m-m_0)\bmod L$ with $k\in\mathbb Z_{>0}$ and $kK\le\lfloor L/2\rfloor$. Then $M=F_L(kd_0/\sigma_X)$.
- **(4.16)** Negative: the wrapping map $\pi(m)=m\bmod L$ on more than $\lfloor L/2\rfloor+1$ positions fails (4.15).

## Consistency checks

1. **Dimensions.** $g_i,t_f,J,\Delta$ are costs. $M^{(2)}\propto g_1g_2/t_f$ and $M^{(3)}\propto g^3L/t_f^2$ are costs; $\gamma=gL/t_f$ and $(\lvert g_1\rvert+\lvert g_2\rvert)L/t_f$ are pure numbers.
2. **Ring $L=3$** (complete graph). $C_1$ has eigenvalues $-2t_f,t_f,t_f$, and $R=(\mathbb 1-\mathbf 1\mathbf 1^{\!\top}/3)/(3t_f)$, so $R(x_1,x_2)=-1/(9t_f)$ and (4.3) gives $M^{(2)}=2N_fg_1g_2/(27t_f)$. (4.5) at $L=3$, $d=1$ gives $\frac{N_fg_1g_2}{9t_f}(2-\tfrac43)$, the same.
3. **Sum over the two ends, ring $L=4$.** From the eigenvectors $(1,\pm1,1,\pm1)/2$, $(1,0,-1,0)/\sqrt2$ and $(0,1,0,-1)/\sqrt2$: $M^{(2)}_{\rm top}(1)=-N_fg_1g_2/(32t_f)$ and $M^{(2)}_{\rm top}(2)=-3N_fg_1g_2/(32t_f)$. (4.5) gives $M^{(2)}_0(1)=N_fg_1g_2/(32t_f)$ and $M^{(2)}_0(2)=3N_fg_1g_2/(32t_f)$. The sums vanish, as (4.13) requires.

## Open issues

- Odd $L$: the highest state is degenerate and is not determined by connection; (4.13) is derived only for even $L$.
- The tori conditions in (4.9) are leading-order estimates with $O(1)$ terms not computed; no rigorous remainder bound is given beyond Kato's sufficient radius $\max_i\lvert g_i\rvert<2t_f\sin^2(\pi/L)$.
- Long range is shown for the hopping class and short range for one gapped field; a general criterion over field contracts is not derived.
- Field objects with $c=-1$, or with pair terms among them, are outside the class.
- Necessity in (4.15) is shown at second order only.
- What remains tied to the stated split: the $g_i$ and their signs, the all-order identity (4.13), and $M$ beyond third order, where it differs from $\mathcal E-\overline{\mathcal E}$ by $\overline M$ (4.17).

## Code

- `code/v2/check_second_order.py` (check, item 1 and (4.13)). It evaluates (4.3) from the matrix $C_1$: by dense diagonalization on rings $L\in\{3,4,16,101,400\}$, compared with (4.5); and by conjugate gradients for $R(\cdot,x_2)$ on square tori $L\in\{64,128,256\}$ and cubic tori $L\in\{32,48,64\}$, compared with (4.6) and (4.7) along axis and diagonal directions with components $\le L/4$. It also evaluates lowest plus highest state on even rings and the top degeneracy at $L=15$. Output: `code/v2/output/second_order.txt`.
- `code/v2/check_exact_ring.py` (check, item 2 and (4.17)). One field object on rings $L=64,256$: the exact lowest eigenvalue by diagonalization and by the roots of (4.10)–(4.12), and $M$ against $M^{(2)}$ (4.5) and $M^{(3)}$ (4.8), for several $\gamma=gL/t_f$ and $d$, plus a case $g_2=2g_1$. It also computes, by diagonalization over all places of the second source (both on one place included), $\overline M$ and $\mathcal E-\overline{\mathcal E}$ of (4.17), against $\gamma^4$ and against $M^{(2)}+M^{(3)}$. Output: `code/v2/output/exact_ring.txt`.
- `code/v2/check_control_decay.py` (check, item 3). The control field on an open chain of $400$ places: (4.3) from $C^c_1$ against (4.14), and the fitted decay length against $\xi$. Output: `code/v2/output/control_decay.txt`.

## Methods used

- Perron–Frobenius theorem; bosonic product states of non-interacting objects
- Rayleigh–Schrödinger perturbation theory (second and third order), reduced resolvent; Hellmann–Feynman theorem
- Invariance under splits of the total contract (A5, M6)
- Graph Laplacian and its pseudo-inverse; Fourier diagonalization on tori
- Lattice–continuum Green-function matching; Kronecker's second limit formula; Ewald constant; Watson's integral
- Rank-two secular equation; ring Green function
- Bipartite sublattice symmetry
- Exact diagonalization, conjugate gradients (code checks)

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