Version 3 · earlier version; the current one is v4 · External review, round 2: minor issues

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

- **Subproject:** 06-ilang-source
- **Package:** 04-profile
- **Version:** v3
- **Mode:** external regeneration
- **Date:** 2026-10-19

## Changes from previous version

- Step 7 and Result (4.14): the field with one shared empty place is now a control at second order only. Its full mutual cost tends to a nonzero constant at fourth order. A new control, the cell field (4.20), has a full mutual cost that falls off faster than any power of $d$ at all orders in a stated range (verification I1).
- Step 5 and Results (4.9), (4.18), (4.19): new pole-free secular equation (4.18) and a proven bound (4.19) on all higher orders, on the ring and both tori. (4.9) is now an estimate of the breakdown scale (I2).
- Step 5: the third-order bracket and the ratio $M^{(3)}/M^{(2)}$ on the ring are given exactly at finite $L$ (I4).
- Step 6 and Result (4.10)–(4.12): the root prescription now uses the pole-cleared form of (4.10) and includes $q=0$. Also proven: $\varepsilon_0(g)<\varepsilon_1$ (I3).
- Step 4 and Results (4.6), (4.7): the radial form and the increase in $\lvert\mathbf d\rvert$ are stated at leading order only (I5).
- Code: new `check_remainder_bound.py`. `check_exact_ring.py` now uses the pole-cleared equation, evaluates (4.12) separately, and adds mixed-sign and $q=0$ cases and the finite-$L$ bracket. `check_control_decay.py` adds the full mutual cost of both control fields. All scripts were re-run in `code/v3/`.
- Open issues and Methods updated accordingly.

## Response to verification

- I1: Accepted and fixed in Step 7. The fourth-order limit of the shared-empty-place field is derived and checked, and (4.14) is restricted to the second-order coefficient. The new cell field (4.20) has $\lvert M\rvert\le C\lvert g_1g_2\rvert F(d)$, with $F$ falling off faster than any power, at all orders for $\lvert g_1\rvert+\lvert g_2\rvert<(\Delta-2J)/2$. Its second-order decay length is $\xi$.
- I2: Accepted and fixed in Step 5. A rank-two Feshbach reduction (4.18), Rouché's theorem and Cauchy estimates give (4.19): $\lvert M-M^{(2)}\rvert\le\frac{4\eta}{1-\eta}\mathcal M$ and $\lvert M-M^{(2)}-M^{(3)}\rvert\le\frac{4\eta^2}{1-\eta}\mathcal M$. Here $\eta=16\ell_0\max_i\lvert g_i\rvert/t_f<1$ is the dimension-dependent small parameter, and $\mathcal M=\max_{x_2}\lvert M^{(2)}\rvert$ is the normalization, which is also the one used near zeros of $M^{(2)}$. (4.9) is presented as an estimate.
- I3: Accepted and fixed in Steps 5 and 6. The difference of the two sides of (4.10), multiplied by $\delta$, equals $\det(\varepsilon-h)/\prod_{k\neq0}(\varepsilon-\varepsilon_k)$, so its zeros below $\varepsilon_1$ are exactly the eigenvalues there, $q=0$ included. The verifier's example and a second $q=0$ case are checked.
- I4: Accepted and fixed in Step 5. The exact finite-$L$ bracket and ratio are displayed and checked against the matrices.
- I5: Accepted and fixed in Step 4 and Results (4.6), (4.7).

## 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; Feshbach (Schur-complement) reduction, Sylvester's determinant identity, Rouché's theorem and Cauchy estimates; for the cell field, the locality of gapped ground states under local perturbations. $\Lambda$ 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/v3/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/v3/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):
$$M^{(2)}_{\rm sq}=-\frac{n\,g_1g_2}{\pi t_f}\Bigl[\ln\frac L{\lvert\mathbf d\rvert}-\kappa_2\Bigr]-\frac{n\,g_1g_2}{t_f}\Bigl[\frac{\lvert\mathbf d\rvert^2}{2L^2}+O(\lvert\mathbf d\rvert^{-2})+O\Bigl(\frac{\lvert\mathbf d\rvert^4}{L^4}\Bigr)\Bigr],\tag{4.6}$$
$$M^{(2)}_{\rm cub}=-\frac{n\,g_1g_2}{2\pi t_f}\Bigl[\frac1{\lvert\mathbf d\rvert}-\frac{c_3}L\Bigr]-\frac{n\,g_1g_2}{t_f}\Bigl[\frac{\lvert\mathbf d\rvert^2}{3L^3}+O(\lvert\mathbf d\rvert^{-3})+O\Bigl(\frac{\lvert\mathbf d\rvert^4}{L^5}\Bigr)\Bigr].\tag{4.7}$$
The leading terms (first brackets) are radial. For $g_1g_2>0$ they are negative in the regime $1\ll\lvert\mathbf d\rvert\ll L$ and increase with $\lvert\mathbf d\rvert$. The exact lattice sums (4.4) are functions of the vector $\mathbf d$, not of $\lvert\mathbf d\rvert$ alone: the corrections $O(\lvert\mathbf d\rvert^{-2})$, $O(\lvert\mathbf d\rvert^{-3})$ and the finite-size terms depend on the direction of $\mathbf d$. The radial form and the increase therefore hold only at leading order, i.e. up to these corrections. Between two separations whose leading terms differ by less than the corrections, the order of the exact values is not fixed. By (4.17) the same holds for $\mathcal E(\mathbf d)-\overline{\mathcal E}$ at second order. The 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. $\ell^{(2)}$ is the zero-mean solution of $\Lambda\ell^{(2)}=\ell$. With $p=d(L-d)$, the second differences of $p$ and $p^2$ are $-2$ and $2L^2+2-12p$. Together with $\sum_dp=L(L^2-1)/6$ and $\sum_dp^2=L(L^4-1)/30$, this gives $\ell^{(2)}_d=\frac{L^2-1}{72L}+\frac{L^4-1}{720L}-\frac{p}{12L}-\frac{p^2}{24L}$. Its leading part $L^3(\tfrac1{30}-w^2)/24$ is the zero-mean periodic solution of $-\mathrm d^2c/\mathrm du^2=\tfrac1{12}-\tfrac w2$, $u=d/L$. The bracket of (4.8) is exactly $\tfrac{L^2}{60}(1-10w+20w^2)-\tfrac1{12}+\tfrac w3+\tfrac1{15L^2}$ (code (ii): it equals the matrix value to $\le2.3\times10^{-13}$ relative for $L\in\{3,4,5,16,64,256\}$). With (4.5):
$$\frac{M^{(3)}}{M^{(2)}}=-\frac{(g_1+g_2)L}{10\,t_f}\,\frac{1-10w+20w^2-5(1-4w)L^{-2}+4L^{-4}}{1-6w-L^{-2}}.$$
- **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 $O(\lvert\mathbf d\rvert/L^2)\,\ell_d$.

**Pole-free secular equation.** By (4.2) one field object suffices. Let $\omega_1=4\sin^2(\pi/L)$ be the smallest nonzero eigenvalue of $\Lambda$, $Q:=\mathbb 1-\lvert\varphi_0\rangle\langle\varphi_0\rvert$, and $\varepsilon=\varepsilon_0+\delta$ with $\delta<t_f\omega_1$. Write $s_0(\delta)$ and $s_d(\delta)$ for the entries of $(\Lambda-\delta/t_f)^{-1}Q$ at $(x_1,x_1)$ and $(x_1,x_2)$. Write an eigenvector of $h=C_1+V$ as $\psi=\alpha\varphi_0+\psi_\perp$. The $Q$-part of the eigen-equation gives $\psi_\perp=-t_f^{-1}(\Lambda-\delta/t_f)^{-1}QV\psi$. So $v:=(\psi(x_1),\psi(x_2))$ obeys $(\mathbb 1+SG/t_f)v=\alpha\lvert X_f\rvert^{-1/2}\mathbf 1$, with $S$ the $2\times2$ matrix of these entries and $G=\operatorname{diag}(g_1,g_2)$. The $\varphi_0$-part gives $\delta\alpha=\lvert X_f\rvert^{-1/2}\mathbf 1^{\top}Gv$. Eliminating $v$ with the adjugate, and with $c_i:=1+g_is_0(\delta)/t_f$, $b:=s_d(\delta)/t_f$:
$$\lvert X_f\rvert\,\delta\,\bigl(c_1c_2-b^2g_1g_2\bigr)=g_1c_2+g_2c_1-2b\,g_1g_2.\tag{4.18}$$
The lowest eigenvector is positive, so $\alpha\neq0$ for it. Sylvester's identity and the matrix determinant lemma, applied to $\mathcal G(\varepsilon)=(\varepsilon-C_1)^{-1}=\lvert\varphi_0\rangle\langle\varphi_0\rvert/\delta-t_f^{-1}(\Lambda-\delta/t_f)^{-1}Q$, show that left side minus right side equals $\lvert X_f\rvert\det(\varepsilon-h)/\prod_{k\neq0}(\varepsilon-\varepsilon_k)$. Its zeros on $\delta<t_f\omega_1$ are therefore exactly the eigenvalues of $h$ below $\varepsilon_1$, $\delta=0$ included. Moreover $\varepsilon_0(g)<\varepsilon_1$: some vector in the span of $\varphi_0$ and the ($2D$-fold) $\varepsilon_1$-eigenvectors vanishes at $x_1,x_2$; its cost is $\le\varepsilon_1$, and it is not the positive lowest eigenvector. Hence, for every real $(g_1,g_2)$, $\varepsilon_0(g)-\varepsilon_0$ is the smallest zero of (4.18) on $\delta<t_f\omega_1$. Code: (4.18) holds at the exact lowest eigenvalue to $\le2.7\times10^{-11}$ relative, on the ring $L=64$, the square torus $16^2$ and the cubic torus $8^3$, for couplings from $(0.01,0.02)$ to $(-10,-10)$ (`code/v3/output/remainder_bound.txt`).

**Bound on all higher orders.** Let $\mathcal M:=2N_f\lvert g_1g_2\rvert\ell_0/(\lvert X_f\rvert t_f)$. By (4.4) and $\lvert\Lambda^+(x,y)\rvert\le\ell_0$, $\mathcal M=\max_{x_2}\lvert M^{(2)}(x_1,x_2)\rvert$. Let $R:=t_f/(16\ell_0)$, $r:=t_f\omega_1/2$, and take complex $\lvert g_i\rvert\le R$ and $\lvert\delta\rvert\le r$. The Fourier modes have $\lvert\varphi_k(x)\rvert^2=1/\lvert X_f\rvert$ and $\lvert\omega_k-\delta/t_f\rvert\ge\omega_k/2$. Hence $\lvert s_0\rvert,\lvert s_d\rvert\le2\ell_0$ and, by $\ell^{(2)}_0\le\ell_0/\omega_1$, $\lvert\partial_\delta s_0\rvert\le4\ell_0/(t_f\omega_1)$. Dividing (4.18) by $c_1c_2$ gives
$$\lvert X_f\rvert\delta=\tau_1+\tau_2+X_{12},\qquad\tau_i:=g_i/c_i,\qquad X_{12}:=\frac{\tau_1\tau_2b\,(-2+(\tau_1+\tau_2)b)}{1-b^2\tau_1\tau_2}.$$
Here $\lvert\tau_i\rvert\le\tfrac87\lvert g_i\rvert$, $\lvert b^2\tau_1\tau_2\rvert\le\tfrac1{49}$ and $\lvert X_{12}\rvert\le\tfrac{128}{21}\lvert g_1g_2\rvert\ell_0/t_f$, so the right side is at most $\tfrac83R$. Since $\ell_0\lvert X_f\rvert\omega_1\ge1$ (the term $k=1$ of $\ell_0$), $\tfrac83R/\lvert X_f\rvert\le r/3$. By Rouché's theorem, (4.18) then has exactly one zero in the disc. That zero is analytic in $(g_1,g_2)$, and for real $g$ it is $\varepsilon_0(g)-\varepsilon_0$, by continuity from $g=0$ and the simplicity of $\varepsilon_0(g)$. Subtracting the one-source equations $\lvert X_f\rvert\delta_i=\tau_i(\delta_i)$ gives
$$\lvert X_f\rvert M/N_f=[\tau_1(\delta)-\tau_1(\delta_1)]+[\tau_2(\delta)-\tau_2(\delta_2)]+X_{12}.$$
With $\lvert\partial_\delta\tau_i\rvert\le\tfrac{256}{49}g_i^2\ell_0/(t_f^2\omega_1)\le\lvert X_f\rvert/49$, this gives $\lvert\delta-\delta_1\rvert\le\tfrac{14}9\lvert g_2\rvert/\lvert X_f\rvert$. Each bracket is then at most $0.51\lvert g_1g_2\rvert\ell_0/t_f$, so $\lvert M\rvert\le3.6\,\mathcal M$ on the polydisc. For real $g$ put $\eta:=\max_i\lvert g_i\rvert/R$. Then $\lambda\mapsto M(\lambda g)/\lambda^2=\sum_k\lambda^{k-2}M^{(k)}$ is analytic for $\lvert\lambda\rvert<1/\eta$ and bounded by $4\mathcal M$, so Cauchy's estimate gives $\lvert M^{(k)}\rvert\le4\mathcal M\eta^{k-2}$. Hence (exact bound; ring, square and cubic torus, every $L\ge3$, every $N_f$, all $x_1,x_2$ including $x_2=x_1$, where $b\to s_0(\delta)/t_f$):
$$\lvert M-M^{(2)}\rvert\le\frac{4\eta}{1-\eta}\,\mathcal M,\qquad\lvert M-M^{(2)}-M^{(3)}\rvert\le\frac{4\eta^2}{1-\eta}\,\mathcal M,\qquad\eta=\frac{16\,\ell_0\max_i\lvert g_i\rvert}{t_f}<1.\tag{4.19}$$
The values of $\ell_0$ are $\frac{L^2-1}{12L}$ (ring), $\frac{\ln L}{2\pi}+O(1)$ (square torus) and $W_3+O(1/L)$ (cubic torus). Thus relative accuracy $\epsilon$, measured against $\mathcal M$, is guaranteed for $\eta\le\epsilon/(4+\epsilon)$. On the ring it suffices that $\max_i\lvert g_i\rvert L/t_f\le\frac{3\epsilon}{4(4+\epsilon)}$, since $16\ell_0<4L/3$. $\mathcal M$ is the normalization that remains meaningful near the zeros of $M^{(2)}$ (ring: $w\approx\tfrac16$), whose position moves at third order. By (4.17), $\overline M$ has no terms of order $<4$, so $\lvert\overline M\rvert\le\max_{x_2}\lvert M-M^{(2)}-M^{(3)}\rvert$. Hence (4.19) holds for $\mathcal E-\overline{\mathcal E}$ with $4$ replaced by $8$. Code, same three graphs, $\eta\in\{0.05,0.25,0.5,0.9\}$, $(g_1,g_2)\propto(1,1),(1,-\tfrac12),(-1,-\tfrac13)$, all classes of $x_2$: the bounds are never violated. The largest ratios observed are $\lvert M-M^{(2)}\rvert/\mathcal M=0.149$ and $\lvert M-M^{(2)}-M^{(3)}\rvert/\mathcal M=0.020$, at $\eta=0.9$.

**Estimate.** The constants in (4.19) are not sharp. Setting $\lvert M^{(3)}\rvert=\epsilon\lvert M^{(2)}\rvert$ in the leading forms above gives the *estimated* breakdown scales (on the ring $\lvert1-10w+20w^2\rvert\le1$, measured against $\max_d\lvert M^{(2)}\rvert$; on the tori against $M^{(2)}(\mathbf d)$):
$$\text{ring: }\frac{(\lvert g_1\rvert+\lvert g_2\rvert)L}{10\,t_f}\le\epsilon\ \ (\text{relative to }\max_d\lvert M^{(2)}\rvert);\quad \text{square: }\frac{(\lvert g_1\rvert+\lvert g_2\rvert)(3\ln L-\ln\lvert\mathbf d\rvert)}{4\pi t_f}\lesssim\epsilon;\quad \text{cubic: }\frac{(\lvert g_1\rvert+\lvert g_2\rvert)\bigl(W_3+\frac1{8\pi\lvert\mathbf d\rvert}\bigr)}{t_f}\lesssim\epsilon.\tag{4.9}$$
Their small parameters are those of (4.19) up to constant factors. By (4.2), neither (4.9) nor (4.19) involves $N_f$ or $n$. At fixed $n$ and $g$, (4.19) guarantees the second-order form for $L\lesssim t_f/\lvert g\rvert$ on the ring and for $\ln L\lesssim t_f/\lvert g\rvert$ on the square torus, and uniformly in $L$ on the cubic torus. Breakdown beyond these scales is shown only on the ring: at $\gamma=3$ the second-order form is off by $25$–$59\,\%$ of the exact value (Step 6), and $ML^2/t_f$ depends only on $\gamma$ and $d/L$ up to relative $O(L^{-2})$. On the tori, (4.9) is an estimate.

### 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. The difference of the two sides of (4.10) has a simple pole at $\varepsilon=-2t_f$ ($q=0$), because its double poles cancel. Multiplied by $L\delta$, with $\delta=\varepsilon+2t_f=4t_f\sin^2(q/2)$, it is the difference of the two sides of (4.18). So $\varepsilon_0(g)$ is the smallest zero on $(-\infty,\varepsilon_1)$ of $\delta\,[(1-g_1\mathcal G_0)(1-g_2\mathcal G_0)-g_1g_2\mathcal G_d^2]$, continued to $q=0$ by its limit $-(g_1+g_2)/L-2g_1g_2(\ell_0-\ell_d)/(Lt_f)$. This zero is the largest root $\theta$ if one exists. Otherwise it is $q=0$ if the limit vanishes, and otherwise the smallest root $q\in(0,2\pi/L)$. $q=0$ is a zero iff $g_1+g_2+2g_1g_2(\ell_0-\ell_d)/t_f=0$; it is the lowest one, for example, at $L=4$, $t_f=1$, $d=2$, $(g_1,g_2)=(1,-\tfrac12)$, with eigenvalue $-2$. For $g_1=g_2=g$ the unique positive lowest eigenvector is even under the reflection exchanging $x_1,x_2$, so $1=g(\mathcal G_0+\mathcal G_d)$; at $q=0$ the cleared form of this equation is $-2g/L\neq0$, so (4.12) has no root there:
$$t_f\sin q\,\sin\frac{qL}2=g\cos\frac{qd}2\cos\frac{q(L-d)}2,\qquad\text{one source: }\ 2t_f\sin q\tan\frac{qL}2=g,\qquad\varepsilon_0=-2t_f\cos q.\tag{4.12}$$
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 smallest zeros of the cleared (4.10) and of (4.12) agree with exact diagonalization to $\le4.4\times10^{-16}$. The cleared (4.10) does so also for unequal and mixed-sign pairs, including two with the root at $q=0$: the example above, and $L=64$, $d=8$ with $g_2$ chosen by the condition above;
- 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/v3/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 (a): one shared empty place (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
$$M^{(2)}_c(d)=-\frac{N_f\,g_1g_2}{J\sinh(1/\xi)}\,e^{-d/\xi},\qquad \xi=\frac1{\operatorname{arccosh}(\Delta/2J)}.\tag{4.14}$$
This is the second-order coefficient; $\xi\to\infty$ as $\Delta\to2J^+$. On a finite chain the ends add relative corrections $O(e^{-2\delta_e/\xi})$, where $\delta_e$ is the distance of the sources from the ends. For $\lvert g_1\rvert+\lvert g_2\rvert<(\Delta-2J)/2$ the lowest state is the lowest eigenvector of $C$ (simple, isolated), so $\mathcal E$ is invariant as in Step 1. Away from the ends the single-source terms do not depend on the places, up to the same corrections. So (4.14) is the second-order part of $\mathcal E(d)-\mathcal E(d')$ for $d'\gg\xi$, up to $O(e^{-d'/\xi})$; 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/v3/output/control_decay.txt`).

*The full mutual cost of (a) does not decay.* Eliminating $X_f$ from the one-object eigen-equation gives exactly $\varepsilon=-v^{\top}(K-\varepsilon)^{-1}v$, with $v=g_1\lvert x_1\rangle+g_2\lvert x_2\rangle$. Hence $\varepsilon=-S_1+S_1S_2+O(g^6)$, with $S_n:=v^{\top}K^{-n}v$. The mutual terms of $S_1S_2$ that contain $K^{-1}(x_1,x_2)$ or $K^{-2}(x_1,x_2)$ vanish as $d\to\infty$. The remaining mutual term is $g_1^2g_2^2[A(x_1)B(x_2)+A(x_2)B(x_1)]$, so
$$M_c(d)\;\xrightarrow{d\to\infty}\;N_fg_1^2g_2^2\bigl[A(x_1)B(x_2)+A(x_2)B(x_1)\bigr]+O(g^6),\qquad A(x):=K^{-1}(x,x)>0,\ B(x):=K^{-2}(x,x)>0.$$
This limit is nonzero for small $g_1g_2\neq0$, because both sources deplete the same place $\varnothing$. Code (iii), $\Delta/J=3$, $g_1=g_2=g$, $d=160$: the exact $M_c$ equals this limit up to relative $-4.5\times10^{-3}$, $-1.8\times10^{-2}$, $-6.7\times10^{-2}$ at $g=0.05,0.1,0.2$ ($O(g^2)$). So (a) is short-range only in its second-order coefficient.

**Control field (b): cells (assumption, M8).** One field object with the places $\{0,1\}^{X_f}$, $X_f=\{1,\dots,L\}$ an open chain. Equivalently, $L$ two-place cells of distinct types with nearest-neighbour pair terms; the types are distinct, so (1.11) imposes no symmetry. Own term $C^{\rm cell}_f=\Delta\sum_yn_y-J\sum_y(\sigma^+_y\sigma^-_{y+1}+\sigma^-_y\sigma^+_{y+1})$, with $\Delta>2J>0$ and $n_y=\lvert1\rangle\langle1\rvert_y$. Coupled operator $\Phi^{\rm cell}_x=\sigma^x_x$. $\hat N=\sum_yn_y$ is conserved. On the sector $\hat N=k$, the Jordan–Wigner map (no strings for the nearest-neighbour terms of an open chain) makes $C^{\rm cell}_f$ the contract of $k$ free fermions with the one-particle matrix $K$ of (a), whose spectrum lies in $(\Delta-2J,\Delta+2J)$. So the lowest state is $\lvert\mathbf 0\rangle$ (cost $0$), simple, with gap $>\Delta-2J$. It is not sharp: $\sigma^x_x\lvert\mathbf 0\rangle=\lvert1_x\rangle$. With $V=g_1\sigma^x_{x_1}+g_2\sigma^x_{x_2}$, $\lVert V\rVert\le\lvert g_1\rvert+\lvert g_2\rvert$. For $\lvert g_1\rvert+\lvert g_2\rvert<(\Delta-2J)/2$ the lowest eigenvalue stays simple, with gap $\ge\Delta-2J-2(\lvert g_1\rvert+\lvert g_2\rvert)$ on all of $[0,g_1]\times[0,g_2]$ (Weyl). So the lowest state is connected and $\mathcal E$ is analytic there. The parity $(-1)^{\hat N}$ maps $V\to-V$, so $\mathcal E$ is even in $g$. At second order, $R\sigma^x_{x_2}\lvert\mathbf 0\rangle=\sum_yK^{-1}(y,x_2)\lvert1_y\rangle$, so $M^{(2)}=-2g_1g_2K^{-1}(x_1,x_2)$: this is (4.14) with $N_f=1$, decay length $\xi$. At all orders, Hellmann–Feynman gives $M=\int_0^{g_2}[\langle\sigma^x_{x_2}\rangle_{g_1,b}-\langle\sigma^x_{x_2}\rangle_{0,b}]\,\mathrm db$, with $\langle\cdot\rangle_{a,b}$ the lowest state of $C^{\rm cell}_f+a\sigma^x_{x_1}+b\sigma^x_{x_2}$. The bracket is the change of a local expectation at $x_2$ when the local term $g_1\sigma^x_{x_1}$ is switched on, along a path of nearest-neighbour contracts with a gap bounded below uniformly in $L$. By the locality of gapped ground states under local perturbations (Lieb–Robinson bound plus spectral gap; Bachmann–Michalakis–Nachtergaele–Sims), it is at most $C\lvert g_1\rvert F(d)$. Here $F$ falls off faster than any power of $d$, and $C,F$ depend only on $J$, $\Delta$ and the gap bound. Hence
$$\lvert M^{\rm cell}(d)\rvert\le C\,\lvert g_1g_2\rvert\,F(d),\qquad M^{\rm cell}=-2g_1g_2K^{-1}(x_1,x_2)+O(g^4),\qquad\lvert g_1\rvert+\lvert g_2\rvert<\tfrac{\Delta-2J}2.\tag{4.20}$$
As in (a), away from the ends $M^{\rm cell}(d)$ is $\mathcal E(d)-\mathcal E(d')$ for $d'\to\infty$, now at all orders, so the decay is a statement on the total cost. Code (iii): chain of $18$ cells (dimension $2^{18}$), $x_1=3$, $d=1,\dots,14$, $g_1=g_2=g$, exact $M$ by sparse diagonalization, for $(\Delta/J,g)=(3,0.1),(3,0.2),(2.5,0.1)$, all inside the range of (4.20). Over $4\le d\le14$, the ratio $M/M^{(2)}$ (with the open-chain $K^{-1}$) is constant to $8\times10^{-4}$, $3.2\times10^{-3}$, $5.6\times10^{-3}$, at the values $0.990$, $0.961$, $0.974$, which are $1-O(g^2)$. The fitted decay lengths of $M$ on $2\le d\le12$ are $1.03921$, $1.03974$, $1.44387$, against $\xi=1.03904$, $1.03904$, $1.44270$. $\lvert M\rvert$ falls to $1.2\times10^{-8}$, $4.7\times10^{-8}$, $7.3\times10^{-7}$ at $d=14$, with no floor in this range. So on $2\le d\le12$ the full mutual cost decays with a fitted length equal to $\xi$ to $\le0.1\%$.

### 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): $M^{(2)}=-\frac{ng_1g_2}{\pi t_f}[\ln\frac L{\lvert\mathbf d\rvert}-\kappa_2]$, $\kappa_2=1.31053$. This leading term is radial; for $g_1g_2>0$ it is negative and increasing in $\lvert\mathbf d\rvert$. The exact second-order sum depends on the direction of $\mathbf d$ through the corrections, so its radial form and its increase hold at leading order only. By (4.17) the same holds for $\mathcal E(\mathbf d)-\overline{\mathcal E}$ at second order. 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): $M^{(2)}=-\frac{ng_1g_2}{2\pi t_f}[\frac1{\lvert\mathbf d\rvert}-\frac{c_3}L]$, $c_3=2.837297$. This leading term is radial; for $g_1g_2>0$ it is negative and increasing in $\lvert\mathbf d\rvert$. The exact second-order sum depends on the direction of $\mathbf d$ through the corrections, so its radial form and its increase hold at leading order only. By (4.17) the same holds for $\mathcal E(\mathbf d)-\overline{\mathcal E}$ at second order. 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, $\propto g_1g_2(g_1+g_2)$. On the ring its bracket is exactly $\frac{L^2}{60}(1-10w+20w^2)-\frac1{12}+\frac w3+\frac1{15L^2}$ for every $L\ge3$.
- **(4.9)** Estimate (not a proven accuracy; from $\lvert M^{(3)}\rvert=\epsilon\lvert M^{(2)}\rvert$ with the leading forms of $M^{(3)}$; lowest state): the second-order form is estimated to hold 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 scales do not involve $N_f$ or $n$. The proven range is (4.19).
- **(4.18)** Theorem (exact, every real $g_1,g_2$; one field object on the ring or a torus, or any $N_f$ by (4.2), lowest state): $\varepsilon_0(g)-\varepsilon_0$ is the smallest zero $\delta<t_f\omega_1$ of the pole-free secular equation $\lvert X_f\rvert\delta(c_1c_2-b^2g_1g_2)=g_1c_2+g_2c_1-2bg_1g_2$, with $c_i=1+g_is_0(\delta)/t_f$, $b=s_d(\delta)/t_f$, and $s$ the entries of $(\Lambda-\delta/t_f)^{-1}Q$. Left side minus right side equals $\lvert X_f\rvert\det(\varepsilon-h)/\prod_{k\neq0}(\varepsilon-\varepsilon_k)$, and $\delta=0$ is included.
- **(4.19)** Theorem (exact bound; ring, square and cubic torus, every $L\ge3$, every $N_f$, all $x_1,x_2$ including $x_2=x_1$, lowest state, stated split): for $\eta=16\ell_0\max_i\lvert g_i\rvert/t_f<1$, $\lvert M-M^{(2)}\rvert\le\frac{4\eta}{1-\eta}\mathcal M$ and $\lvert M-M^{(2)}-M^{(3)}\rvert\le\frac{4\eta^2}{1-\eta}\mathcal M$, where $\mathcal M=2N_f\lvert g_1g_2\rvert\ell_0/(\lvert X_f\rvert t_f)=\max_{x_2}\lvert M^{(2)}\rvert$. So relative accuracy $\epsilon$ against $\mathcal M$ holds for $\eta\le\epsilon/(4+\epsilon)$; on the ring it suffices that $\max_i\lvert g_i\rvert L/t_f\le\frac{3\epsilon}{4(4+\epsilon)}$. For $\mathcal E-\overline{\mathcal E}$ the same holds with $4\to8$.
- **(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 every real $(g_1,g_2)$, $\varepsilon_0(g)$ is the smallest zero on $(-\infty,\varepsilon_1)$ of (4.10) multiplied by $\delta=\varepsilon+2t_f$, continued to $q=0$; it lies at $q=0$ iff $g_1+g_2+2g_1g_2(\ell_0-\ell_d)/t_f=0$ and no root $\theta>0$ exists. (4.12) has no root at $q=0$ for $g\neq0$. 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 coefficient only; field with one shared empty place on a chain, $\lvert g_1\rvert+\lvert g_2\rvert<(\Delta-2J)/2$, sources away from the ends, lowest state connected to $\lvert\varnothing\rangle^{\otimes N_f}$): $M^{(2)}_c=-\frac{N_fg_1g_2}{J\sinh(1/\xi)}e^{-d/\xi}$ 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 $O(e^{-d'/\xi})$. The full mutual cost of this field does not decay: as $d\to\infty$ it tends to $N_fg_1^2g_2^2[A(x_1)B(x_2)+A(x_2)B(x_1)]+O(g^6)$, with $A=K^{-1}$, $B=K^{-2}$ on the diagonal, which is nonzero for small $g_1g_2\neq0$.
- **(4.20)** Example (all orders, $\lvert g_1\rvert+\lvert g_2\rvert<(\Delta-2J)/2$; cell field $\{0,1\}^{X_f}$ on an open chain, $C^{\rm cell}_f=\Delta\sum n_y-J\sum(\sigma^+_y\sigma^-_{y+1}+{\rm h.c.})$, $\Phi_x=\sigma^x_x$, lowest state $\lvert\mathbf 0\rangle$, sources away from the ends): $\lvert M^{\rm cell}\rvert\le C\lvert g_1g_2\rvert F(d)$, with $F$ falling off faster than any power of $d$ and $C,F$ independent of $L$. Its second-order part is $-2g_1g_2K^{-1}(x_1,x_2)$, i.e. (4.14) with $N_f=1$ and decay length $\xi$, and the remainder is $O(g^4)$. Numerically ($18$ cells, $d\le14$), the fitted decay length of the full $M$ equals $\xi$ to $\le0.1\%$. The ratio $M/M^{(2)}=1-O(g^2)$ is constant in $d$ to $\le6\times10^{-3}$ for $4\le d\le14$.
- **(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. $M^{(2)}\propto g_1g_2/t_f$, $M^{(3)}\propto g^3L/t_f^2$ and $\mathcal M$ are costs. $\gamma=gL/t_f$, $(\lvert g_1\rvert+\lvert g_2\rvert)L/t_f$, $\eta=16\ell_0\max_i\lvert g_i\rvert/t_f$, $s_{0,d}$ and $g_is_0/t_f$ in (4.18) are pure numbers. Both sides of (4.18) are costs.
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 proven range (4.19) has constants that are not sharp. The breakdown scales (4.9) are estimates; breakdown beyond them is shown only on the ring (Step 6).
- For the cell field (4.20), $C$ and $F$ are not explicit, and the decay length $\xi$ of the full mutual cost is established only numerically; only its second-order part is shown to have it exactly.
- Long range is shown for the hopping class, and short range at all orders for the cell field. The field with one shared empty place is short-range only in its second-order coefficient. 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/v3/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/v3/output/second_order.txt`.
- `code/v3/check_exact_ring.py` (check, item 2, Step 5 ring bracket and (4.17)). One field object on rings $L=64,256$: the exact lowest eigenvalue by diagonalization, by the smallest zero of the pole-cleared (4.10) with (4.11), and by (4.12). $M$ is compared with $M^{(2)}$ (4.5) and $M^{(3)}$ (4.8) for several $\gamma=gL/t_f$ and $d$. It adds unequal and mixed-sign pairs, including two with the root at $q=0$, and the exact finite-$L$ bracket of (4.8) for $L\in\{3,4,5,16,64,256\}$. 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/v3/output/exact_ring.txt`.
- `code/v3/check_remainder_bound.py` (check, item 2, (4.18) and (4.19)). One field object on the ring $L=64$, the square torus $16^2$ and the cubic torus $8^3$. It computes the residual of (4.18) at the exact lowest eigenvalue for small and large couplings, and the ratios $\lvert M-M^{(2)}\rvert/\mathcal M$ and $\lvert M-M^{(2)}-M^{(3)}\rvert/\mathcal M$ over all classes of $x_2$ against the bounds of (4.19), for $\eta\in\{0.05,0.25,0.5,0.9\}$ and three sign patterns. It also treats the pole case $L=4$. Output: `code/v3/output/remainder_bound.txt`.
- `code/v3/check_control_decay.py` (check, item 3). (a) The shared-empty-place field on an open chain of $400$ places: (4.3) from $C^c_1$ against (4.14), the fitted decay length against $\xi$, and the exact full $M_c$ at $d\le160$ against its fourth-order limit. (b) The cell field on $18$ cells: the exact full $M$ by sparse diagonalization against $M^{(2)}$, and its fitted decay length against $\xi$. Output: `code/v3/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; Feshbach (Schur-complement) reduction, Sylvester's determinant identity, matrix determinant lemma
- Rouché's theorem, Cauchy estimates for analytic functions of the couplings
- Bipartite sublattice symmetry
- Jordan–Wigner map; Weyl's inequality; locality of gapped ground states under local perturbations (Lieb–Robinson bound plus gap)
- Exact diagonalization (dense and sparse), conjugate gradients (code checks)

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