Version 1 · earlier version; the current one is v3 · External review, round 1: major errors

# Typical or forced: couplings and field contracts of the made field

- **Subproject:** 06-ilang-source
- **Package:** 08-typical-or-forced
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-19

## Setup and assumptions

**Item 1.** One Model D body ($M_k=m+d_k>0$, $d_k=2\pi k/(N\lambda_0)$ the $N$ distinct eigenvalues of $D$) is coupled to the field of 04-profile by $\gamma\sum_xQ_x\otimes\Phi_x$, with $Q_x$ in class C (question). A real function of $D$ is the list of its $N$ values, written $f_k:=f(d_k)$, so class C $\cong\mathbb R^{5N}$. The inputs are (5.3)–(5.5) of 05-made-field-scale, (17.11) of 05-ilang-free-fall/17-typical-or-forced and (4.16) of 05-ilang-free-fall/04-branch-fall, as quoted in the question. Assumptions:
- **T1 (test body).** As in 05-made-field-scale, the body evolves under the partial expectation of the total contract in the field state, $C_{\rm eff}=C_{\rm D}+\gamma\sum_xN_\varphi(x)Q_x+E_\varphi$. Its own change of the field is neglected (first order in $\gamma$, A15). $C_{\rm eff}$ is used as it stands: the local value $\gamma N_\varphi(x_0)$ is not expanded, and $M_k+\gamma f_kN_\varphi(x_0)>0$.
- **T2 (fall).** The fall is the acceleration of a branch start at rest at the band point $p=0$, at first order in the gradients of $N_\varphi$ (the regime of (4.16)).
- **T3 (field states).** "Every field state" includes states with $N_\varphi(x+1)\neq N_\varphi(x)$ at any $x$, and slowly varying profiles with any local value $N_0$ in an interval and any small slope $N'\neq0$ (product states of field objects placed at chosen sites, and their superpositions).

**Item 2.** Class F as in the question; $\phi_x:=\lvert x\rangle\langle x\rvert\otimes\phi$ on one object, $\Phi_x=\sum_j\phi_x^{(j)}$. Translation invariance and finite range give $h=\sum_a\sum_x\lvert x+a\rangle\langle x\rvert\otimes h_a$ with finitely many $r\times r$ blocks $h_a$, and Hermitian Bloch matrices $h(k)=\sum_ae^{-ik\cdot a}h_a$ (trigonometric polynomials), with eigenvalues $\epsilon_0(k)\le\epsilon_1(k)\le\dots$ and eigenvectors $w_\ell(k)$. The inputs are (3.6), (4.4) and (4.20), as quoted. Assumptions:
- **F1 (lowest band).** Over the continuous zone, $\min_{k,\ell}\epsilon_\ell(k)=:\epsilon_0$ is attained only at one $k_0$ and only by $\ell=0$, with $\epsilon_0(k_0+q)-\epsilon_0=q^{\mathsf T}Aq+O(q^3)$, $A>0$. Moreover $k_0\in\frac{2\pi}{L}\mathbb Z^D$ for the sides $L$ considered. Write $w_0:=w_0(k_0)$ and $\bar\phi:=\langle w_0\vert\phi\vert w_0\rangle$. Then $u_0=\lvert X_f\rvert^{-1/2}\sum_xe^{ik_0\cdot x}\lvert x\rangle\otimes w_0$ is the unique lowest eigenvector of $h$, as class F requires.
- **F2 (state, limit).** The field state is the lowest state (M9, named). $L\to\infty$ is taken at fixed $r,h,\phi$ and fixed density $n=N_f/\lvert X_f\rvert$. Since $\chi_0/(2n)$ does not depend on $N_f$ (8.8), the range criterion is a property of $(h,\phi)$. Distances are graph distances on $X_f$.

## Derivation

### Step 1. Effective level blocks in class C

$C_{\rm D}$ and every $Q_x$ act on the clock label only through functions of $D$, so $C_{\rm eff}$ commutes with $D$ and the level $k$ is conserved. On level $k$, the terms $\Pi_x[\dots]$ give on-site terms. The identity $\sum_xN(x)\frac12(A_x+A_{x-1})=\sum_xA_x\bar N_x$, with $\bar N_x:=\frac12(N(x)+N(x+1))$, rescales the bond $(x,x+1)$. With $N=N_\varphi$:
$$B_{x,k}=M_k\sigma_3+\gamma N(x)\,W_k+E_\varphi,\quad W_k:=f_k\sigma_3+g_k\mathbb 1_2+h_{1,k}\sigma_1+h_{2,k}\sigma_2,\qquad \tau_{x,x+1;k}=\tau\bigl(1+\gamma b_k\bar N_x\bigr),\tag{8.1}$$
where $\frac12\tau_{x,x+1;k}(-i\sigma_1)$ is the hopping on $(x,x+1)$. For couplings (a) and (c) this is (5.3).

### Step 2. R3 at every place

By (8.1), $B_{x+1,k}-B_{x,k}=\gamma\,\Delta N(x)\,W_k$. If $\Delta N(x)\neq0$, a level-independent $\Gamma(x)$ with $\gamma\Delta N(x)W_k=M_k\Gamma(x)$ for all $k$ exists iff $W_k/M_k$ does not depend on $k$. By T3 such states exist at every $x$. Conversely, $W_k=M_kW_*$ gives R3 in every state, with $\Gamma(x)=\gamma\Delta N(x)W_*$. Since $\{\mathbb 1_2,\sigma_i\}$ is a basis:
$$\text{R3 at every place in every field state}\iff(f,g,h_1,h_2)=(\alpha,\beta,\eta_1,\eta_2)\,(m\mathbb 1_N+D),\qquad W_*=\alpha\sigma_3+\beta\mathbb 1_2+\eta_1\sigma_1+\eta_2\sigma_2,\tag{8.2}$$
with real $\alpha,\beta,\eta_1,\eta_2$ and $b$ unrestricted.

### Step 3. Rest and fall at $p=0$

Freeze the level-$k$ blocks at $x_0$, with $N_0:=N(x_0)$ and $N':=N'(x_0)$. The motion term gives $-\tau\sin p\,\sigma_1$ on $\sum_xe^{ipx}\lvert x\rangle$, so $H_k(p)=v_k+\mathbf d_k(p)\cdot\boldsymbol\sigma$, with $\mathbf d_k=(-\tau_k\sin p+\gamma N_0h_{1,k},\ \gamma N_0h_{2,k},\ \mu_k)$, $\mu_k=M_k+\gamma f_kN_0$, $v_k=\gamma g_kN_0+E_\varphi$ and $\tau_k=\tau(1+\gamma b_kN_0)$. Branch $\sigma=\pm$ has $E_{k,\sigma}=v_k+\sigma\lvert\mathbf d_k\rvert$.

(i) At $p=0$, $\partial_pE_{k,\sigma}=\sigma d_1\partial_pd_1/\lvert\mathbf d_k\rvert$, which gives the velocity in (8.3). It vanishes for every level in every state iff $h_1=0$.

(ii) Let $h_1=0$. On level $k$, the rotation $U=\sum_x\lvert x\rangle\langle x\rvert\otimes e^{-i\theta_k(x)\sigma_1/2}$ takes $\mu_k\sigma_3+\gamma Nh_{2,k}\sigma_2$ to $\tilde\mu_k\sigma_3$ for a suitable angle $\theta_k(x)$, where $\tilde\mu_k:=(\mu_k^2+\gamma^2N^2h_{2,k}^2)^{1/2}$. $U$ commutes with $\sigma_1$ and with every position reading. It turns the bond operator $-i\sigma_1$ into $-i\sigma_1\cos\frac{\delta\theta}2-\sin\frac{\delta\theta}2\,\mathbb 1_2$, with $\delta\theta=\theta_k(x+1)-\theta_k(x)$. The change of the $\sigma_1$ part is of second order in the gradients. The branch-diagonal hop is of first order, and its variation in $x$ is of second order. At first order, the rotated blocks therefore have the form of (4.16), with $G_k=\tilde\mu_k'$, $\mathcal F_k=-v_k'$ and the local hopping $\tau_k$:
$$v_{k,\sigma}(0)=-\frac{\sigma\,\tau_k\,\gamma N_0\,h_{1,k}}{\lvert\mathbf d_k(0)\rvert};\qquad h_1=0:\quad a_{k,\sigma}(0)=-\frac{\tau_k^2}{\tilde\mu_k}\bigl(\sigma v_k'+\tilde\mu_k'\bigr)\Big|_{x_0}.\tag{8.3}$$

### Step 4. Level-independent fall: the set $\mathcal S_1$

On (8.2) with $\eta_1=0$ we have $\tilde\mu_k=M_k\,s(N)$, with $s(N):=[(1+\gamma\alpha N)^2+(\gamma\eta_2N)^2]^{1/2}$, and $v_k'=\gamma\beta M_kN'$. In (8.3) the factor $M_k$ cancels, which gives the fall in (8.4). The fraction there vanishes identically in $N_0$ for both $\sigma$ iff $\alpha=\beta=\eta_2=0$. Otherwise it is nonzero for some $\sigma$ on an open set of $N_0$. Level independence then requires $(1+\gamma b_kN_0)^2$ to be independent of $k$ on an interval, i.e. $b_k=b_0$ (polynomial identity in $N_0$). With (i):
$$\begin{aligned}
&\mathcal S_1=\bigl\{h_1=0,\ (f,g,h_2)=(\alpha,\beta,\eta_2)(m\mathbb 1_N+D),\ (\alpha,\beta,\eta_2)\neq0,\ b=b_0\mathbb 1_N\bigr\},\qquad\mathcal S_1^0=\{f=g=h_1=h_2=0\},\\
&a_{k,\sigma}(0)=-\tau^2(1+\gamma b_0N_0)^2\,\gamma N'\,\frac{\sigma\beta\,s(N_0)+\alpha(1+\gamma\alpha N_0)+\gamma\eta_2^2N_0}{s(N_0)^2}\quad\text{on }\mathcal S_1;\qquad a_{k,\sigma}(0)=0\ \text{on }\mathcal S_1^0 .
\end{aligned}\tag{8.4}$$

### Step 5. Pure local scale $\mathcal S_2$ and the requirement

"The hopping changes by the same local factor as the on-site blocks" means two things: $B_{x,k}-E_\varphi=\nu(x)M_k\sigma_3$ with one function $\nu$, and $\tau_{x,x+1;k}=\tau\frac12(\nu(x)+\nu(x+1))$. By (8.1), the first holds in every state iff $W_k=\alpha M_k\sigma_3$ (then $\nu=1+\gamma\alpha N$), and the second iff $b_k=\alpha$. Define the body's own contract at $x$, $C_{{\rm D},x}:=\Pi_xB+\frac12(A_x+A_{x-1})$, so that $\sum_xC_{{\rm D},x}=C_{\rm D}$. The two conditions select the couplings $Q_x=\alpha C_{{\rm D},x}$. Also, $\sum_xQ_x=\mathbb 1_X\otimes W(D)+A\,b(D)$; its place-diagonal and off-diagonal parts equal those of $\alpha C_{\rm D}=\alpha A+\mathbb 1_X\otimes\alpha\sigma_3(m+D)$ iff $W=\alpha\sigma_3(m+D)$ and $b=\alpha$. Hence
$$\mathcal S_2=\{Q_x=\alpha\,C_{{\rm D},x},\ \alpha\neq0\}=\Bigl\{\textstyle\sum_xQ_x=\alpha\,C_{\rm D},\ \alpha\neq0\Bigr\},\qquad C_{\rm eff}=\sum_x\nu(x)\,C_{{\rm D},x}+E_\varphi,\qquad a(0)=-\tau^2\gamma\alpha N'\,\nu(x_0),\tag{8.5}$$
with $\mathcal S_2\subset\mathcal S_1$ ($\beta=\eta_2=0$, $b_0=\alpha$; the fall follows from (8.4)). The conditions of the larger sets can also be written with the terms of $C_{\rm D}$:
- (8.2): the on-site part of $Q_x$ is $W_*\,\lvert\Pi_xB\rvert$, with one branch operator $W_*$ for all levels, where $\lvert\Pi_xB\rvert=\Pi_x\otimes\mathbb 1_2\otimes(m+D)$ is the modulus of the body's own on-site term;
- (8.4) adds two conditions: $W_*$ has no part along $\sigma_1$, the branch operator of the motion term; and the bond part of $Q_x$ is $b_0$ times the body's own motion term at $x$.

### Step 6. Dimensions

Each set is described by linear conditions on $(f,g,h_1,h_2,b)\in\mathbb R^{5N}$:
$$\dim\mathcal S_{\rm R3}=N+4,\qquad\dim\mathcal S_1=4\ (3\text{ for }\beta=0),\qquad\dim\mathcal S_1^0=N,\qquad\dim\mathcal S_2=1,\qquad\dim(\text{class C})=5N .\tag{8.6}$$
Here $\mathcal S_{\rm R3}$ is the set (8.2). For $N\ge2$ the codimensions $4N-4$, $5N-4$, $4N$ and $5N-1$ are positive, so every set lies in a proper linear subspace.

### Step 7. Breaking at first order

Expand around $P_0=\bigl((\alpha,\beta,0,0)(m+D),\,b_0\bigr)\in\mathcal S_1$, which contains (a), (c) and $\mathcal S_2$. The deviation is $(\delta f,\delta g,\delta h_1,\delta h_2,\delta b)$. Write $\nu_0:=1+\gamma\alpha N_0$, $\rho_0:=1+\gamma b_0N_0$ and $\delta W_k:=\delta f_k\sigma_3+\delta g_k\mathbb 1_2+\delta h_{1,k}\sigma_1+\delta h_{2,k}\sigma_2$. The block difference follows exactly from (8.1). The rest velocity and the fall follow from (8.3) by expanding $\mu_k=M_k\nu_0+\gamma N_0\delta f_k$ and $\tau_k^2=\tau^2(\rho_0^2+2\rho_0\gamma N_0\delta b_k)+\dots$ ($\delta h_2$ enters $\tilde\mu_k$ only quadratically):
$$\begin{aligned}
&B_{x+1,k}-B_{x,k}=M_k\,\gamma\Delta N(x)\,W_*+\gamma\Delta N(x)\,\delta W_k,\qquad v_{k,\sigma}(0)=-\sigma\tau\,\frac{\rho_0}{\nu_0}\,\gamma N_0\,\frac{\delta h_{1,k}}{M_k},\\
&a_{k,\sigma}(0)=-\tau^2\gamma N'\Bigl[\frac{\rho_0^2(\alpha+\sigma\beta)}{\nu_0}+\frac{\rho_0^2}{\nu_0^2}\,\frac{(1-\sigma\beta\gamma N_0)\,\delta f_k+\sigma\nu_0\,\delta g_k}{M_k}+\frac{2\rho_0\gamma N_0(\alpha+\sigma\beta)}{\nu_0}\,\delta b_k\Bigr].
\end{aligned}\tag{8.7}$$

### Step 8. Response in class F

Without pair terms, $C_f=\sum_jh^{(j)}$. On the symmetric space ($c=+1$), its lowest state is $\varphi_0=u_0^{\otimes N_f}$, unique by F1. $\Phi_x$ is a sum of one-object operators, so $\Phi_x\varphi_0$ lies in the span of $\varphi_0$ and of the symmetrized states $\varphi_{(n)}$ with one object in an excited eigenvector $u_n$ of $h$. These have $\langle\varphi_{(n)}\vert\Phi_x\vert\varphi_0\rangle=\sqrt{N_f}\langle u_n\vert\phi_x\vert u_0\rangle$ and cost $\epsilon_n-\epsilon_0>0$. Inserted in (3.6), with Bloch eigenvectors $u_{k\ell}=\lvert X_f\rvert^{-1/2}\sum_xe^{ik\cdot x}\lvert x\rangle\otimes w_\ell(k)$ and $\langle u_0\vert\phi_x\vert u_{k\ell}\rangle=\lvert X_f\rvert^{-1}e^{i(k-k_0)\cdot x}\langle w_0\vert\phi\vert w_\ell(k)\rangle$, this gives, with $q:=k-k_0$ and $d:=x-y$:
$$\chi_0(x,y)=\frac{2n}{\lvert X_f\rvert}\sum_{q}\cos(q\cdot d)\,S_L(q),\qquad S_L(q)=\sum_{\ell:\,(q,\ell)\neq(0,0)}\frac{\lvert\langle w_0\vert\phi\vert w_\ell(k_0+q)\rangle\rvert^2}{\epsilon_\ell(k_0+q)-\epsilon_0}=\bigl\langle w_0\big\vert\phi\,(h(k_0+q)-\epsilon_0)^{-1}\phi\big\vert w_0\bigr\rangle\ \ (q\neq0).\tag{8.8}$$
The source $g\Phi_{x_s}$ moves the lowest state by $-gQ_0\Phi_{x_s}\varphi_0$ at first order. The made field is therefore $N(y)=n\bar\phi-g\chi_0(y,x_s)+O(g^2)$, where $N_{\varphi_0}=N_f\langle u_0\vert\phi_y\vert u_0\rangle=n\bar\phi$. The cross term of second-order perturbation theory gives the mutual cost of two sources, $-g_1g_2\chi_0(x_1,x_2)$. Summing (8.8) over $y$ gives $\sum_y\chi_0(x,y)=2n\,S_L(0)$. The code check confirms (8.8) against direct diagonalization (`check_log.txt`: deviation $\le6\times10^{-12}$ on rings and tori).

### Step 9. Singular part and the short-range criterion

By F1, $h(k)-\epsilon_0$ is invertible for real $k\neq k_0$, so $S$ is real-analytic on the torus minus $q=0$. Near $k_0$, $\epsilon_0(k)$ is an isolated simple eigenvalue, and its spectral projector $P_0(k)$ is analytic:
$$S(q)=\frac{\lvert F(q)\rvert^2}{\epsilon_0(k_0+q)-\epsilon_0}+S_{\rm an}(q),\qquad\lvert F(q)\rvert^2:=\langle w_0\vert\phi P_0(k_0+q)\phi\vert w_0\rangle,\quad\lvert F(0)\rvert^2=\bar\phi^2,\tag{8.9}$$
with $\lvert F\rvert^2$ and $S_{\rm an}$ analytic. The set of $q$ is symmetric, so only the even part $S_{\rm e}(q)=\frac12(S(q)+S(-q))$ enters (8.8). Suppose $S_{\rm e}$ extends analytically to $q=0$. Then its Fourier coefficients decay exponentially in the graph distance (shift each $q_\mu$ into the strip of analyticity). The finite sum (8.8) is the sum of their periodic images plus a $d$-independent term of order $\lvert X_f\rvert^{-1}$. Hence the differences of $\chi_0(\cdot,x_s)$ beyond distance $d$ are bounded by $Ce^{-d/\xi}$ uniformly in $L$: the field is short range. Otherwise the leading non-analytic homogeneous term of $S_{\rm e}$ at $q=0$ fixes the large-distance form (Steps 10–11).

### Step 10. $\bar\phi\neq0$: Coulomb-type long range

Now $S_{\rm e}(q)=\bar\phi^2/(q^{\mathsf T}Aq)+[\text{even terms of degree }0]+\dots$, because the terms of degree $-1$ are odd. The Fourier integral of $1/(q^{\mathsf T}Aq)$ is the Green function $\mathcal G_A$ of $-\nabla^{\mathsf T}A\nabla$ (standard, up to a constant for $D=1,2$). An even function of degree $0$ gives $O(\lvert d\rvert^{-D})$ for $d\neq0$. In $D=1$, $S-\bar\phi^2/(Aq^2)$ is analytic up to an odd term. In the limit F2:
$$\begin{aligned}
&\chi_0(y,x_s)-\chi_0(y',x_s)=2n\bar\phi^2\bigl[\mathcal G_A(y-x_s)-\mathcal G_A(y'-x_s)\bigr]+R,\\
&\mathcal G_A(d)=-\frac{\lvert d\rvert}{2A}\ (D=1),\qquad-\frac{\ln(d^{\mathsf T}A^{-1}d)}{4\pi\sqrt{\det A}}\ (D=2),\qquad\frac{1}{4\pi\sqrt{\det A}\,(d^{\mathsf T}A^{-1}d)^{1/2}}\ (D=3),
\end{aligned}\tag{8.10}$$
with $R=O(e^{-\min(\lvert d\rvert,\lvert d'\rvert)/\xi})$ for $D=1$ and $R=O(\lvert d\rvert^{-D}+\lvert d'\rvert^{-D})$ for $D=2,3$, where $d=y-x_s$ and $d'=y'-x_s$. Along an axis, $\mathcal G_A(d)-\mathcal G_A(2d)$ is proportional to $d$, $\ln2$ and $1/d$ respectively, which is not exponentially small, so the field is long range. The made field is $N(y)-N(y')=-g\times$(8.10).

On F1, $w_0$ and therefore $\bar\phi$ depend continuously on $(h_a,\phi)$, and $\phi\to\phi+\epsilon\mathbb 1_r$ shifts $\bar\phi$ by $\epsilon$ at fixed $h$. Hence $\{\bar\phi\neq0\}$ is open and dense in F1.

The code check confirms (8.10). On the rings, the differences match the lattice form of $\bar\phi^2/(q^{\mathsf T}Aq)$ to $5\times10^{-5}$, and the oscillation $\Omega(L/4)$ grows $\propto L$. On the tori they match to within 2%.

### Step 11. $\bar\phi=0$

Then $\lvert F(q)\rvert^2=q^{\mathsf T}Cq+O(q^3)$, with $C:=\frac12\nabla_q\nabla_q^{\mathsf T}\lvert F\rvert^2\big|_0\ge0$. In a smooth gauge $F(q)=\mathbf c\cdot q+O(q^2)$, so $C=\mathrm{Re}\,\mathbf c\,\mathrm{Re}\,\mathbf c^{\mathsf T}+\mathrm{Im}\,\mathbf c\,\mathrm{Im}\,\mathbf c^{\mathsf T}$ has rank $\le2$.

For $D=1$, write $\lvert F\rvert^2=q^2v(q)$ and $\epsilon_0(k_0+q)-\epsilon_0=q^2u(q)$ with $v,u$ analytic and $u(0)=A>0$. Then $S$ is analytic at $0$ and the field is short range (Step 9).

For $D=2,3$, the leading term of $S_{\rm e}$ is $q^{\mathsf T}Cq/q^{\mathsf T}Aq$, of degree 0; the next even term has degree 2. With $q_\mu q_\nu\leftrightarrow-\partial_\mu\partial_\nu$:
$$\bar\phi=0,\ D=2,3:\qquad\chi_0(y,x_s)=-2n\sum_{\mu,\nu}C_{\mu\nu}\,\partial_\mu\partial_\nu\mathcal G_A(y-x_s)+c_L+O(\lvert y-x_s\rvert^{-D-1}),\tag{8.11}$$
where $c_L$ does not depend on $y$. The leading term is homogeneous of degree $-D$. It vanishes for all $d\neq0$ iff $q^{\mathsf T}Cq/q^{\mathsf T}Aq$ is constant, i.e. $C=cA$; for $D=3$ the rank bound then forces $C=0$.

The code check confirms both cases. For the 1D member with $\bar\phi=0$ and no conserved sector, $\lvert\chi_0(d)-\chi_0(L/2)\rvert$ decays at rate $1.03749$, which equals the distance of the nearest complex zero of $\det(h(k)-\epsilon_0)$ from the real axis. For the 2D member of the same construction, $d^2(\chi_0(d)-\chi_0(\text{corner}))/(2n)$ is about $-0.020$ along axis 1 and $+0.024$ to $+0.026$ along axis 2 at $d=8,10$. This matches the lattice form of (8.11) to within 5% and the $L\to\infty$ values $-0.0194$ and $+0.0233$.

### Step 12. Conditions stated through the contract

(i) **Long range.** Every member of class F conserves the number of field objects, $\sum_xn_x=N_f$ with $n_x:=\sum_j(\lvert x\rangle\langle x\rvert\otimes\mathbb 1_r)^{(j)}$. This number is carried by a band that becomes gapless as $L\to\infty$ (F1). By Step 10 the Coulomb-type long range is present iff $\Phi_x$ carries the density of this conserved number in the lowest state:
$$\bar\phi=\frac{\langle\varphi_0\vert\Phi_x\vert\varphi_0\rangle}{\langle\varphi_0\vert n_x\vert\varphi_0\rangle}\neq0\iff\lim_{q\to0}\frac{\lvert X_f\rvert}{\sqrt{N_f}}\,\bigl\lvert\langle\varphi_{q}\vert\Phi_x\vert\varphi_0\rangle\bigr\rvert=\lvert\bar\phi\rvert\neq0,\tag{8.12}$$
where $\varphi_q$ is the lowest-band one-object excitation at momentum $k_0+q$; the right-hand form follows from $\lvert\langle u_q\vert\phi_x\vert u_0\rangle\rvert=\lvert X_f\rvert^{-1}\lvert F(q)\rvert$. The right-hand form does not change under $\Phi_x\to\Phi_x+c\mathbb 1$ (M6).

(ii) **Short range, sufficient condition (Z).** Suppose there is a Hermitian $Z$ on $\mathbb C^r$ with $[h,\mathbb 1\otimes Z]=0$, so that the charge $\sum_jZ^{(j)}$ is conserved by $C_f$. Suppose further that $w_0$ lies in an eigenspace of $Z$ with projector $P_Z$, and $P_Z\phi P_Z=0$ ($\Phi_x$ changes the charge). Then $[h(k),Z]=0$, by continuity $w_0(k)\in P_Z$ near $k_0$, and $\phi w_0\in(1-P_Z)$. Hence $S(q)=\langle w_0\vert\phi(1-P_Z)(h(k_0+q)-\epsilon_0)^{-1}(1-P_Z)\phi\vert w_0\rangle$. By F1, the bands of $h(k)$ on $(1-P_Z)$ stay above $\epsilon_0$ on the whole zone. So $S$ is analytic on the torus, and the field is short range in every $D$.

### Step 13. A short-range member

Take $r=2$, $h=(C_1+2Dt_f)\otimes\mathbb 1_2+\Delta\,\mathbb 1\otimes\lvert2\rangle\langle2\rvert$ with $t_f,\Delta>0$, and $\phi=\sigma_1$; (Z) holds with $Z=\sigma_3$. Then $h(k)=\mathrm{diag}(t_f\omega(k),\,t_f\omega(k)+\Delta)$, $k_0=0$, $w_0=\lvert1\rangle$, and F1 holds with $A=t_f\mathbb 1$. Moreover $S_L(q)=1/(t_f\omega(q)+\Delta)$ for every $q$, including $q=0$. The 1D lattice sum is standard:
$$\chi_0(x,y)=\frac{2n}{\lvert X_f\rvert}\sum_q\frac{\cos(q\cdot d)}{t_f\omega(q)+\Delta};\qquad D=1:\ \chi_0=\frac{n}{t_f\sinh\kappa}\,\frac{\cosh\kappa(\frac L2-d)}{\sinh\frac{\kappa L}2},\quad\cosh\kappa=1+\frac{\Delta}{2t_f},\ 0\le d\le L.\tag{8.13}$$
For every $D$, shifting each $q_\mu$ by $\pm i\eta$ with $\cosh\eta<1+\Delta/(2Dt_f)$ keeps $\mathrm{Re}(t_f\omega+\Delta)>0$. Hence $\lvert\chi_0(x,y)\rvert\le C_\eta e^{-\eta\,\mathrm{dist}(x,y)}$ uniformly in $L$. The code check confirms the 1D form to $2\times10^{-13}$.

## Result

Class C, test body (T1), every field state (T3); the fall at first order in the gradients (T2), exact in the local value $\gamma N_\varphi(x_0)$.
- **(8.1)** The effective level-$k$ blocks are $B_{x,k}=M_k\sigma_3+\gamma N(x)W_k+E_\varphi$ and $\tau_{x,x+1;k}=\tau(1+\gamma b_k\bar N_x)$.
- **(8.2)** R3 holds at every place in every field state iff $(f,g,h_1,h_2)=(\alpha,\beta,\eta_1,\eta_2)(m\mathbb 1_N+D)$, with $b$ free; then $\Gamma(x)=\gamma\Delta N(x)W_*$.
- **(8.3)** The start at $p=0$ has velocity $v_{k,\sigma}(0)=-\sigma\tau_k\gamma N_0h_{1,k}/\lvert\mathbf d_k(0)\rvert$, so it is at rest in every state iff $h_1=0$. For $h_1=0$ the fall is $a_{k,\sigma}(0)=-\tau_k^2(\sigma v_k'+\tilde\mu_k')/\tilde\mu_k$.
- **(8.4)** The couplings with R3 at every place and a fall from rest at $p=0$ that is level-independent on both branches in every field state form $\mathcal S_1\cup\mathcal S_1^0$. Here $\mathcal S_1$ is given by $h_1=0$, $(f,g,h_2)=(\alpha,\beta,\eta_2)(m+D)\neq0$ and $b=b_0\mathbb 1_N$, with the fall of (8.4). On $\mathcal S_1^0$ ($f=g=h_1=h_2=0$) the fall vanishes. The two branches fall alike iff $\beta=0$. Control: the rest-only coupling (r) lies outside $\mathcal S_1$.
- **(8.5)** The subset on which, in addition, the on-site blocks scale by one local factor $\nu$ and the hopping by $\frac12(\nu(x)+\nu(x+1))$ is $\mathcal S_2=\{Q_x=\alpha C_{{\rm D},x}\}=\{\sum_xQ_x=\alpha C_{\rm D}\}$, $\alpha\neq0$. On $\mathcal S_2$, $C_{\rm eff}=\sum_x\nu(x)C_{{\rm D},x}+E_\varphi$ and $a(0)=-\tau^2\gamma\alpha N'\nu(x_0)$. Coupling (a) lies in $\mathcal S_1$ but not in $\mathcal S_2$; coupling (c) lies in $\mathcal S_2$. The sets (8.2), $\mathcal S_1$ and $\mathcal S_2$ are selected by requirements written with the terms of $C_{\rm D}$ (Step 5); for $\mathcal S_2$ the requirement is that a uniform field rescales the body's own contract.
- **(8.6)** $\dim\mathcal S_{\rm R3}=N+4$, $\dim\mathcal S_1=4$ (3 with $\beta=0$), $\dim\mathcal S_1^0=N$, $\dim\mathcal S_2=1$, inside $\dim(\text{class C})=5N$. For $N\ge2$ every set lies in a proper linear subspace, so each is nowhere dense and of measure zero: exceptional, not typical. Every member of class C is admissible, so within class C the language does not select these couplings (M10).
- **(8.7)** At first order in a deviation from $P_0\in\mathcal S_1$ (with $\eta_2=0$), R3 breaks by $\gamma\Delta N(x)\,\delta W_k$; this defect is exact and is absent iff $\delta W_k/M_k$ is level-independent. The rest at $p=0$ breaks by $v_{k,\sigma}(0)\propto\sigma\gamma N_0\delta h_{1,k}/M_k$. The fall breaks through $\delta f_k/M_k$ and $\delta g_k/M_k$ (order $\gamma N'$) and through $\delta b_k$ (order $\gamma^2N_0N'$). $\delta h_2$ enters the fall only at second order.

Class F under F1, lowest state (F2); $\chi_0$ is exact at finite $L$, the made field is of first order in the source coupling $g$, and the large-distance forms hold as $L\to\infty$ at fixed $n$.
- **(8.8)** $\chi_0(x,y)=\frac{2n}{\lvert X_f\rvert}\sum_q\cos(q\cdot d)S_L(q)$, with $S_L(q)=\langle w_0\vert\phi(h(k_0+q)-\epsilon_0)^{-1}\phi\vert w_0\rangle$ for $q\neq0$. The made field is $N(y)=n\bar\phi-g\chi_0(y,x_s)+O(g^2)$.
- **(8.9)** $S=\lvert F\rvert^2/(\epsilon_0(k_0+q)-\epsilon_0)+S_{\rm an}$, and the field is short range whenever the even part of $S$ is analytic at $q=0$.
- **(8.10)** If $\bar\phi\neq0$, the field is long range: $\chi_0(y,x_s)-\chi_0(y',x_s)=2n\bar\phi^2[\mathcal G_A(y-x_s)-\mathcal G_A(y'-x_s)]+R$, which is linear in $D=1$, logarithmic in $D=2$ and $1/\lvert d\rvert$ in $D=3$. $\{\bar\phi\neq0\}$ is open and dense in F1, so long range is typical. For $r=1$ every member with $\phi\neq0$ is long range.
- **(8.11)** If $\bar\phi=0$: in $D=1$ the field is short range. In $D=2,3$, $\chi_0=-2n\sum C_{\mu\nu}\partial_\mu\partial_\nu\mathcal G_A+c_L+O(\lvert d\rvert^{-D-1})$, a power law $\lvert d\rvert^{-D}$ (long range) unless $C=cA$ (for $D=3$: $C=0$). Short range is therefore confined to the closed set $\{\bar\phi=0\}$, which has empty interior: in $D=1$ it fills this set, and in $D=2,3$ it requires in addition $C=cA$.
- **(8.12)** The condition that selects the Coulomb-type long range is that $\Phi_x$ carries the density of the conserved number of field objects, whose band is gapless: $\bar\phi\neq0$, in the M6-invariant form (8.12). A sufficient condition for short range in every $D$ is (Z): a conserved charge $\sum_jZ^{(j)}$ that $\Phi_x$ changes within the sector of the lowest state, with the other sectors gapped.
- **(8.13)** A short-range member of class F exists: $r=2$, $h=(C_1+2Dt_f)\otimes\mathbb 1_2+\Delta\,\mathbb 1\otimes\lvert2\rangle\langle2\rvert$, $\phi=\sigma_1$, $\Delta,t_f>0$. It has the exact form (8.13) and $\lvert\chi_0\rvert\le C_\eta e^{-\eta\,\mathrm{dist}}$ uniformly in $L$ for $\cosh\eta<1+\Delta/(2Dt_f)$. Controls: this member for the long-range statement (8.10); the members with $\bar\phi\neq0$ for the short-range statements (8.11)–(8.12).

## Consistency checks

1. **Couplings (a), (c), (r) and dimensions.** (8.1) reproduces (5.3) for (a) and (c). Coupling (a) is $\alpha=1$, $b_0=0$; (8.4) gives $a=-\tau^2\gamma N'/\nu(x_0)$, which is also what (4.16) gives with (5.3). Coupling (c) is $\alpha=b_0=1\in\mathcal S_2$, with $a=-\tau^2\gamma N'\nu(x_0)$. For (r), $W_k/M_k=m\sigma_3/M_k$ is level-dependent (not R3), in agreement with (5.5), and (8.3) gives exactly $a=-\tau^2\gamma mN'/(M_k+\gamma mN_0)$, which is (5.5). $\gamma N$ is dimensionless, and $a$ has the dimension of $\tau^2$. Passed.
2. **Class of 04-profile.** For $r=1$, $\phi=1$, $h=C_1$: $k_0=0$, $\bar\phi=1$, $A=t_f\mathbb 1$, and $S_L(q)=1/(t_f\omega(q))$ with $S_L(0)=0$. So (8.8) gives $\chi_0=\frac{2n}{t_f}\Lambda^+$, and the mutual cost $-g_1g_2\chi_0$ is (4.4), together with its sum rule $\sum_y\chi_0=2nS_L(0)=0$. (8.10) then gives $2n\mathcal G_{\mathbb 1}/t_f$. Passed.
3. **Short-range member against (4.20).** For (8.13), $-g_1g_2\chi_0=-2ng_1g_2K_F^{-1}(x_1,x_2)$ with $K_F=C_1+(2Dt_f+\Delta)\mathbb 1$. At $n=1$ this is the form of the leading term $-2g_1g_2K^{-1}$ of (4.20), with $(\Delta+2t_f,t_f)$ in place of $(\Delta,J)$, and $\Delta>0$ corresponds to $\Delta>2J$. The dimensions match: $[\chi_0]=[\phi]^2/[\text{cost}]$, the same as $n\bar\phi^2\mathcal G_A$. Passed.

## Open issues

- Members of class F outside F1 are not treated: a minimum at a momentum incommensurate with $L$, a degenerate or quartic minimum, several minimizers. A quartic minimum with $\bar\phi=0$ could be long range even in $D=1$.
- In $D=2,3$, short range on $\{\bar\phi=0,\ C=cA\}$ is decided by higher orders of (8.9), which are not characterized here; (Z) is sufficient only.
- Item 1 is derived at the band point $p=0$ and at first order in the gradients. The second-order effects of $h_1,h_2$ and the band point $p=\pi$ are not treated. T3 (which profiles field states realize) is assumed for the field of 04-profile.
- "Exceptional" and "typical" refer to the natural parametrizations of classes C and F (coefficients $f_k,\dots$; blocks $h_a$, $\phi$). The language supplies no measure on descriptions.

## Code

- `code/v1/check_response.py` (role: check, item 2). One field object ($N_f=1$, $n=1/\lvert X_f\rvert$) of class F with $r=2$ and Bloch matrix $h(k)=-2t\sum_\mu\cos k_\mu+\Delta\lvert2\rangle\langle2\rvert+\lambda\sin k_1\,\sigma_1$, with $t=1$ and $\Delta=1.5$. It computes $\chi_0(x,x_s)$ of (3.6) by full diagonalization of $h$ and compares it with (8.8), (8.10), (8.11) and (8.13). Members on rings $L=40,80,160$: long range ($\lambda=0.5$, $\phi=\begin{pmatrix}1&0.3\\0.3&-0.5\end{pmatrix}$, $\bar\phi=1$); $\bar\phi=0$ without a conserved sector ($\lambda=0.5$, $\phi=\sigma_1$); and the (Z) member (8.13) ($\lambda=0$, $\phi=\sigma_1$). On square tori $L=20,32$: the long-range member and the $\bar\phi=0$ member.
- Outputs:
  - `code/v1/output/check_log.txt`: all compared numbers;
  - `code/v1/output/chi_ring.csv`: $\chi_0/(2n)$ and $\Omega(d)/(2n)$ against $d$ on the rings, where $\Omega(d)$ is the max−min of $\chi_0$ over distances $\ge d$;
  - `code/v1/output/chi_torus.csv`: $\chi_0/(2n)$ along both axes of the tori, with the lattice form of the prediction;
  - `code/v1/output/decay.png`: $\Omega(d)/(2n)$ on the rings, and $d^2(\chi_0(d)-\chi_0(\text{corner}))/(2n)$ on the torus.

## Methods used

- partial expectation (mean field) for a test body; level decomposition by functions of $D$
- local Bloch Hamiltonian; local $\sigma_1$ rotation; first-order gradient (semiclassical) band dynamics
- linear algebra of Pauli bases; dimension and codimension counting of linear subspaces
- Rayleigh–Schrödinger perturbation theory; second quantization for non-interacting identical objects
- Bloch decomposition; analytic perturbation of an isolated eigenvalue; Paley–Wiener (analyticity versus exponential decay)
- Fourier transforms of homogeneous functions; anisotropic lattice Green functions
- numerical full diagonalization and FFT (check)

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