Version 2 · earlier version; the current one is v3 · External review, round 2: minor issues

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

- **Subproject:** 06-ilang-source
- **Package:** 08-typical-or-forced
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-19

## Changes from previous version

- Setup T1, Steps 3, 4, 6, 7: item 1 is answered at first order in $\gamma$; $b$ and $h_2$ drop out of the fall; $\mathcal S_1$ now imposes $\beta=0$ (dimension $N+2$); the larger set $\mathcal S_1^{\pm}$ ($\beta$ free) is kept apart; $\mathcal S_1^0$ is removed (its members now lie in $\mathcal S_1$, at $\alpha=\eta_2=0$). (I1, I2)
- Step 5: $\mathcal S_2\subset\mathcal S_1$ with $b=\alpha\mathbb 1_N$; the requirement for $\mathcal S_1$ is restated ($W_*$ anticommutes with $\sigma_1$). (I1, I2)
- Setup item 2: F1 replaced by F0 (coefficient space $P$ with its topology; class F requires uniqueness at every side, no commensurability) and the generic set $U=$ G1 ∩ G2; F2 states the order of limits. (I3, I4)
- Step 8: (8.8) for every member of class F, at the grid momentum $k_L$ of the lowest state; finite-$L$ convergence radius of the expansion in $g$. (I3, I4)
- Step 9: new necessary condition (8.14) for short range; (8.9) restricted to G1. (I3)
- Step 10: long range proved on $U$ for every $k_0$ through (8.14); for $r=1$ for every member with $\phi\neq0$; new genericity statement (8.15); (8.10) with its order of limits; numerical statements corrected. (I3, I4, I5)
- Step 11: restricted to G1 with $k_0=0$, inside the complement of (8.15); numerical statements corrected. (I3, I5)
- Step 12: (8.12) stated through limits $L\to\infty$; (Z) holds uniformly in $L$ for every $k_0$. Step 13: membership in class F stated. (I3)
- Code: `code/v2/check_response.py` adds the torus $L=64$ through (8.8) and logs maximal deviations over stated windows. (I5)
- Result, consistency check 1, Open issues, Methods used: updated accordingly.

## Response to verification

- **I1:** Accepted and fixed in Setup T1 and Steps 3, 4, 6, 7. The sets are now at first order in $\gamma$, where $a_{k,\sigma}(0)=-\tau^2\gamma N'(f_k+\sigma g_k)/M_k+O(\gamma^2)$ and $b$, $h_2$ do not enter. (8.3) is kept as a property of $C_{\rm eff}$ as it stands, and it is no longer used to impose conditions.
- **I2:** Accepted and fixed in Steps 4 and 6. $\mathcal S_1$ imposes $\beta=0$; the set with a level-independent fall on each branch separately is $\mathcal S_1^{\pm}$.
- **I3:** Accepted and fixed in Setup F0 and Steps 8–12, within the scope fixed by the dated note of the question. Class F is taken as defined, with the topology of its coefficient space. Long range is proved on the open dense set $U$ for every $k_0$, commensurate or not, and the complement of $U$ is shown closed with empty interior (8.15). The verifier's examples: $2t[1-\cos(k-\theta)]$ with $\phi\neq0$ lies in $U$ and is long range; the quartic minimum lies in the complement. The classification inside the complement (quartic or degenerate minima, $\bar\phi=0$ with $C=cA$) is not asked by the note; (8.11) is kept only as a statement for G1 with $k_0=0$.
- **I4:** Accepted and fixed in Setup F2 and Steps 8, 10. The classification concerns the Taylor coefficient $\chi_0$ at each finite $L$, before $L\to\infty$. The finite-$L$ range of the expansion is $\lvert g\rvert<\Delta_L/(2N_f\lVert\phi\rVert)$ with $\Delta_L\le C/L^2$; no $L$-uniform regime is claimed. (8.10) is the limit $L\to\infty$ at fixed distances ($1\ll\lvert d\rvert\ll L$).
- **I5:** Accepted and fixed in Steps 10, 11 and the code. The maximal deviations are now given over stated windows, separately against the lattice form at the same $L$ and against the $L\to\infty$ value.

## 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, first order in $\gamma$).** 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, which is an error of order $\gamma^2$ (A15). Item 1 is therefore answered at first order in $\gamma$: the sets (8.2), (8.4), (8.5), their dimensions (8.6) and the breaking (8.7) keep the terms of order $\gamma$ and drop those of order $\gamma^2$. (8.1) and (8.3) are written for $C_{\rm eff}$ as it stands (exact in $\gamma N_\varphi$, with $M_k+\gamma f_kN_\varphi(x_0)>0$). They are properties of $C_{\rm eff}$, not of the full description beyond first order.
- **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 $h=\sum_{a\in R}\sum_x\lvert x+a\rangle\langle x\rvert\otimes h_a$ with finitely many $r\times r$ blocks $h_a=h_{-a}^\dagger$, 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 $w_\ell(k)$. At side $L$, $k$ runs over the grid $\frac{2\pi}L\mathbb Z^D$ (mod $2\pi$). The inputs are (3.6), (4.4) and (4.20), as quoted. Assumptions:
- **F0 (coefficient topology).** Fix $D$, $r$ and a finite range $R\supseteq\{0,\pm e_1,\dots,\pm e_D\}$. A member is a point $p=((h_a)_{a\in R},\phi)$ of the real vector space $P$ of these coefficients, with its Euclidean topology. It belongs to class F iff $h$ has a unique lowest eigenvector at every side $L$. That eigenvector is then a Bloch state $u_0=\lvert X_f\rvert^{-1/2}\sum_xe^{ik_L\cdot x}\lvert x\rangle\otimes w_L$ at a grid momentum $k_L$, with $w_L:=w_0(k_L)$, $\epsilon_L:=\epsilon_0(k_L)$, $\bar\phi_L:=\langle w_L\vert\phi\vert w_L\rangle$. $\Delta_L>0$ is the distance of $\epsilon_L$ from the rest of the spectrum of $h$.
- **G1, G2 (generic set).** G1: 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 $\epsilon_0(k_0+q)-\epsilon_0=q^{\mathsf T}Aq+O(q^3)$, $A>0$. G2: $\bar\phi:=\langle w_0\vert\phi\vert w_0\rangle\neq0$, with $w_0:=w_0(k_0)$. $U\subset P$ is the set where G1 and G2 hold. No commensurability of $k_0$ with the grid is assumed.
- **F2 (state, order, limit).** The field state is the lowest state (M9, named). The made field is the term of order $g$ of the Taylor expansion at $g=0$, taken at each finite $L$ ($\chi_0$ of (3.6)). $L\to\infty$ is taken afterwards, at fixed $p$ 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 $p$. 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). At first order in $\gamma$ it is $-\sigma\tau\gamma N_0h_{1,k}/M_k+O(\gamma^2)$, so it vanishes for every level in every state iff $h_1=0$. At $p=0$, $h_1$ changes $\lvert\mathbf d_k\rvert$, its $x$-derivative and $\partial_p^2E_{k,\sigma}$ only at order $\gamma^2$. At first order it therefore enters only through this velocity.

(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- and branch-independent fall: the set $\mathcal S_1$

At first order in $\gamma$: $\tilde\mu_k=M_k+\gamma f_kN_0+O(\gamma^2)$ ($h_2$ enters $\tilde\mu_k$ only quadratically), $\tilde\mu_k'=\gamma f_kN'+O(\gamma^2)$, $v_k'=\gamma g_kN'$ and $\tau_k^2=\tau^2+O(\gamma)$. With $h_1=0$, (8.3) becomes $a_{k,\sigma}(0)=-\tau^2\gamma N'(f_k+\sigma g_k)/M_k+O(\gamma^2)$. This is also (4.16) with $G_k=\gamma f_kN'$, $\mathcal F_k=-\gamma g_kN'$. Neither $b$ nor $h_2$ enters at this order. States with $N'\neq0$ exist (T3), so the fall is the same for all levels and both branches iff $(f_k+\sigma g_k)/M_k=\alpha$ for all $k$ and $\sigma$, i.e. $g=0$ and $f=\alpha(m\mathbb 1_N+D)$. With (i) and (8.2):
$$\begin{aligned}
&\mathcal S_1=\bigl\{h_1=0,\ g=0,\ (f,h_2)=(\alpha,\eta_2)(m\mathbb 1_N+D),\ b\ \text{free}\bigr\},\qquad a_{k,\sigma}(0)=-\tau^2\gamma\alpha N'+O(\gamma^2);\\
&\mathcal S_1^{\pm}=\bigl\{h_1=0,\ (f,g,h_2)=(\alpha,\beta,\eta_2)(m\mathbb 1_N+D),\ b\ \text{free}\bigr\},\qquad a_{k,\sigma}(0)=-\tau^2\gamma N'(\alpha+\sigma\beta)+O(\gamma^2).
\end{aligned}\tag{8.4}$$
$\mathcal S_1^{\pm}$ is the set with R3 and rest. On it the fall is level-independent on each branch, but $a_{k,+}-a_{k,-}=-2\tau^2\gamma\beta N'$, so $\mathcal S_1=\mathcal S_1^{\pm}\cap\{\beta=0\}$. On $\mathcal S_1$ the fall vanishes at this order iff $\alpha=0$.

### 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
$$\mathcal S_2=\{Q_x=\alpha\,C_{{\rm D},x},\ \alpha\neq0\}=\Bigl\{\textstyle\sum_xQ_x=\alpha\,C_{\rm D},\ \alpha\neq0\Bigr\},\qquad C_{\rm eff}=\sum_x\nu(x)\,C_{{\rm D},x}+E_\varphi,\qquad a(0)=-\tau^2\gamma\alpha N'+O(\gamma^2),\tag{8.5}$$
with $\mathcal S_2\subset\mathcal S_1$ ($\eta_2=0$, $b=\alpha\mathbb 1_N$; 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;
- $\mathcal S_1$ in (8.4): in addition, $W_*$ anticommutes with $\sigma_1$, the branch operator of the motion term, since $\{W_*,\sigma_1\}=2\beta\sigma_1+2\eta_1\mathbb 1_2$. The bond part of $Q_x$ is free.

### Step 6. Dimensions

Each set is described by linear conditions on $(f,g,h_1,h_2,b)\in\mathbb R^{5N}$:
$$\dim\mathcal S_{\rm R3}=N+4,\qquad\dim\mathcal S_1^{\pm}=N+3,\qquad\dim\mathcal S_1=N+2,\qquad\dim\mathcal S_2=1,\qquad\dim(\text{class C})=5N .\tag{8.6}$$
Here $\mathcal S_{\rm R3}$ is the set (8.2). For $N\ge2$ the codimensions $4N-4$, $4N-3$, $4N-2$ and $5N-1$ are positive, so every set lies in a proper linear subspace.

### Step 7. Breaking at first order

Expand around $P_0=\bigl((\alpha,0,0,\eta_2)(m+D),\,b\bigr)\in\mathcal S_1$; $\mathcal S_1$ contains (a), (c) and $\mathcal S_2$. The deviation is $(\delta f,\delta g,\delta h_1,\delta h_2,\delta b)$, with $\delta W_k:=\delta f_k\sigma_3+\delta g_k\mathbb 1_2+\delta h_{1,k}\sigma_1+\delta h_{2,k}\sigma_2$. The block difference follows exactly from (8.1). The rest velocity follows from Step 3 (i), and the fall from Step 4, which at first order in $\gamma$ is linear in $(f,g)$ and does not contain $h_1$, $h_2$, $b$:
$$\begin{aligned}
&B_{x+1,k}-B_{x,k}=M_k\,\gamma\Delta N(x)\,W_*+\gamma\Delta N(x)\,\delta W_k,\qquad v_{k,\sigma}(0)=-\sigma\tau\gamma N_0\,\frac{\delta h_{1,k}}{M_k}+O(\gamma^2),\\
&a_{k,\sigma}(0)=-\tau^2\gamma N'\Bigl[\alpha+\frac{\delta f_k+\sigma\,\delta g_k}{M_k}\Bigr]+O(\gamma^2).
\end{aligned}\tag{8.7}$$
So R3 breaks iff $\delta W_k/M_k$ depends on $k$. The fall becomes level-dependent through $\delta f_k/M_k$ and branch-dependent through $\delta g_k/M_k$. $\delta b$ and $\delta h_2$ do not enter at order $\gamma$.

### 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 F0. $\Phi_x$ 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 $\epsilon_n-\epsilon_L>0$. 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 $\langle u_0\vert\phi_x\vert u_{k\ell}\rangle=\lvert X_f\rvert^{-1}e^{i(k-k_L)\cdot x}\langle w_L\vert\phi\vert w_\ell(k)\rangle$, this gives, with $q:=k-k_L$ on the grid and $d:=x-y$:
$$\chi_0(x,y)=\frac{2n}{\lvert X_f\rvert}\sum_{q}\cos(q\cdot d)\,S_L(q),\qquad S_L(q)=\sum_{\ell:\,(q,\ell)\neq(0,0)}\frac{\lvert\langle w_L\vert\phi\vert w_\ell(k_L+q)\rangle\rvert^2}{\epsilon_\ell(k_L+q)-\epsilon_L}=\bigl\langle w_L\big\vert\phi\,(h(k_L+q)-\epsilon_L)^{-1}\phi\big\vert w_L\bigr\rangle\ \ (q\neq0).\tag{8.8}$$
Every term of $S_L$ is $\ge0$. 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 $N(y)=n\bar\phi_L-g\chi_0(y,x_s)+O(g^2)$, where $N_{\varphi_0}=N_f\langle u_0\vert\phi_y\vert u_0\rangle=n\bar\phi_L$. At fixed $L$ the lowest state is analytic in $g$ for $\lvert g\rvert<\Delta_L/(2N_f\lVert\phi\rVert)$ (isolated simple eigenvalue, $\lVert\Phi_{x_s}\rVert\le N_f\lVert\phi\rVert$; standard), which is the range of the $O(g^2)$ term. On G1, $\Delta_L\le C'/L^2$ (Step 10), so this range shrinks with $L$. No $L$-uniform small-coupling regime is claimed, and the range classification concerns $\chi_0$ (F2). 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. Two criteria on $S_L$, and the singular part

The grid is symmetric, so only the even part $S_{L,\rm e}(q):=\frac12(S_L(q)+S_L(-q))$ enters (8.8), and $\sum_ye^{-iq\cdot(y-x_s)}\chi_0(y,x_s)=2nS_{L,\rm e}(q)$ for $q\neq0$.

(i) **Necessary.** Let the field be short range, $\lvert\chi_0(y,x_s)-\chi_0(y',x_s)\rvert\le2nCe^{-d/\xi}$ for all $L$ and all $y,y'$ at distance $\ge d$ from $x_s$. Take $y'$ at maximal distance; then the bound holds with $d=\lvert y-x_s\rvert$ for every $y$. Since $\sum_ye^{-iq\cdot(y-x_s)}=0$ for $q\neq0$, the constant $\chi_0(y',x_s)$ may be subtracted, and with $c_D(d+1)^{D-1}$ bounding the number of places at distance $d$:
$$\text{short range}\ \Longrightarrow\ \bigl\lvert S_{L,\rm e}(q)\bigr\rvert\le C\sum_{d\ge0}c_D(d+1)^{D-1}e^{-d/\xi}=:K<\infty\qquad\text{for all }L\text{ and all grid }q\neq0.\tag{8.14}$$

(ii) **Sufficient.** Suppose that for $L\ge L_0$, $S_{L,\rm e}$ is the restriction to the grid of a $2\pi$-periodic function, analytic and bounded by $K'$ on the strip $\lvert\mathrm{Im}\,q_\mu\rvert<\eta$, with $K',\eta$ independent of $L$. Then its Fourier coefficients decay as $e^{-\eta'\lvert d\rvert}$ for every $\eta'<\eta$, uniformly in $L$ (shift each $q_\mu$ into the strip). The finite sum (8.8) is the sum of their periodic images plus a $d$-independent term of order $\lvert X_f\rvert^{-1}$ (from $q=0$). Hence the field is short range; the finitely many $L<L_0$ are absorbed into $C$.

(iii) **Singular part on G1.** Let $S(q):=\langle w_0\vert\phi(h(k_0+q)-\epsilon_0)^{-1}\phi\vert w_0\rangle$ for $q\neq0$. If $k_0=0$, then $k_L=k_0$, $\epsilon_L=\epsilon_0$ and $S_L=S$ on the grid for every $L$. By G1, $h(k)-\epsilon_0$ 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. For $k_0=0$, by (ii), the field is short range whenever $S_{\rm e}$ extends analytically to $q=0$. Otherwise the leading non-analytic homogeneous term of $S_{\rm e}$ at $q=0$ fixes the large-distance form (Steps 10–11).

### Step 10. $\bar\phi\neq0$: long range on $U$, and genericity

**Long range.** Let $p\in U\cap{}$F. By G1, $\epsilon_0(k)-\epsilon_0\ge c_1\lvert k-k_0\rvert^2$ on the zone (quadratic near $k_0$, bounded below elsewhere; torus distance). The grid point nearest to $k_0$ lies within $\pi\sqrt D/L$, so $\epsilon_L-\epsilon_0\le C/L^2$ and $\lvert k_L-k_0\rvert\le c/L$. With $q_L:=\frac{2\pi}Le_1$, this gives $\epsilon_0(k_L\pm q_L)-\epsilon_L\le C'/L^2$ (hence $\Delta_L\le C'/L^2$), and $\lvert\langle w_L\vert\phi\vert w_0(k_L\pm q_L)\rangle\rvert^2\to\bar\phi^2$ by continuity of $w_0(k)$ near $k_0$. Keeping only the lowest-band term of the nonnegative sum (8.8), $S_{L,\rm e}(q_L)\ge\bar\phi^2L^2/(4C')$ for large $L$. This is unbounded, so by (8.14) the made field is long range, for every $k_0$, commensurate with the grid or not. For $r=1$ the same holds for every member of class F with $\phi\neq0$: then $S_L(\pm q_L)=\phi^2/(\epsilon(k_L\pm q_L)-\epsilon_L)$, and $0<\epsilon(k_L\pm q_L)-\epsilon_L\le\lVert\nabla\epsilon\rVert_\infty\,2\pi/L$.

**Leading form.** Take $L\to\infty$ along sides with $L\lvert k_L-k_0\rvert\to0$. This holds for every side if $k_0=0$, and such sides exist for every $k_0$ (Dirichlet's simultaneous approximation). Then $\epsilon_0(k_L+q)-\epsilon_L=q^{\mathsf T}Aq\,(1+o(1))$ for $\lvert q\rvert=O(1/L)$ and $\ge c\lvert q\rvert^2$ for all grid $q\neq0$. So the Riemann sums (8.8) for differences $\chi_0(y,x_s)-\chi_0(y',x_s)$ at fixed $y,y'$ converge to the integrals with $S$ of (8.9), dominated by $C\lvert q\rvert^2/\lvert q\rvert^2$. 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 $L\to\infty$ at fixed $y,y'$ (taken first; at finite $L$ the form describes $1\ll\lvert d\rvert,\lvert d'\rvert\ll L$):
$$\begin{aligned}
&\chi_0(y,x_s)-\chi_0(y',x_s)=2n\bar\phi^2\bigl[\mathcal G_A(y-x_s)-\mathcal G_A(y'-x_s)\bigr]+R,\\
&\mathcal G_A(d)=-\frac{\lvert d\rvert}{2A}\ (D=1),\qquad-\frac{\ln(d^{\mathsf T}A^{-1}d)}{4\pi\sqrt{\det A}}\ (D=2),\qquad\frac{1}{4\pi\sqrt{\det A}\,(d^{\mathsf T}A^{-1}d)^{1/2}}\ (D=3),
\end{aligned}\tag{8.10}$$
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. The made field is $N(y)-N(y')=-g\times$(8.10).

**Genericity.** G1 is open in $P$: the lowest eigenvalue is continuous in $(k,p)$, simple and analytic near $k_0$ with positive Hessian, and above $\epsilon_0+c$ away from a neighbourhood of $k_0$; a small change of $p$ keeps one nondegenerate minimum near $k_0$ (implicit function theorem) and none elsewhere. On G1, $w_0$ and $\bar\phi$ are continuous, so $U$ is open. $U$ is dense: given $p$, a minimizer $k_*$ of $\epsilon_0$ and a lowest eigenvector $v$ of $h(k_*)$, add the nearest-neighbour term with Bloch matrix $\delta\,w(k)\mathbb 1_r$, $w(k)=\sum_\mu(1-\cos(k_\mu-k_{*\mu}))$, and the on-site term $-\delta'\lvert v\rangle\langle v\rvert$, with $\delta,\delta'>0$ small. The new lowest eigenvalue is $\ge\epsilon_0(k)-\delta'+\delta w(k)>\epsilon_0(k_*)-\delta'$ for $k\neq k_*$. At $k_*$ it equals $\epsilon_0(k_*)-\delta'$ and is simple ($v$ is an eigenvector, and $v^\perp$ is untouched). Its Hessian is $\ge\delta\mathbb 1$. So G1 holds, and $\phi\to\phi+\epsilon\mathbb 1_r$ ($\bar\phi\to\bar\phi+\epsilon$) gives G2. The same perturbation, with $k_*$ a minimizer on the grid of side $L$, makes the lowest eigenvalue at side $L$ simple, which is an open condition. So class F is a countable intersection of open dense subsets of $P$, dense by Baire's theorem. A nonempty relatively open subset $V\cap{}$F of class F therefore meets $U$, because $V\cap U$ is open and nonempty. Hence, in the topology of F0:
$$U\cap\text{F is open and dense in class F; its complement F}\setminus U\text{ (G1 fails, or }\bar\phi=0\text{) is closed in F with empty interior; on G1}\cap\text{F, short range requires }\bar\phi=0.\tag{8.15}$$

The code check confirms (8.10) (`check_log.txt`). On the rings $L=40,80,160$, the ratio of the differences $\chi_0(d)-\chi_0(L/2)$ to those of the lattice form of $\bar\phi^2/(q^{\mathsf T}Aq)$ at the same $L$ differs from 1 by at most $4.8\times10^{-5}$ over $6\le d\le L/2-6$, and the oscillation $\Omega(L/4)=1.5,3.0,6.0$ grows $\propto L$. On the tori, the same ratio over $3\le d\le L/4$ differs from 1 by at most $2.2\%$ ($L=20$), $1.6\%$ ($L=32$) and $1.1\%$ ($L=64$, through (8.8)). The maximum is at $d=3$; at $d=L/4=16$ ($L=64$) the deviation is below $0.2\%$, as expected from the remainder $R$.

### Step 11. $\bar\phi=0$ on G1 with $k_0=0$ (inside the complement)

The members with $\bar\phi=0$ lie in the complement of (8.15). Their classification is not asked (question, note of 2026-10-19). For members of G1 with $k_0=0$ ($S_L=S$ for every $L$), the leading order of (8.9) gives the following. $\lvert F(q)\rvert^2=q^{\mathsf T}Cq+O(q^3)$, 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 (ii)).

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 (`check_log.txt`) illustrates both cases. For the 1D member with $\bar\phi=0$ and no conserved sector, the fitted decay rate of $\lvert\chi_0(d)-\chi_0(L/2)\rvert$ over $4\le d\le14$ is $1.03774$ ($L=40$) and $1.03749$ ($L=80,160$). The distance of the nearest complex zero of $\det(h(k)-\epsilon_0)$ from the real axis is $1.03749$. For the 2D member of the same construction, $Y(d):=d^2(\chi_0(d)-\chi_0(\text{corner}))/(2n)$ is negative along axis 1 and positive along axis 2, as (8.11) requires. Against the lattice form of the leading term of (8.11) at the same $L$, the maximal relative deviation over $8\le d\le L/2-2$ is $7.4\%$ ($L=20$), $6.1\%$ ($L=32$) and $6.1\%$ ($L=64$, through (8.8)), and $2.0\%$ over $16\le d\le30$ at $L=64$. It decreases with $d$, consistent with the remainder of (8.11). Against the $L\to\infty$ values $-0.01937$ and $+0.02325$, the deviation is at most $2.8\%$ over $8\le d\le12$ at $L=64$. For $d\gtrsim L/4$, $Y(d)$ grows through the finite-size images (order $(d/L)^2$).

### 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 $n_x:=\sum_j(\lvert x\rangle\langle x\rvert\otimes\mathbb 1_r)^{(j)}$. On G1 this number is carried by a band that becomes gapless as $L\to\infty$. By Step 10, on G1 the made field is long range, with the Coulomb-type form (8.10), when $\Phi_x$ carries the density of this conserved number in the lowest state:
$$\bar\phi=\lim_{L\to\infty}\frac{\langle\varphi_0\vert\Phi_x\vert\varphi_0\rangle}{\langle\varphi_0\vert n_x\vert\varphi_0\rangle}\neq0\iff\lim_{L\to\infty}\frac{\lvert X_f\rvert}{\sqrt{N_f}}\,\bigl\lvert\langle\varphi_{q_L}\vert\Phi_x\vert\varphi_0\rangle\bigr\rvert=\lvert\bar\phi\rvert\neq0,\tag{8.12}$$
where $\varphi_q$ is the lowest-band one-object excitation at momentum $k_L+q$ and $q_L=\frac{2\pi}Le_1$. The ratio equals $\bar\phi_L\to\bar\phi$. The right-hand form follows from $\lvert\langle u_q\vert\phi_x\vert u_0\rangle\rvert=\lvert X_f\rvert^{-1}\lvert\langle w_0(k_L+q)\vert\phi\vert w_L\rangle\rvert$ and the continuity of $w_0$ near $k_0$. It 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 $\phi w_L\in(1-P_Z)$ for large $L$. Hence $S_L(q)=\langle w_L\vert\phi(1-P_Z)(h(k_L+q)-\epsilon_L)^{-1}(1-P_Z)\phi\vert w_L\rangle$. By G1, the bands of $h(k)$ on $(1-P_Z)$ stay above $\epsilon_0+g_Z$, $g_Z>0$, on the whole zone, and $\epsilon_L\to\epsilon_0$. So for large $L$, $S_L$ is the restriction of a function analytic and bounded on a strip of fixed width, uniformly in $L$. By Step 9 (ii) the field is short range in every $D$, for every $k_0$. Such members have $\bar\phi=0$.

### 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 $h(k)=\mathrm{diag}(t_f\omega(k),\,t_f\omega(k)+\Delta)$, $k_0=0$, $w_0=\lvert1\rangle$, G1 holds with $A=t_f\mathbb 1$, and the lowest eigenvector is unique at every side (class F). 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 sets, dimensions and breaking at first order in $\gamma$, the fall at first order in the gradients (T2) at the band point $p=0$. (8.1) and (8.3) hold for $C_{\rm eff}$ as it stands (exact in $\gamma N_\varphi$).
- **(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)** For $C_{\rm eff}$, the start at $p=0$ has velocity $v_{k,\sigma}(0)=-\sigma\tau_k\gamma N_0h_{1,k}/\lvert\mathbf d_k(0)\rvert$, which is $-\sigma\tau\gamma N_0h_{1,k}/M_k+O(\gamma^2)$; 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)** At first order in $\gamma$ the fall from rest is $a_{k,\sigma}(0)=-\tau^2\gamma N'(f_k+\sigma g_k)/M_k+O(\gamma^2)$; $b$ and $h_2$ do not enter. The couplings with R3 at every place, rest at $p=0$, and a fall equal across levels and across the two branches in every field state form $\mathcal S_1=\{h_1=0,\ g=0,\ (f,h_2)=(\alpha,\eta_2)(m\mathbb 1_N+D),\ b\text{ free}\}$, with $a_{k,\sigma}(0)=-\tau^2\gamma\alpha N'+O(\gamma^2)$ (zero at this order iff $\alpha=0$). The larger set $\mathcal S_1^{\pm}=\{h_1=0,\ (f,g,h_2)=(\alpha,\beta,\eta_2)(m\mathbb 1_N+D),\ b\text{ free}\}$ (R3 and rest) has a level-independent fall on each branch, $a_{k,\sigma}(0)=-\tau^2\gamma N'(\alpha+\sigma\beta)+O(\gamma^2)$, which differs between the branches iff $\beta\neq0$. Control: the rest-only coupling (r) lies outside $\mathcal S_1^{\pm}$.
- **(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 $a(0)=-\tau^2\gamma\alpha N'+O(\gamma^2)$. 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 $\mathcal S_1$, the on-site part of $Q_x$ is $W_*\lvert\Pi_xB\rvert$, with one branch operator $W_*$ that anticommutes with the branch operator $\sigma_1$ of the motion term. For $\mathcal S_2$, a uniform field rescales the body's own contract.
- **(8.6)** $\dim\mathcal S_{\rm R3}=N+4$, $\dim\mathcal S_1^{\pm}=N+3$, $\dim\mathcal S_1=N+2$, $\dim\mathcal S_2=1$, 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 $\gamma$ and in a deviation from $P_0\in\mathcal S_1$, 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 $v_{k,\sigma}(0)=-\sigma\tau\gamma N_0\delta h_{1,k}/M_k$. The fall becomes $a_{k,\sigma}(0)=-\tau^2\gamma N'[\alpha+(\delta f_k+\sigma\delta g_k)/M_k]$: it is level-dependent through $\delta f_k/M_k$ and branch-dependent through $\delta g_k/M_k$. $\delta b$ and $\delta h_2$ do not enter at order $\gamma$.

Class F (F0, with the Euclidean topology of the coefficient space $P$), lowest state (F2). $\chi_0$ is exact at each finite $L$. The made field is its term of order $g$ at fixed $L$, valid for $\lvert g\rvert<\Delta_L/(2N_f\lVert\phi\rVert)$. Large-distance forms hold for $L\to\infty$ at fixed distances.
- **(8.8)** For every member of class F, $\chi_0(x,y)=\frac{2n}{\lvert X_f\rvert}\sum_q\cos(q\cdot d)S_L(q)$, with $S_L(q)=\langle w_L\vert\phi(h(k_L+q)-\epsilon_L)^{-1}\phi\vert w_L\rangle\ge0$ for $q\neq0$. The made field is $N(y)=n\bar\phi_L-g\chi_0(y,x_s)+O(g^2)$.
- **(8.14)** If the made field is short range, $S_{L,\rm e}(q)$ is bounded uniformly in $L$ and in the grid momenta $q\neq0$.
- **(8.9)** On G1, $S=\lvert F\rvert^2/(\epsilon_0(k_0+q)-\epsilon_0)+S_{\rm an}$, with $\lvert F(0)\rvert^2=\bar\phi^2$. For $k_0=0$, the field is short range whenever the even part of $S$ is analytic at $q=0$.
- **(8.10)** On $U$ (G1 and $\bar\phi\neq0$), the made field is long range for every $k_0$. For $L\to\infty$ along sides with $L\lvert k_L-k_0\rvert\to0$ (all sides if $k_0=0$), at fixed $y,y'$: $\chi_0(y,x_s)-\chi_0(y',x_s)=2n\bar\phi^2[\mathcal G_A(y-x_s)-\mathcal G_A(y'-x_s)]+R$. This is linear in $D=1$, logarithmic in $D=2$ and $1/\lvert d\rvert$ in $D=3$. For $r=1$, every member of class F with $\phi\neq0$ is long range.
- **(8.15)** $U\cap{}$F is open and dense in class F. Its complement (G1 fails, or $\bar\phi=0$) is closed in class F with empty interior. Long range is therefore typical, and short range exceptional. On G1, short range requires $\bar\phi=0$.
- **(8.11)** Inside the complement, for members of G1 with $k_0=0$ and $\bar\phi=0$ (not a classification of the complement): in $D=1$ the field is short range. In $D=2,3$, $\chi_0=-2n\sum C_{\mu\nu}\partial_\mu\partial_\nu\mathcal G_A+c_L+O(\lvert d\rvert^{-D-1})$ as $L\to\infty$, a power law $\lvert d\rvert^{-D}$ (long range) unless $C=cA$ (for $D=3$: $C=0$).
- **(8.12)** On G1, long range follows from the condition 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). This condition holds on the open dense set $U$. On G1, a sufficient condition for short range in every $D$ and for every $k_0$ is (Z): a conserved charge $\sum_jZ^{(j)}$ that $\Phi_x$ changes within the sector of the lowest state, with the other sectors gapped (then $\bar\phi=0$).
- **(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 of $U$ 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 $\alpha=1$, $b=0$, in $\mathcal S_1$. (8.4) gives $a=-\tau^2\gamma N'+O(\gamma^2)$, the first-order part of $-\tau^2\gamma N'/\nu(x_0)$, which is what (4.16) gives with (5.3). Coupling (c) is $\alpha=1$, $b=\mathbb 1_N$, in $\mathcal S_2$, with the same first-order fall. For (r), $W_k/M_k=m\sigma_3/M_k$ is level-dependent (not R3), in agreement with (5.5). (8.4) gives $a=-\tau^2\gamma mN'/M_k+O(\gamma^2)$, the first-order part of (5.5), and (8.3) reproduces (5.5) exactly for $C_{\rm eff}$. $\gamma N$ 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

- The members of class F in the complement of $U$ are not classified, as the question's dated note allows: degenerate, quartic or multiple minima (for $r\ge2$), $\bar\phi=0$ with $k_0\neq0$, and $\bar\phi=0$, $C=cA$ in $D=2,3$, where higher orders of (8.9) decide. (Z) is only sufficient.
- The made field is the finite-$L$ Taylor coefficient. Its convergence radius $\Delta_L/(2N_f\lVert\phi\rVert)$ shrinks as $L^{-2}$ on G1, and no statement uniform in $L$ beyond $\chi_0$ is derived.
- Item 1 is derived at first order in $\gamma$, at the band point $p=0$ and at first order in the gradients. At order $\gamma^2$, $b$ and $h_2$ enter the fall ((8.3) for $C_{\rm eff}$), but that order also needs the body's own change of the field, which is not treated. The band point $p=\pi$ is not treated. T3 (which profiles field states realize) is assumed for the field of 04-profile.
- "Exceptional" and "typical" refer to the Euclidean topology of the coefficient spaces of classes C and F. The language supplies no measure on descriptions.

## Code

- `code/v2/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 $\Delta=1.5$ ($k_0=0$). 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 $L=20,32$ (full diagonalization) and $L=64$ (through (8.8) only): the long-range member and the $\bar\phi=0$ member. It logs the maximal deviations over the distance windows quoted in Steps 10 and 11, against the lattice form of the leading term at the same $L$ and, for the $\bar\phi=0$ member, against its $L\to\infty$ value.
- Outputs:
  - `code/v2/output/check_log.txt`: all compared numbers;
  - `code/v2/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/v2/output/chi_torus.csv`: $\chi_0/(2n)$ along both axes of the tori, with the lattice form of the prediction;
  - `code/v2/output/decay.png`: $\Omega(d)/(2n)$ on the rings, and $d^2(\chi_0(d)-\chi_0(\text{corner}))/(2n)$ on the torus $L=64$.

## 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 (convergence radius); Paley–Wiener (analyticity versus exponential decay)
- Fourier sum rule bounding the response by the decay; genericity in a coefficient space (implicit function theorem, Baire category theorem); Dirichlet simultaneous approximation
- 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