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

# Speed of the made field: front, formation of the static mark and long range

- **Subproject:** 07-ilang-weight
- **Package:** 08-field-speed
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-21

## Changes from previous version

- Setup, Step 1: the source is a fixed term of the contract (A5), started in the lowest state without it. The gradual switching, withdrawn by the question's note of 2026-10-21, is removed: v1 Step 6, (8.12), (8.13) and the gradual parts of the Result, twin and code (I1, I2).
- Setup (Regime), Step 2, new Step 8: the order of limits and the sense of every result are stated. The new exact bound (8.18) on the finite-coupling remainder makes (8.5) hold at every order and gives sufficient conditions on $g_s$ for (8.7), (8.10), (8.11) and (8.14). The expansion parameter of (8.2) is stated (I5).
- Step 3: (8.6) extended to the axes in every $D$, with a lower bound on the front maximum; the arrival claim of (8.7) restricted to the axes, with only the lower bound in other directions (I6).
- Step 5: the averaging lengths from (8.11) labelled sufficient upper bounds; the saturation claim removed (I7).
- Step 6 (v1 Step 7): (8.14) proved from a fixed-$\zeta$ time at which the criterion fails, now for $\varepsilon<1$ (I4). The harmonic control is replaced by a pinned harmonic lattice with a lowest state, a uniform mode without drift and a stated truncation, (8.15) (I3). The lowest state of the control (8.16) is stated.
- Step 7 (v1 Step 8): $\tau<2t_f$ stated as sufficient; full-band maximum and the exact condition per level, (8.17) (I8).
- Code: gradual runs removed; checks of (8.18), of (8.5) at every order and of (8.6) on the cube added; rerun into `code/v2/output/`.

## Response to verification

- I1: Accepted. The question's note of 2026-10-21 withdraws the gradual switching (A5 excludes $\lambda$-dependent contracts). Its steps, results, twin entries and code are removed. The sudden switching is a fixed contract with the source, started in the lowest state without it (Setup, Step 1). One line in Open issues.
- I2: Accepted; settled by the same note: (8.13) is removed.
- I3: Accepted and fixed in Step 6. The control is now a pinned harmonic lattice ($m_h>0$), truncated by compression to $M$ places per object. For every $g_s$ it has a unique lowest state with the source and the static mark $-g_sK^{-1}e_{x_s}$, and its uniform mode oscillates with frequency $m_h$ without drift. The truncation errors vanish as $M\to\infty$ on every finite window (min-max, gap $m_h$, Duhamel). Formation at the front is shown with an explicit bound, (8.15).
- I4: Accepted and fixed in Step 6: (8.14) now follows from a fixed $\zeta^*$ with $|1-g_D(\zeta^*)|>\varepsilon$, where (8.9) applies; no ballistic-time asymptotics are used.
- I5: Accepted and fixed in Setup (Regime), Step 2 and Step 8. Every result is labelled exact or first order ($g_s\to0$ first), with the order of the further limits. The exact bound (8.18) gives sufficient conditions on $g_s$, uniform in $L$, for (8.7), (8.10), (8.11) and (8.14). For (8.2) the expansion parameter $g_s\Lambda^+(0)/t_f$, which grows with $L$ for $D=1,2$, is stated.
- I6: Accepted and fixed in Step 3: the leading front response along the axes for every $D$, with a lower bound on its maximum in the front window, (8.6). The arrival claim (8.7) is restricted to the axes; in other directions only the lower bound (no early arrival) is claimed.
- I7: Accepted and fixed in Step 5: the lengths from (8.11) are labelled sufficient upper bounds, and the saturation claim is removed.
- I8: Accepted and fixed in Step 7. $\tau<2t_f$ is stated as sufficient. The full-band maximum $\sqrt{m_a^2+\tau^2}-m_a$ and the exact per-level condition $\tau^2<4t_f(t_f+m_a)$ are given, (8.17).

## Setup and assumptions

- **Class** (question; M8). $N_f\ge1$ identical field objects ($c=+1$) on the torus $X_f=\mathbb Z_L^D$, $D\in\{1,2,3\}$, $L$ even, $|X_f|=L^D$; no pair terms among them; own term $C_1=-t_fA$, $A$ the nearest-neighbour adjacency, $t_f>0$; $\Phi_x$ the number of field objects at $x$, $n=N_f/|X_f|$. $\Lambda=2D\mathbb 1-A$ has the eigenvalues $\omega(\mathbf k)=\sum_\mu4\sin^2(k_\mu/2)$, $\mathbf k\in(2\pi/L)\mathbb Z_L^D$.
- **Source and start** (assumption; M8, M9, A11). The contract contains the source term $g_s\Phi_{x_s}$, $g_s$ real, at every $\lambda$ (A5). At $\lambda=0$ the field is in its lowest state without the source (the lowest eigenvector of the contract at $g_s=0$); $\lambda\ge0$ (A9), with the evolution (1.9). This is the sudden switching of the question. The gradual switching is withdrawn by the question's note.
- **Made field.** $\delta\rho(x,\lambda)=\langle\Phi_x\rangle_\lambda-n$; the static mark $\delta\rho_{\rm st}$ is the same quantity in the lowest state of the contract with the source. Both are fixed by the state and by the operators $\Phi_x$, which are symmetric in the identical objects (M6).
- **Distances.** $r=x-x_s$ with $|r_\mu|\le L/2$; $\rho=\lVert r\rVert_\infty$, $|r|$ the Euclidean length, $d=\sum_\mu|r_\mu|$ the graph distance. Along a lattice axis $d=\rho=|r|$.
- **Regime and order of limits** (A13). A result marked *first order* is a statement about the first-order coefficient $\delta\rho^{(1)}=g_s\,\partial_{g_s}\delta\rho|_{g_s=0}$. The limit $g_s\to0$ is taken first, at fixed $L$, $r$, $\lambda$. The further limits are then taken in the order stated at each result: $L\to\infty$ (or $\lambda$ below the wrap time), then $|r|\to\infty$ along a stated path (fixed $2t_f\lambda/\rho$, or fixed $\zeta$), then $\varepsilon\to0$ where stated. Exact are the first part of (8.1), (8.5) and (8.18). Step 8 gives sufficient conditions on $g_s$ under which the first-order results hold for the exact response.
- **Inputs.** (4.4) of 06-ilang-source/04-profile:
$$M^{(2)}(x_1,x_2)=-\frac{2n\,g_1g_2}{t_f}\,\Lambda^+(x_1,x_2),\qquad \Lambda^+(x,y)=\frac1{\lvert X_f\rvert}\sum_{\mathbf k\neq0}\frac{e^{i\mathbf k\cdot(x-y)}}{\omega(\mathbf k)},\qquad \sum_{x_2\in X_f}M^{(2)}(x_1,x_2)=0;$$
(4.4) of 07-ilang-weight/04-weight-of-bond, with $\ell(\mathbf r)=\Lambda^+(x,x+\mathbf r)$: $\ell(0)-\ell(\mathbf r)=\frac{\ln|\mathbf r|}{2\pi}+\frac{2\gamma_E+3\ln2}{4\pi}+O(|\mathbf r|^{-2})-\frac{|\mathbf r|^2}{4L^2}$ for $D=2$, and $=G_0-\frac1{4\pi|\mathbf r|}+O(|\mathbf r|^{-3})-\frac{|\mathbf r|^2}{6L^3}$, $G_0=0.2527310$, for $D=3$.

## Derivation

### Step 1. Exact reduction and first-order response

Without pair terms and with $c=+1$, $C=\sum_{i=1}^{N_f}h_i$ on the symmetric subspace, $h=h_0+g_s|x_s\rangle\langle x_s|$, $h_0=-t_fA$. The lowest state of $h_0$ is $\psi_0=|X_f|^{-1/2}\sum_x|x\rangle$, nondegenerate (Perron–Frobenius, connected graph), with eigenvalue $-2Dt_f$; the start is $\psi_0^{\otimes N_f}$. The lowest state of $h$ is nondegenerate for every $g_s$ by the same argument, so the static mark is well defined. The evolution of a sum of one-object terms keeps a product of identical states a product, $\Psi(\lambda)=\psi(\lambda)^{\otimes N_f}$ with $\psi(\lambda)=e^{-ih\lambda}\psi_0$. Hence $\langle\Phi_x\rangle_\lambda=N_f|\psi(x,\lambda)|^2$ exactly: the made field of $N_f$ objects is $N_f$ times that of one, at every order in $g_s$. First-order Dyson expansion around $e^{2iDt_f\lambda}\psi_0$, with $\langle x_s|\psi_0\rangle=|X_f|^{-1/2}$:

$$\delta\rho(x,\lambda)=2ng_s\int_0^\lambda\operatorname{Im}\mathcal U(x-x_s,s)\,\mathrm ds+O(g_s^2),\qquad \mathcal U(r,s)=\langle x_s+r|e^{-ist_f\Lambda}|x_s\rangle=\frac1{|X_f|}\sum_{\mathbf k}e^{i\mathbf k\cdot r-it_f\omega(\mathbf k)s}.\tag{8.1}$$

The lowest state of $h$ is, at first order, $\psi_0-\sum_{\mathbf k\ne0}|\mathbf k\rangle\langle\mathbf k|g_s|x_s\rangle\langle x_s|\psi_0\rangle/(t_f\omega(\mathbf k))$, so

$$\delta\rho_{\rm st}(r)=-\frac{2ng_s}{t_f}\Lambda^+(r).\tag{8.2}$$

With $\operatorname{Im}\mathcal U=-|X_f|^{-1}\sum_{\mathbf k}\cos(\mathbf k\cdot r)\sin(t_f\omega s)$ and $\int_0^\lambda\sin(t_f\omega s)\,\mathrm ds=(1-\cos t_f\omega\lambda)/(t_f\omega)$, for $\lambda\ge0$:

$$\delta\rho(r,\lambda)=\delta\rho_{\rm st}(r)+\frac{2ng_s}{t_f}R(r,\lambda),\qquad R(r,\lambda)=\frac1{|X_f|}\sum_{\mathbf k\ne0}\frac{\cos(\mathbf k\cdot r)\cos(t_f\omega(\mathbf k)\lambda)}{\omega(\mathbf k)}.\tag{8.3}$$

$R(r,0)=\Lambda^+(r)$, so $\delta\rho(r,0)=0$; $R$ is a finite sum of cosines and does not decay.

### Step 2. Free propagator, Kapteyn bound and finite coupling

By the Jacobi–Anger expansion $e^{iz\cos k}=\sum_mi^mJ_m(z)e^{imk}$ and Poisson summation over the torus,

$$\mathcal U(r,s)=\sum_{\mathbf m\in\mathbb Z^D}\mathcal U_\infty(r+L\mathbf m,s),\qquad \mathcal U_\infty(r,s)=e^{-2iDt_fs}\prod_{\mu=1}^Di^{r_\mu}J_{r_\mu}(2t_fs).\tag{8.4}$$

Kapteyn's inequality gives, for integer $m\ge0$ and $0\le u\le m$, $|J_m(u)|\le e^{-m\eta(u/m)}$ with $\eta(z)=\operatorname{artanh}\sqrt{1-z^2}-\sqrt{1-z^2}$, which decreases from $\infty$ to $0$ on $(0,1]$ and equals $\tfrac13(2(1-z))^{3/2}(1+O(1-z))$ near $z=1$. With $|J_m|\le1$ for the other factors, $|\mathcal U_\infty(r,s)|\le e^{-\rho\,\eta(2t_fs/\rho)}$ for $2t_fs\le\rho$; $R\eta(c/R)$ increases with $R$ (its derivative is $\eta+\sqrt{1-z^2}>0$).

**Finite coupling.** Expand $\psi(\lambda)=e^{-ih\lambda}\psi_0$ in the Dyson series in $g_s|x_s\rangle\langle x_s|$ around $h_0$. Since $e^{-ih_0s}\psi_0=e^{2iDt_fs}\psi_0$, the $k$-th term at $x=x_s+r$ is $(-ig_s)^k|X_f|^{-1/2}$ times an integral over $0<s_1<\dots<s_k<\lambda$ of a phase, of $k-1$ factors $\langle x_s|e^{-ih_0u}|x_s\rangle$ of modulus $\le1$, and of $\langle x|e^{-ih_0(\lambda-s_k)}|x_s\rangle$, of modulus $|\mathcal U(r,\lambda-s_k)|$. With $B(r,\lambda)=\max_{0\le s\le\lambda}|\mathcal U(r,s)|\le1$ and the simplex volume $\lambda^k/k!$, $a=|X_f|^{1/2}e^{-2iDt_f\lambda}\psi(x,\lambda)$ obeys $|a-1|\le(e^{|g_s|\lambda}-1)B$ and $|a-1-a^{(1)}|\le(e^{|g_s|\lambda}-1-|g_s|\lambda)B$, $a^{(1)}$ the first-order term. Since $\delta\rho=n(2\operatorname{Re}(a-1)+|a-1|^2)$, $\delta\rho^{(1)}=2n\operatorname{Re}a^{(1)}$ and $B^2\le B$:

$$|\delta\rho(r,\lambda)|\le n\bigl(e^{2|g_s|\lambda}-1\bigr)B(r,\lambda),\qquad |\delta\rho-\delta\rho^{(1)}|\le n\bigl(e^{2|g_s|\lambda}-1-2|g_s|\lambda\bigr)B(r,\lambda)\le2n(g_s\lambda)^2e^{2|g_s|\lambda}B(r,\lambda),\tag{8.18}$$

exactly, for every $g_s$, $L$, $r$ and $\lambda\ge0$, with $\delta\rho^{(1)}$ the first-order term of (8.1). Code (ring $L=256$, $g_s/t_f=\pm0.02$; cube $L=16$, $g_s/t_f=0.05$; $|g_s|\lambda\le1$): wherever the bounds exceed the round-off ($10^{-8}$ in units $2n|g_s|/t_f$), the exact remainder reaches at most $0.25$ of the second bound and $|\delta\rho|$ at most $0.98$ of the first; elsewhere neither exceeds its bound by more than $10^{-15}$.

### Step 3. The front (item 1)

**Ahead of the front.** Let $\rho\le L/2$ and $2t_f\lambda<\rho$. Every image ($\mathbf m\ne0$) has $\rho_{\mathbf m}=\lVert r+L\mathbf m\rVert_\infty\ge L-\rho\ge\rho$, and $e^{-\rho_{\mathbf m}\eta(2t_fs/\rho_{\mathbf m})}$ increases with $s$. So $B(r,\lambda)\le\sum_{\mathbf m}e^{-\rho_{\mathbf m}\eta(2t_f\lambda/\rho_{\mathbf m})}$, and (8.18) gives, at every order in $g_s$,

$$|\delta\rho(r,\lambda)|\le n\bigl(e^{2|g_s|\lambda}-1\bigr)\sum_{\mathbf m}e^{-\rho_{\mathbf m}\eta(2t_f\lambda/\rho_{\mathbf m})}\qquad(2t_f\lambda<\rho\le L/2).\tag{8.5}$$

At first order the prefactor is $2n|g_s|\lambda$. At fixed $z=2t_f\lambda/\rho<1$ the bound is exponentially small in $\rho$ if $|g_s|/t_f<\eta(z)$; the $\mathbf m=0$ term dominates. For every start of the $N_f$ objects, at first order, $\Delta\langle\Phi_x\rangle=-ig_s\int_0^\lambda\langle[\Phi^H_x(\lambda),\Phi^H_{x_s}(\lambda')]\rangle\mathrm d\lambda'$ is a sum of one-object commutators of norm $\le2|\mathcal U(r,\lambda-\lambda')|$, so the first-order (8.5) holds with $2N_f|g_s|\lambda$ as prefactor.

**At the front, along an axis** (first order). Let $r=\rho\mathbf e_\mu$ and $\rho\le(1-\delta)L/2$ for a fixed $\delta\in(0,1)$, or $L\to\infty$ first. Then in the window below the images have $\rho_{\mathbf m}\ge\rho(1+\delta)/(1-\delta)$ and are exponentially small by Step 2. On the axis $\mathcal U_\infty=e^{-iDu}i^\rho J_\rho(u)J_0(u)^{D-1}$, $u=2t_fs$, and (8.1) is $\frac{ng_s}{t_f}\int_0^{2t_f\lambda}\operatorname{Im}[\dots]\mathrm du$. The part $u<\rho/2$ is exponentially small (Kapteyn). For larger $u$, $J_0(u)^{D-1}=(\frac2{\pi u})^{\frac{D-1}2}[\cos^{D-1}(u-\frac\pi4)+O(u^{-1})]$, and $e^{-iDu}\cos^{D-1}(u-\frac\pi4)=\sum_{j=0}^{D-1}c_je^{-i(2j+1)u}$ with $c_j=2^{1-D}\binom{D-1}je^{-i(D-1-2j)\pi/4}$. Each term oscillates with frequency $2j+1\ge1$, while the local frequency of $J_\rho$ is $\sqrt{1-\rho^2/u^2}<1$, so the integrand has no stationary point. One integration by parts leaves the boundary term at $u=2t_f\lambda$ and integrals of $J_\rho'$ against the same exponentials, which are $O(\rho^{-2/3})$ relative to it ($J_\rho'=O(\rho^{-2/3})$ near $u=\rho$). With the Airy form $J_\rho(\rho+\xi(\rho/2)^{1/3})=(2/\rho)^{1/3}[\mathrm{Ai}(-\xi)+O(\rho^{-2/3})]$ and $p_D=\frac13+\frac{D-1}2$:

$$\delta\rho(\rho\mathbf e_\mu,\lambda)=\frac{ng_s}{t_f}\Bigl[\Bigl(\frac2\rho\Bigr)^{1/3}\Bigl(\frac2{\pi\rho}\Bigr)^{\frac{D-1}2}\mathrm{Ai}\Bigl(\frac{\rho-2t_f\lambda}{(\rho/2)^{1/3}}\Bigr)P_D(2t_f\lambda)+O(\rho^{-p_D-1/3})\Bigr],\qquad P_D(u)=\operatorname{Im}\Bigl[i^{\rho+1}\sum_{j=0}^{D-1}\frac{c_j\,e^{-i(2j+1)u}}{2j+1}\Bigr],\tag{8.6}$$

for $|2t_f\lambda-\rho|=O(\rho^{1/3})$. For $D=1$, $P_1(u)=\cos(\frac{\rho\pi}2-u)$, and the envelope reaches $0.6749\,(n|g_s|/t_f)\,\rho^{-1/3}$ at $2t_f\lambda=\rho+1.019(\rho/2)^{1/3}$. For every $D$, $P_D$ is a real trigonometric polynomial with frequencies $1,3,\dots,2D-1$ whose frequency-1 part has amplitude $|c_0|=2^{1-D}$. By Parseval its mean square over a period is $\ge2^{1-2D}$, so its maximum modulus on any interval of length $2\pi$ is $\ge2^{1/2-D}$. Ai varies on the scale $(\rho/2)^{1/3}\gg2\pi$ and has the maximum $0.5357$ at $-1.019$. Hence the maximum of $|\delta\rho|$ on $|2t_f\lambda-\rho-1.019(\rho/2)^{1/3}|\le\pi$ is at least $b_D(n|g_s|/t_f)\rho^{-p_D}(1-o(1))$, with $b_D=0.6749\cdot2^{1/2-D}(2/\pi)^{(D-1)/2}$ ($b_1=0.477$, $b_2=0.190$, $b_3=0.0760$). Off the axes the stationary points $\sin k_\mu=r_\mu/(2t_f\lambda)$ give contributions of order $(n|g_s|/t_f)|\mathcal U_\infty|$. That their sum does not cancel in the front window is not shown.

**Criterion (front).** Fix $\varepsilon>0$ and $p>p_D$. The change arrives at $r$ at $\lambda_a(r)=\min\{\lambda\ge0:|\delta\rho(r,\lambda)|\ge\varepsilon\,(n|g_s|/t_f)\,\rho^{-p}\}$. By the first-order (8.5), with $2n|g_s|\lambda<n|g_s|\rho/t_f$, the threshold is not reached while $\rho\,\eta(2t_f\lambda/\rho)>(p+1)\ln\rho+\ln(1/\varepsilon)+o(1)$, in every direction. That is, it is not reached while $1-2t_f\lambda/\rho>\frac12[3((p+1)\ln\rho+\ln(1/\varepsilon))/\rho]^{2/3}(1+o(1))$. Along an axis, by (8.6), the change exceeds $\frac12b_D\rho^{-p_D}>\varepsilon\rho^{-p}$ within $O(\rho^{1/3})$ behind $\rho=2t_f\lambda$, for large $\rho$. Hence, at first order and as $\rho\to\infty$,

$$\lambda_a(r)\ge\frac{\rho}{2t_f}-O\Bigl(\frac{\rho^{1/3}(\ln\rho)^{2/3}}{t_f}\Bigr)\ (\rho\le L/2),\qquad \lambda_a(\rho\mathbf e_\mu)=\frac{\rho}{2t_f}+O\Bigl(\frac{\rho^{1/3}(\ln\rho)^{2/3}}{t_f}\Bigr)\ (\rho\le(1-\delta)L/2):\quad v_f=2t_f .\tag{8.7}$$

$v_f$ is in place labels per unit $\lambda$; along an axis $d=\rho$. In other directions only the lower bound is derived: no change arrives before $\rho/(2t_f)(1-o(1))$, that is before $d/(2Dt_f)(1-o(1))$ in graph distance. Code (ring $L=16384$ with $\varepsilon=0.1$, $p=\frac12$; cube $L=160$ along an axis with $p=\frac32$):
- Fitted speeds: $2.036\,t_f$ (ring, $\rho=64$–$384$) and $2.06\,t_f$ (cube, $\rho=12$–$64$).
- On the ring, $2t_f\lambda_a/\rho$ rises from $0.87$ at $\rho=16$ to $0.971$ at $\rho=384$, as the correction in (8.7) allows.
- Ahead of the front, $|\delta\rho|$ stays below $0.02$ of the bound (8.5).
- The front peak decays as $\rho^{-0.340}$ on the ring (prediction $-\frac13$) and as $\rho^{-1.322}$ on the cube (prediction $-\frac43$). In units $n|g_s|/t_f$, peak $\times\rho^{p_D}$ is $0.68$–$0.72$ on the ring ($\rho\ge64$) and $0.157$–$0.188$ on the cube, above $b_1=0.477$ and $b_3=0.0760$.
- (8.6) deviates from the exact change by $0.44,0.33,0.27,0.24$ of the peak at $\rho=64,128,256,384$ on the ring, decreasing like $\rho^{-1/3}$, and by $0.40$–$0.49$ at $\rho=12$–$20$ and $0.24$–$0.32$ at $\rho=48$–$64$ on the cube.

### Step 4. Behind the front: scaling form

Take the first-order coefficient and $L\to\infty$ first, so that $\mathcal U\to\mathcal U_\infty$ and the $\mathbf k$-sums become integrals; then (8.3) reads $\delta\rho=-\frac{2ng_s}{t_f}\int\frac{\mathrm d^D\mathbf k}{(2\pi)^D}\cos(\mathbf k\cdot r)\frac{1-\cos t_f\omega\lambda}{\omega}$, convergent at $\mathbf k=0$ for every $D$. Let $|r|\to\infty$ at fixed $\zeta=|r|^2/(4t_f\lambda)$ (behind the front: $\zeta<|r|/2$, true for large $|r|$). At leading order only $|\mathbf k|=O(1/|r|)$ contributes, where $\omega=\mathbf k^2(1+O(\mathbf k^2))$: elsewhere the integrand oscillates on the scale $1/|r|$, and the other critical points of $\omega$ ($k_\mu\in\{0,\pi\}$) contribute $O((t_f\lambda)^{-D/2})=O(|r|^{-D})$, smaller by $|r|^{-2}$. With $(1-e^{-ia\mathbf k^2})/\mathbf k^2=i\int_0^ae^{-ia'\mathbf k^2}\mathrm da'$ and the Fresnel integral $\int\frac{\mathrm d^D\mathbf k}{(2\pi)^D}e^{i\mathbf k\cdot r-ia'\mathbf k^2}=(4\pi ia')^{-D/2}e^{i|r|^2/4a'}$:

$$\delta\rho(r,\lambda)\simeq-\frac{2ng_s}{t_f}\operatorname{Re}W_D(|r|,it_f\lambda),\qquad W_D(r,\Theta)=\int_0^\Theta\frac{e^{-r^2/4s}}{(4\pi s)^{D/2}}\mathrm ds,\tag{8.8}$$

continued along $s=ia'$, $0\le a'\le t_f\lambda$. Substituting $u=r^2/4s$, $\partial_rW_D(r,\Theta)=-\Gamma(\frac D2,\frac{r^2}{4\Theta})/(2\pi^{D/2}r^{D-1})$, and $r^2/(4it_f\lambda)=-i\zeta$. As $\Theta\to\infty$ the gradient of (8.8) tends to that of the long-range part $\delta\rho^\infty_{\rm st}$ of the static mark, $\partial_{|r|}\delta\rho^\infty_{\rm st}=\frac{2ng_s}{t_f}\frac{s_D}{|r|^{D-1}}$, $s_D=\frac{\Gamma(D/2)}{2\pi^{D/2}}$ ($s_1=\frac12$, $s_2=\frac1{2\pi}$, $s_3=\frac1{4\pi}$). Hence

$$\frac{\partial_{|r|}\delta\rho(r,\lambda)}{\partial_{|r|}\delta\rho^\infty_{\rm st}(r)}\simeq g_D(\zeta)=\frac{\operatorname{Re}\Gamma(D/2,-i\zeta)}{\Gamma(D/2)}=\begin{cases}\operatorname{Re}\operatorname{erfc}\sqrt{-i\zeta},&D=1,\\ \cos\zeta,&D=2,\\ \operatorname{Re}\bigl[\operatorname{erfc}\sqrt{-i\zeta}+2\sqrt{-i\zeta/\pi}\;e^{i\zeta}\bigr],&D=3,\end{cases}\tag{8.9}$$

with the principal root. From $\gamma(s,w)=w^s/s-w^{s+1}/(s+1)+\dots$: $1-g_1=\sqrt{2\zeta/\pi}\,(1+O(\zeta))$, $1-g_2=\frac{\zeta^2}2(1+O(\zeta^2))$, $1-g_3=-\frac{4\zeta^{3/2}}{3\sqrt{2\pi}}(1+O(\zeta))$. For $\zeta\gg1$, $g_1\to0$, $g_2$ oscillates between $\pm1$, and $g_3=2\sqrt{\zeta/\pi}\cos(\zeta-\pi/4)+o(1)$: there the made field is a ripple of local wavenumber $|r|/(2t_f\lambda)$, not the static mark. In $D=3$ the value itself obeys $\delta\rho/\delta\rho^\infty_{\rm st}\simeq\operatorname{Re}\operatorname{erfc}\sqrt{-i\zeta}=1-\sqrt{2\zeta/\pi}(1+O(\zeta))$. Code (window before the wrap time): for $\zeta\le0.5$ the exact gradient ratio follows (8.9) within $0.006$ on the ring ($|r|=6$–$40$, $L=16384$), and within $0.04$ to $0.008$ on the cube along an axis ($|r|=3$–$9$, $L=160$), decreasing with $|r|$.

### Step 5. Formation of the static mark (item 2)

**Criterion (formation).** The long-range part at distance $|r|$ is formed to accuracy $\varepsilon$ from $\lambda_f$ on if $|1-\partial_{|r|}\delta\rho/\partial_{|r|}\delta\rho_{\rm st}|\le\varepsilon$ for every $\lambda\ge\lambda_f$ in the regime of Step 4 ($\delta\rho_{\rm st}$ replaced by its long-range part $\delta\rho^\infty_{\rm st}$). By (8.9) this holds iff $\zeta\le\zeta_D(\varepsilon)$, the largest $\zeta$ below which $|1-g_D|\le\varepsilon$:

$$\lambda_{\rm form}(|r|;\varepsilon)=\frac{|r|^2}{4t_f\,\zeta_D(\varepsilon)},\qquad \zeta_1=\frac{\pi\varepsilon^2}2\bigl(1+O(\varepsilon^2)\bigr),\quad \zeta_2=\sqrt{2\varepsilon}\bigl(1+O(\varepsilon)\bigr),\quad \zeta_3=\Bigl(\frac{3\sqrt{2\pi}\,\varepsilon}4\Bigr)^{2/3}\bigl(1+O(\varepsilon^{2/3})\bigr),\tag{8.10}$$

independent of $L$. In $D=3$ the value at $|r|$ comes within $\varepsilon$ of $\delta\rho^\infty_{\rm st}$ at $\lambda=|r|^2/(2\pi\varepsilon^2t_f)$. In $D=1,2$ the level does not form on the infinite lattice: at fixed $|r|$, $\operatorname{Re}W_1=\sqrt{t_f\lambda/2\pi}-|r|/2+o(1)$ and $\operatorname{Re}W_2=\frac1{4\pi}\bigl(\ln\frac{4t_f\lambda}{|r|^2}-\gamma_E\bigr)+o(1)$ grow without bound. Code: on the ring ($L=16384$, $\varepsilon=0.2$, $\zeta_1=0.0604$ from (8.9), leading order $0.0628$), $\lambda_{\rm form}$ at $|r|=6$–$40$ lies within $3.4\%$ of (8.10), with fitted exponent $1.98$. On the cube along an axis ($L=160$, $\varepsilon=0.1$, $\zeta_3=0.407$, leading order $0.328$), it lies within $5\%$, with exponent $2.03$ and prefactor $0.586$ against $1/(4\zeta_3)=0.614$.

**The torus.** By (8.4), $\delta\rho_L(r,\lambda)=\sum_{\mathbf m}\delta\rho_\infty(r+L\mathbf m,\lambda)$ exactly at first order; by (8.5) the image terms are exponentially small for $\lambda<(L-\rho)/(2t_f)$, the wrap time. So (8.8)–(8.10) hold on the torus for $\lambda$ below the wrap time, and the formation (8.10) happens before it when $|r|^2<2\zeta_D(\varepsilon)(L-\rho)$. After it the wrapped changes add to (8.3) a part that does not decay. Only the $\lambda$-average converges, exactly: $\omega(\mathbf k)>0$ for $\mathbf k\ne0$ and $\frac1\Lambda\int_0^\Lambda\cos(t_f\omega\lambda)\mathrm d\lambda=\frac{\sin t_f\omega\Lambda}{t_f\omega\Lambda}$, so, at first order,

$$\Bigl|\frac1\Lambda\int_0^\Lambda\delta\rho(r,\lambda)\,\mathrm d\lambda-\delta\rho_{\rm st}(r)\Bigr|\le\frac{2n|g_s|}{t_f}\,\frac{S_2}{t_f\Lambda}\xrightarrow{\Lambda\to\infty}0,\qquad S_2=\frac1{|X_f|}\sum_{\mathbf k\ne0}\frac1{\omega(\mathbf k)^2}=O(L^{4-D}),\tag{8.11}$$

since $S_2$ is dominated by $|\mathbf k|=O(1/L)$, where $\omega^{-2}=O(L^4)$; $S_2=\frac{L^3}{720}(1+O(L^{-2}))$ for $D=1$. Hence the averaged field is within $\varepsilon|\delta\rho_{\rm st}(r)|$ for every $\Lambda\ge S_2/(\varepsilon t_f|\Lambda^+(r)|)$: of order $L^2/(60\varepsilon t_f)$ ($D=1$, $|r|\ll L$, $\Lambda^+\simeq L/12$), $L^2/(\varepsilon t_f\ln(L/|r|))$ ($D=2$) and $L|r|/(\varepsilon t_f)$ ($D=3$). These are sufficient upper bounds on the averaging length, not formation scales: no matching lower bound is derived. In $D=3$, for instance, the field at $|r|$ forms already on the scale (8.10) when $|r|^2<2\zeta_3(\varepsilon)(L-\rho)$. Code (ring $L=1024$, units $2ng_s/t_f$): at $\Lambda=1,4,16\,L^2/t_f$ the running average of the exact evolution deviates from its $\Lambda\to\infty$ limit by $1.34,0.25,0.025$, against the bound $1.42,0.36,0.089$.

### Step 6. Long range and finite speed (item 3)

Take the first-order coefficient with $L\to\infty$ first, and fix $\varepsilon<1$. There is a fixed $\zeta^*>\zeta_D(\varepsilon)$ with $|1-g_D(\zeta^*)|>\varepsilon$: $g_1\to0$ as $\zeta\to\infty$, $g_2(\pi)=-1$, and $g_3$ is unbounded (Step 4). Let $\lambda^*(|r|)=|r|^2/(4t_f\zeta^*)$. At fixed $\zeta^*$, (8.9) gives that the gradient ratio at $(|r|,\lambda^*)$ tends to $g_D(\zeta^*)$, so for all large $|r|$ the criterion of Step 5 fails at $\lambda^*$. The criterion requires the bound at every $\lambda\ge\lambda_{\rm form}$, so $\lambda_{\rm form}>\lambda^*$. Only the fixed-$\zeta$ regime of (8.9) is used. On the torus the same holds when $\lambda^*$ is below the wrap time, $|r|^2<2\zeta^*(L-\rho)$. Hence:

$$\text{in the class, at first order, for every fixed }\beta>0\text{ and }\varepsilon<1:\quad \lambda_{\rm form}(|r|;\varepsilon)>\frac{|r|^2}{4t_f\zeta^*}>\frac{\beta|r|}{t_f}\ \text{ for all large }|r|.\tag{8.14}$$

The long-range part forms only on $\lambda\propto|r|^2/t_f$, although the front moves with the finite speed $2t_f$.

**Control for items 2 and 3** (wider class: field objects with more places and pair terms among them; $D=3$). One field object of its own type $T_x$ per place $x$ (distinct types, so (1.11) imposes no condition). Each object has the places $0,\dots,M-1$, read as the $M$ lowest levels of an oscillator of a fixed frequency, with operators $q_x,p_x$. Let

$$H=\sum_x\tfrac12p_x^2+\frac{\kappa_h}2\sum_{\langle xy\rangle}(q_x-q_y)^2+\frac{m_h^2}2\sum_xq_x^2+g_s\,q_{x_s},\qquad\kappa_h,m_h>0,$$

on the untruncated oscillators, and let the contract be its compression $C_M=\Pi H\Pi$, $\Pi$ the projector onto the levels $<M$ of every object. Each term of $H$ acts on one or two objects, so $C_M$ is a sum of pair terms (A5) on a finite space (A1). The start is the lowest state of $C_M$ at $g_s=0$, and the made field is $\langle q_x\rangle_\lambda$.

*Untruncated.* $H=\frac12p\cdot p+\frac12q\cdot Kq+g_sq_{x_s}$ with $K=\kappa_h\Lambda+m_h^2\ge m_h^2$. Completing the square, $H$ has for every $g_s$ a unique Gaussian lowest state, with the static mark $\langle q\rangle_{\rm st}=-g_sK^{-1}e_{x_s}$ exactly. Its gap is $m_h$, the lowest normal-mode frequency ($\Omega_{\mathbf k}=(\kappa_h\omega(\mathbf k)+m_h^2)^{1/2}$). The Heisenberg equations are linear, $\mathrm d^2\langle q\rangle/\mathrm d\lambda^2=-K\langle q\rangle-g_se_{x_s}$, with $\langle q\rangle=\langle p\rangle=0$ at $\lambda=0$. So $\langle q\rangle(\lambda)=-g_sK^{-1}(1-\cos K^{1/2}\lambda)\,e_{x_s}$, exactly for every $g_s$. The uniform mode has frequency $m_h$: it oscillates with amplitude $\le2|g_s|/(|X_f|m_h^2)$ and does not drift.

*Truncation* ($M\to\infty$ first, at fixed $L$, $g_s$ and a window $\lambda\le\Lambda_1$). The lowest states $\Psi_0$ of $H$ (with and without source) and the states $e^{-iH\lambda}\Phi_0$, $\lambda\le\Lambda_1$, are Gaussian with all moments finite, uniformly on the window. Here $\Phi_0$ is the lowest state at $g_s=0$. Their weight on the levels $\ge M-2$ of any object, with $H$ applied to it, tends to 0 as $M\to\infty$; on vectors without such weight $\Pi H\Pi$ acts as $H$ (each term changes a level by at most 2). By min-max $E_0(C_M)\ge E_0(H)$. The trial vector $\Pi\Psi_0$ gives $E_0(C_M)\le E_0(H)+\delta_M$ with $\delta_M\to0$. The gap then gives $1-|\langle\Psi_0|\Psi_M\rangle|^2\le\delta_M/m_h$ for the lowest eigenvector $\Psi_M$ of $C_M$. By Duhamel, $\lVert e^{-iC_M\lambda}\Pi\Phi_0-\Pi e^{-iH\lambda}\Phi_0\rVert\le\int_0^\lambda\lVert\Pi H(1-\Pi)e^{-iHs}\Phi_0\rVert\mathrm ds\to0$. So the static mark and the made field of the truncated description tend to those of $H$ on the window.

*Formation* (then $L\to\infty$, then $|r|\gg1$ with $m_Y=m_h/\sqrt{\kappa_h}\ll1$, leading order). With $\Omega_{\mathbf k}\simeq\sqrt{\kappa_h}(\mathbf k^2+m_Y^2)^{1/2}$, $\langle q\rangle-\langle q\rangle_{\rm st}=(g_s/\kappa_h)G(|r|,\sqrt{\kappa_h}\lambda)$ with $G(r,s)=\int\frac{\mathrm d^3\mathbf k}{(2\pi)^3}e^{i\mathbf k\cdot r}\frac{\cos(s\sqrt{\mathbf k^2+m_Y^2})}{\mathbf k^2+m_Y^2}$. $-\partial_sG$ is the retarded function of this dispersion, $\frac{\delta(s-r)}{4\pi r}-\frac{m_Y}{4\pi}\theta(s-r)\frac{J_1(m_Y\sqrt{s^2-r^2})}{\sqrt{s^2-r^2}}$, and $G(r,\infty)=0$ (Riemann–Lebesgue, $m_Y>0$). Hence $G(r,s)=G(r,0)=e^{-m_Yr}/(4\pi r)$ for $s<r$. For $s>r$, with $s'=r\cosh\theta$, $|G|\le\frac{m_Y}{4\pi}\int_0^\infty|J_1(m_Yr\sinh\theta)|\mathrm d\theta\le\frac{m_Y}{4\pi}\bigl(\tfrac12\sqrt{1+m_Y^2r^2}+3\cdot0.7858\bigr)$. This uses $|J_1(x)|\le x/2$ up to $m_Yr\sinh\theta_0=1$, and Landau's bound $|J_1(x)|\le0.7858\,x^{-1/3}$ beyond, where $m_Yr\sinh\theta\ge e^{\theta-\theta_0}$. With $\langle q\rangle_{\rm st}(r)=-\frac{g_s}{\kappa_h}\frac{e^{-m_Y|r|}}{4\pi|r|}$:

$$\langle q\rangle(r,\lambda)=0\quad(\sqrt{\kappa_h}\,\lambda<|r|),\qquad \bigl|\langle q\rangle(r,\lambda)-\langle q\rangle_{\rm st}(r)\bigr|\le8.4\,m_Y|r|\,\bigl|\langle q\rangle_{\rm st}(r)\bigr|\quad(\sqrt{\kappa_h}\,\lambda>|r|,\ m_Y|r|\le1).\tag{8.15}$$

So at every $|r|$ with $1\ll|r|\le\varepsilon/(8.4\,m_Y)$ the mark is formed to accuracy $\varepsilon$ for every $\lambda>|r|/\sqrt{\kappa_h}$: the formation scale is linear in $|r|$, at the front speed $\sqrt{\kappa_h}$. The range $\varepsilon/(8.4\,m_Y)$ grows without bound as $m_h\to0$ at fixed $\varepsilon$. On it $\langle q\rangle_{\rm st}=-\frac{g_s}{4\pi\kappa_h|r|}(1+O(m_Y|r|))$, the long-range form $1/(4\pi|r|)$ of (8.2) in $D=3$. On the lattice $\Omega_{\mathbf k}$ differs from the continuum form by relative $O(\mathbf k^2)$. This changes the field behind the front by relative $O(|r|^{-2})$, and smooths the step over $O(|r|^{1/3})$ (cubic term of $\Omega$ at small $\mathbf k$, as in Step 3), which keeps the scale linear. Here the answers of items 2 and 3 differ from (8.10), (8.14).

**Control for item 1** (wider class: own terms of unbounded range). On the ring, $C_1'=-t_f\sum_{x\ne y}|x-y|^{-\alpha}|x\rangle\langle y|$ ($|x-y|$ the ring distance, $\alpha>1$) has the uniform lowest state. With the source, its one-object contract has negative off-diagonal entries on a connected graph, so its lowest state is nondegenerate for every $g_s$ (Perron–Frobenius): the static mark is well defined. For $t_fs\ll1$, $\operatorname{Im}\mathcal U(r,s)=t_fs|r|^{-\alpha}(1+O((t_fs)^2))$ uniformly in $r$, since powers of a kernel decaying like $|r|^{-\alpha}$, $\alpha>1$, decay like $|r|^{-\alpha}$. So at a fixed small $\lambda_1$, at first order,

$$\delta\rho(r,\lambda_1)=\frac{n\,g_s\,t_f\lambda_1^2}{|r|^{\alpha}}\bigl(1+O((t_f\lambda_1)^2)\bigr).\tag{8.16}$$

The argument of (8.18) holds for this contract, with $B\le c\,t_f\lambda_1|r|^{-\alpha}$. So the exact change differs from (8.16) by a relative $O(|g_s|\lambda_1)$, uniformly in $r$. It exceeds the front threshold $\varepsilon(n|g_s|/t_f)\rho^{-p}$ for all large $\rho$ when $p>\alpha$. The arrival time stays bounded, and the change ahead of any line $\lambda=\rho/v$ is only power-law small. There is no finite front speed.

### Step 7. Speeds (item 4)

The motion term of a Model D body is, in $k$-space, $A(k)=-\tau\sin k\,\sigma_1\otimes\mathbb 1_N$, with group velocities $\mp\tau\cos k$, largest $\tau$. Its propagator has the entries $J_r(\tau\lambda)$ (Jacobi–Anger with $\sin k$), so without the rest and clock term the body's own front is at $|x|=\tau\lambda$, with the bound of Step 2. With the rest and clock term the bands of level $a$ are $\pm(\tau^2\sin^2k+m_a^2)^{1/2}$, with $m_a=m+2\pi a/(N\lambda_0)\ge m>0$ the eigenvalues of $m\mathbb 1_N+D$ (here $D$ is the clock operator of Model D). With $s=\sin^2k$ the group velocities obey $v^2=\tau^4s(1-s)/(\tau^2s+m_a^2)$, which is largest at $s=m_a/(R_a+m_a)$, $R_a=(m_a^2+\tau^2)^{1/2}$. There $v_{\max,a}=R_a-m_a<\tau$. The body's position label is the field place of the couplings ($\sum_x|x\rangle\langle x|\otimes G\otimes\Phi_x$), along one axis of $X_f$, where $v_f=2t_f$ by (8.7). Since $R_a-m_a<2t_f$ iff $\tau^2<4t_f(t_f+m_a)$:

$$\frac{v_f}{\tau}=\frac{2t_f}{\tau};\qquad \tau<2t_f\ \text{is sufficient for every group velocity of the body to be below }v_f;\qquad \text{all group velocities of level }a\text{ are below }v_f\ \text{iff}\ \tau^2<4t_f(t_f+m_a).\tag{8.17}$$

For all levels the condition is $\tau^2<4t_f(t_f+m)$. A packet moves with an average of group velocities, so under these conditions it is slower than the front; a particular packet can be slower still.

### Step 8. Order of limits and finite coupling

All results except the first part of (8.1), (8.5) and (8.18) concern the first-order coefficient (Setup). For the exact response at a small nonzero $g_s$, (8.18) gives sufficient conditions that do not depend on $L$. On the windows used below $|g_s|\lambda\le\frac12$, where $2n(g_s\lambda)^2e^{2|g_s|\lambda}\le2e\,n(g_s\lambda)^2$.
- **Front (8.7).** If $|g_s|/t_f\le\varepsilon\rho^{-(p+2)}/(4e)$, the remainder on $\lambda\le\rho/t_f$ is at most half the threshold. The exact arrival time then lies between the first-order arrival times for the thresholds $\varepsilon/2$ and $3\varepsilon/2$, and both obey (8.7), whose leading terms do not depend on $\varepsilon$.
- **Formation (8.10).** On a window $\lambda\le\Lambda_1$ with $|g_s|t_f\Lambda_1^2\le\varepsilon s_D|r|^{1-D}/(2e)$, the remainder of the gradient (central difference) is at most $\varepsilon/2$ of the static long-range gradient $\frac{2n|g_s|}{t_f}s_D|r|^{1-D}$. Then the exact gradient is formed to accuracy $\varepsilon$ on $[\lambda_{\rm form}(\varepsilon/2),\Lambda_1]$. Since $\Lambda_1\ge\lambda_{\rm form}\propto|r|^2/t_f$, this needs $|g_s|/t_f=O(\varepsilon\zeta_D^2|r|^{-(D+3)})$; (8.18) does not extend the window to $\lambda\to\infty$.
- **Average (8.11).** The remainder averaged over $[0,\Lambda]$ is at most $\frac{2e}3n(g_s\Lambda)^2$. With $\Lambda=2S_2/(\varepsilon t_f|\Lambda^+(r)|)$ the first-order deviation is at most $\frac\varepsilon2|\delta\rho_{\rm st}(r)|$, and the remainder is at most the other half if $|g_s|/t_f\le3\varepsilon^3|\Lambda^+(r)|^3/(8eS_2^2)$.
- **Negative (8.14).** At $\lambda^*$ the remainder of the gradient is below half the margin $|1-g_D(\zeta^*)|-\varepsilon$ if $|g_s|/t_f\le8s_D\zeta^{*2}(|1-g_D(\zeta^*)|-\varepsilon)|r|^{-(D+3)}/e$. Then the exact gradient also fails the criterion at $\lambda^*$.
- **Static mark (8.2).** The source is rank one, so the lowest state of $h$ is $\psi\propto(h_0-E)^{-1}|x_s\rangle$ with $1+g_s\langle x_s|(h_0-E)^{-1}|x_s\rangle=0$. Expanding in $E+2Dt_f=g_s/|X_f|+\dots$ gives (8.2) as the leading term of a series in $g_s\Lambda^+(0)/t_f$. The other parameter, $g_sL^{2-D}/(4\pi^2t_f)$, is at most of the same order. So (8.2) needs $|g_s|\Lambda^+(0)/t_f\ll1$, with $\Lambda^+(0)\simeq L/12$ ($D=1$), $\simeq\ln L/(2\pi)$ ($D=2$) and $\to G_0$ ($D=3$): nonuniform in $L$ for $D=1,2$. No bound on this remainder is derived.
- **Controls.** (8.15) is exact in $g_s$ for the untruncated $H$; (8.16) holds for the exact change when $|g_s|\lambda_1\ll1$ (Step 6).

## Result

Class: the field class of the setup ($N_f\ge1$, $D=1,2,3$, torus of even side $L$), the fixed contract with the source $g_s\Phi_{x_s}$, started in the lowest state without it. *First order* means the first-order coefficient in $g_s$ ($g_s\to0$ first, at fixed $L$, $r$, $\lambda$); the further limits are those stated at each result.

- **(8.1)** Exactly, $\langle\Phi_x\rangle_\lambda=N_f|\psi(x,\lambda)|^2$: the made field of $N_f$ objects is $N_f$ times that of one, at every order. At first order $\delta\rho$ is the time integral (8.1) of the free propagator.
- **(8.2)** First order: the static mark (in the lowest state with the source, nondegenerate for every $g_s$) is $-(2ng_s/t_f)\Lambda^+(r)$. It is the leading term in $g_s\Lambda^+(0)/t_f$, which grows with $L$ for $D=1,2$.
- **(8.3)** First order, $\lambda\ge0$: $\delta\rho=\delta\rho_{\rm st}+(2ng_s/t_f)R$. On the torus $R$ is a finite sum of cosines and does not decay.
- **(8.4)** The propagator is the image sum of products of Bessel functions, with $|\mathcal U_\infty|\le e^{-\rho\eta(2t_fs/\rho)}$ for $2t_fs\le\rho$.
- **(8.18)** Exactly, for every $g_s$, $L$, $r$, $\lambda\ge0$: $|\delta\rho|\le n(e^{2|g_s|\lambda}-1)B$. Also $|\delta\rho-\delta\rho^{(1)}|\le n(e^{2|g_s|\lambda}-1-2|g_s|\lambda)B\le2n(g_s\lambda)^2e^{2|g_s|\lambda}B$, with $B=\max_{s\le\lambda}|\mathcal U(r,s)|\le1$. Sufficient conditions (Step 8, with $|g_s|\lambda\le\frac12$ on the window used) for the exact response:
  - (8.7): $|g_s|/t_f\le\varepsilon\rho^{-(p+2)}/(4e)$;
  - (8.10), on $[\lambda_{\rm form}(\varepsilon/2),\Lambda_1]$: $|g_s|t_f\Lambda_1^2\le\varepsilon s_D|r|^{1-D}/(2e)$;
  - (8.11): $|g_s|/t_f\le3\varepsilon^3|\Lambda^+(r)|^3/(8eS_2^2)$;
  - (8.14): $|g_s|/t_f\le8s_D\zeta^{*2}(|1-g_D(\zeta^*)|-\varepsilon)|r|^{-(D+3)}/e$.
- **(8.5)** Item 1, ahead of the front, at every order: for $2t_f\lambda<\rho\le L/2$, $|\delta\rho|\le n(e^{2|g_s|\lambda}-1)\sum_{\mathbf m}e^{-\rho_{\mathbf m}\eta(2t_f\lambda/\rho_{\mathbf m})}$. At fixed $z=2t_f\lambda/\rho<1$ this is exponentially small in $\rho$ if $|g_s|/t_f<\eta(z)$. At first order the prefactor is $2n|g_s|\lambda$, and the bound holds for every start with $2N_f|g_s|\lambda$.
- **(8.6)** Item 1, change at the front along an axis, first order, $\rho\to\infty$ with $|2t_f\lambda-\rho|=O(\rho^{1/3})$ and $\rho\le(1-\delta)L/2$: the Airy form (8.6) with the trigonometric factor $P_D$. Its maximum over the front window is at least $b_D(n|g_s|/t_f)\rho^{-p_D}(1-o(1))$, $p_D=\frac13+\frac{D-1}2$, $b_D=0.6749\cdot2^{1/2-D}(2/\pi)^{(D-1)/2}$. For $D=1$ the envelope maximum is $0.6749\,(n|g_s|/t_f)\rho^{-1/3}$.
- **(8.7)** Item 1: the front speed is finite. Criterion $|\delta\rho|\ge\varepsilon(n|g_s|/t_f)\rho^{-p}$, $p>p_D$; first order, $\rho\to\infty$. In every direction ($\rho\le L/2$), $\lambda_a\ge\rho/(2t_f)-O(\rho^{1/3}(\ln\rho)^{2/3}/t_f)$. Along the axes ($\rho\le(1-\delta)L/2$), $\lambda_a=\rho/(2t_f)+O(\rho^{1/3}(\ln\rho)^{2/3}/t_f)$, so $v_f=2t_f$ place labels per unit $\lambda$. Off the axes the arrival time is not bounded from above.
- **(8.8), (8.9)** Behind the front (first order; $L\to\infty$ first; $|r|\to\infty$ at fixed $\zeta=|r|^2/(4t_f\lambda)$; leading order) the made field is $-(2ng_s/t_f)\operatorname{Re}W_D(|r|,it_f\lambda)$. Its gradient is $g_D(\zeta)=\operatorname{Re}\Gamma(D/2,-i\zeta)/\Gamma(D/2)$ times that of the long-range static part.
- **(8.10)** Item 2, first order. The long-range part (the gradient) at distance $|r|$ is formed to accuracy $\varepsilon$ from $\lambda_{\rm form}=|r|^2/(4t_f\zeta_D(\varepsilon))$, with $\zeta_1=\frac{\pi\varepsilon^2}2(1+O(\varepsilon^2))$, $\zeta_2=\sqrt{2\varepsilon}(1+O(\varepsilon))$, $\zeta_3=(3\sqrt{2\pi}\varepsilon/4)^{2/3}(1+O(\varepsilon^{2/3}))$. This scale is independent of $L$ and holds on the torus up to the wrap time $(L-\rho)/(2t_f)$. In $D=3$ the value at $|r|$ forms at $|r|^2/(2\pi\varepsilon^2t_f)$ (relative to $\delta\rho^\infty_{\rm st}$). In $D=1,2$ the level grows without bound on the infinite lattice.
- **(8.11)** Item 2, $\lambda$-average on the torus, first order. The running average of $\delta\rho$ tends to $\delta\rho_{\rm st}$, with error at most $(2n|g_s|/t_f)S_2/(t_f\Lambda)$, $S_2=O(L^{4-D})$. It is within $\varepsilon|\delta\rho_{\rm st}(r)|$ for every $\Lambda\ge S_2/(\varepsilon t_f|\Lambda^+(r)|)$. These sufficient lengths are of order $L^2/(60\varepsilon t_f)$, $L^2/(\varepsilon t_f\ln(L/|r|))$ and $L|r|/(\varepsilon t_f)$ for $D=1,2,3$; no lower bound is derived. The instantaneous field does not converge as $\lambda\to\infty$, since $R$ is a nonconstant finite sum of cosines.
- **(8.14)** Item 3, negative. In the class, at first order, with $L\to\infty$ first (or $|r|^2<2\zeta^*(L-\rho)$), the long-range part does not form on any scale linear in $|r|$. For every fixed $\beta>0$ and $\varepsilon<1$, $\lambda_{\rm form}>|r|^2/(4t_f\zeta^*)>\beta|r|/t_f$ for all large $|r|$, for any fixed $\zeta^*$ with $|1-g_D(\zeta^*)|>\varepsilon$.
- **(8.15)** Control for items 2 and 3, in a wider class: field objects with more places and nearest-neighbour pair terms ($D=3$ pinned harmonic lattice, compressed to $M$ places per object, source coupled to $q_{x_s}$). The untruncated contract has, for every $g_s$, a unique lowest state with the static mark $-g_sK^{-1}e_{x_s}$, and an exact linear evolution whose uniform mode does not drift. Its truncation converges as $M\to\infty$ on every finite window. With $M\to\infty$, then $L\to\infty$, then $|r|\gg1$, $m_Y\ll1$, at leading order: $\langle q\rangle=0$ ahead of the front $|r|=\sqrt{\kappa_h}\lambda$, and behind it $\langle q\rangle$ is within $8.4\,m_Y|r|$ (relative) of the static mark, for $m_Y|r|\le1$. So on $1\ll|r|\le\varepsilon/(8.4\,m_Y)$ the mark forms at the front, linearly in $|r|$.
- **(8.16)** Control for item 1, in a wider class: own term with hopping $\propto|x-y|^{-\alpha}$, $\alpha>1$, on the ring, with a nondegenerate lowest state with the source. At fixed small $\lambda_1$ the change is $ng_st_f\lambda_1^2|r|^{-\alpha}$ at first order, and for the exact response up to a relative $O(|g_s|\lambda_1)$. For $p>\alpha$ the arrival time stays bounded, so there is no finite front speed.
- **(8.17)** Item 4: $v_f/\tau=2t_f/\tau$, both in place labels per unit $\lambda$, for a body moving along an axis of the field. $\tau<2t_f$ is sufficient for every group velocity of a Model D body to be below the front speed $2t_f$. Level $a$ has the full-band maximum $\sqrt{m_a^2+\tau^2}-m_a$, and all its group velocities are below $2t_f$ iff $\tau^2<4t_f(t_f+m_a)$.

## Consistency checks

1. **Static limit.** The $\lambda$-average (8.11) gives (8.2), and $\Theta\to\infty$ in the gradient of (8.8) gives the gradient of its long-range part. By Hellmann–Feynman ($\partial E/\partial g_2=\langle\Phi_{x_2}\rangle$), $M^{(2)}(x_1,x_2)=g_2\,\delta\rho_{{\rm st},1}(x_2)$, which is (4.4) of 06-ilang-source/04-profile. The gradient limit of (8.9) is the $d$-dependence of (4.4) of 07-ilang-weight/04-weight-of-bond: $-\ln|r|/2\pi$ for $D=2$, $1/(4\pi|r|)$ for $D=3$. Code: the exact $\lambda\to\infty$ average differs from $-\Lambda^+$ by $1.1\cdot10^{-3}$ (ring $L=1024$) and $3.7\cdot10^{-4}$ (cube $L=32$) of $|\Lambda^+(0)|$. This is the second order, of the size of $g_s\Lambda^+(0)/t_f$ ($8.5\cdot10^{-4}$ on the ring). Passed.
2. **Sum rule.** $\sum_x\delta\rho(x,\lambda)=0$ at every $\lambda$, from (8.3) ($\sum_r\cos\mathbf k\cdot r=0$ for $\mathbf k\ne0$), as $\sum_{x_2}M^{(2)}=0$ in (4.4). Code: $|\sum_x\delta\rho|\le9\cdot10^{-11}$ in units $2ng_s/t_f$. Passed.
3. **Dimensions.** $g_s$, $t_f$, $\tau$, $m_a$, $\sqrt{\kappa_h}$ and $m_h$ have the units of $C$, and $\lambda$ the inverse units. $\delta\rho$, $g_s/t_f$, $g_s\lambda$, $t_f\lambda$, $\zeta$ and $m_Y|r|$ are numbers; $v_f$ and $\tau$ are place labels per unit $\lambda$; the scales (8.10) are $\propto|r|^2/t_f$, that of (8.15) is $|r|/\sqrt{\kappa_h}$. Passed.

## Open issues

- The gradual switching of the question is withdrawn by its note of 2026-10-21 (A5 excludes $\lambda$-dependent contracts) and is not treated.
- On the torus, the size of the wrapped changes after the wrap time is not derived; the instantaneous formation statements hold up to the wrap time, or with $L\to\infty$ first.
- Off the axes, a lower bound on the front maximum, and hence an upper bound on the arrival time, is not derived. $D=2$ is not checked numerically (the code covers the ring and the cube, as the question requires).
- At finite $g_s$ the static mark (8.2) is given only with its expansion parameter $g_s\Lambda^+(0)/t_f$, without a remainder bound. The instantaneous formation at finite $g_s$ is controlled by (8.18) only on finite windows. For starts other than the lowest state only the first-order (8.5) is derived.
- In the control (8.15) the lattice corrections near the front and the truncation errors are argued, not bounded explicitly. Its range of linear formation is $|r|\le\varepsilon/(8.4\,m_Y)$.
- Since the long-range part forms only on $|r|^2/t_f$, it is open how far out the long-range mark of a moving body is formed. Moving sources are out of scope.
- The meaning of (8.17) rests on identifying the body's position label with the field place along an axis. The relation to the distances of 03-ilang-space is out of scope.
- Whether a field with linear long-wavelength frequencies, as in (8.15), is selected by anything in the language is item 5, and is not addressed (M10).

## Code

- `code/v2/check_front_formation.py` (role: check). It evolves one field object ($N_f=1$, which suffices by (8.1)) exactly from its lowest state without the source, under the fixed contract with the source (sudden switching). The evolution runs in the sector invariant under the point group of the torus fixing $x_s$, which the state never leaves. Constant contracts are propagated by Chebyshev expansion, which agrees with dense diagonalization to $2\cdot10^{-15}$ (cube $L=16$). Lattices: rings $L=256$, $1024$, $16384$; cubic tori $L=16$, $32$, $160$. Couplings: $g_s/t_f=10^{-5}$ (rings $L\ge1024$) and $10^{-3}$ (cubes $L\ge32$), where the exact evolution differs from (8.3) by $3.7\cdot10^{-5}$ and $7.9\cdot10^{-4}$ of $|\Lambda^+(0)|$. For the finite-coupling bounds it uses $g_s/t_f=\pm0.02$ (ring $L=256$) and $0.05$ (cube $L=16$). It checks (8.3), (8.5)–(8.7), (8.9)–(8.11) and (8.18), with the numbers quoted in Steps 2–5. The gradual switching of v1 is removed.
- Outputs: `code/v2/output/check_front_formation.json` (all numbers) and `code/v2/output/check_front_formation.log` (run log).

## Methods used

- Reduction of non-interacting identical objects to one object; first-order Dyson expansion and a bound on the Dyson series; Hellmann–Feynman; rank-one resolvent
- Fourier analysis on tori, Poisson summation, Jacobi–Anger expansion
- Kapteyn's inequality, Airy asymptotics of Bessel functions, stationary phase, integration by parts, Parseval
- Fresnel–Gaussian integrals, analytic continuation of the integrated heat kernel, incomplete Gamma function
- Heisenberg equations of a quadratic contract, retarded Klein–Gordon function, Landau's Bessel bound; min-max principle, compression truncation, Duhamel formula (control)
- Group velocities of the Model D bands
- Numerical check: exact evolution in the symmetric sector, Chebyshev propagation, dense diagonalization

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