Version 2 · current · External review, round 2: correct

# Weight and fall of a body with its own mark in the field of a held source

- **Subproject:** 06-ilang-source
- **Package:** 07-dressed-fall
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-19

## Changes from previous version

- Setup: (6.4) of 06-own-mark is quoted in full (dated note of the question), replacing the sufficient inequality for $\sigma=+$ (I3).
- Step 5: Regime R rewritten. R1 is now a self-contained isolation gap $\delta_{k,\sigma}$ of the band, derived from (6.4), with a smallness condition on the coupling, and a stated class of starts. R2 is now the coefficient of $\tau^2$ of a sequential expansion, with the two independent small-motion conditions (I1, I2, I3).
- Step 5: (7.7) is now the exact acceleration of the projected lattice contract at order $\tau^2$, with the envelope sum $\sum_j\phi_{j+1}\phi_{j-1}W'(j)$. It replaces the local Ehrenfest form $-E''W'(x_0)$. The central slope is now derived, from the hops by two places of the band, and is no longer read from (5.7) (I1).
- Step 6: $W'$ is evaluated at every place of the packet and averaged. In (7.8) and (7.9) this adds the factor $c_\phi$ and the packet term $12s_\phi^2$ in the geometric factor, and (7.8) gets the remainder $O(\tau^4\gamma g_s)$ (I1, I2).
- Result (7.7)–(7.9), consistency check 2, Open issues, Code: changed accordingly. `check_weight.py` adds a check of (7.7) and (7.8) for packets.

## Response to verification

- I1: Accepted and fixed in Steps 5 and 6. The acceleration is now derived exactly for the projected lattice contract at order $\tau^2$ and first order in $g_s$, (7.7). It keeps the packet-dependent force average: the factor $c_\phi=\langle\cos2\hat P\rangle$ and the variance term $12s_\phi^2$ in (7.8) and (7.9). No width approximation is used.
- I2: Accepted and fixed in Step 5 (R2) and (7.8). The result is stated as the coefficient of $\tau^2$, with remainder $O(\tau^4\gamma g_s)$, and with the conditions $\tau\ll M_k$ and $\tau^2\ll M_k\delta_{k,\sigma}$, named as independent.
- I3: Accepted and fixed in Step 5 (R1), using (6.4) as now quoted. R1 gives the gap $\delta_{k,\sigma}$ for both branches and every $P$, and the smallness of the coupling and the source against it.

## Setup and assumptions

**Description (assumptions of the package, M8, M9).**
- *Field.* $N_f\ge1$ identical field objects ($c=+1$) on the ring $\mathbb Z_L$, $L\ge5$, no pair terms among them, own term $C_1=-t_f\sum_x(\lvert x\rangle\langle x+1\rvert+\lvert x+1\rangle\langle x\rvert)$, $t_f>0$; $\Phi_x$ the number of field objects at $x$, $n=N_f/L$. With $\Lambda$ the ring Laplacian, $C_1=-2t_f+t_f\Lambda$; $G:=\Lambda^+$, and $\omega_{\min}=4\sin^2(\pi/L)$ is the smallest nonzero eigenvalue of $\Lambda$. *State:* the lowest state adapted to the bodies present.
- *Source.* A held body (A15) at $x_s$, in an eigenvector of its coupled operator with eigenvalue $g_s$: pair term $g_s\Phi_{x_s}$.
- *Body.* Model D on the same ring, level $k$ with half-gap $M_k=m+d_k>0$, branch $\sigma=\pm1$, coupling (a) or (c) as in (6.2) of 06-own-mark. Records at $\kappa\to0$. Its own change of the field is kept.
- *Shorthand.* $g_b:=\sigma\gamma M_k$; $P_y:=\lvert y\rangle\langle y\rvert$; $r:=(x-x_s)\bmod L\in\{0,\dots,L-1\}$, $u(x):=r(L-r)$; $G_0:=G(x,x)$; for a function $f$ on the ring, $f'(x_0):=\frac12[f(x_0+1)-f(x_0-1)]$ (central lattice slope).
- *Weakness (W).* $|g_b|+|g_s|<t_f\omega_{\min}/2$. All expansions at fixed $L,N_f,t_f,M_k$.
- *Regime R* of item 2: stated in Step 5.

**Inputs (quoted in question.md).**
- (4.2) of 04-profile: $E(g)=N_f\varepsilon_0(g)$ for the field with held point terms. (4.4) of 04-profile: $M^{(2)}(x_1,x_2)=-\frac{2ng_1g_2}{t_f}\Lambda^+(x_1,x_2)$.
- (5.8) of 05-made-field-scale: $\delta N(y)=-\frac{2ng_s}{t_f}\Lambda^+(y,x_s)+O(N_fg_s^2/(t_f^2\omega_{\min}^2))$. (5.7): $a_{k,\sigma}(0)=-\tau_{\rm loc}(x_0)^2\gamma N'(x_0)/\nu(x_0)$, $\nu=1+\gamma N_\varphi$, $\tau_{\rm loc}=\tau$ (a), $\tau\nu$ (c). Its second form, with $r\propto\nu$, reads $N'$ as the central slope; we use the same slope throughout.
- (6.2) of 06-own-mark: the total contract on level $k$ for (a) and (c). (6.7): $E_{k,\sigma}(P)|_{\tau=0}=\sigma M_k+N_f\varepsilon_0(\sigma\gamma M_k)$. (6.10): $E''_{k,\sigma}(0)=\sigma\frac{\tau^2}{M_k}(1-\gamma n+\gamma^2n^2)$ (a), $\sigma\frac{\tau^2}{M_k}(1+\gamma n)$ (c), $+\Delta''_{k,\sigma}+O(\gamma^3)$, $\Delta''_{k,\sigma}=O(\gamma^2)$. (6.4): with $\omega_k(p)=\sqrt{M_k^2+\tau^2\sin^2p}$, the start $(P,\sigma,k)$ is a simple eigenvalue of its sector at $\gamma=0$ iff $\sigma\omega_k(P)\ne\sigma'\omega_k(P-Q)+E_{\rm exc}$ for every excited field eigenstate $(Q,E_{\rm exc})$ and $\sigma'=\pm$.

## Derivation

### Step 1. The held body is an exactly decoupled block

For $\tau=0$ the motion term vanishes, and the extra term of (c) in (6.2) is proportional to $\tau$; (a) and (c) give the same total contract. Every remaining term is diagonal in the body's position label and commutes with $\sigma_3$ and with $D$, and the source stays on its eigenvector. The total contract therefore leaves invariant the block of body place $x$, branch $\sigma$ and level $k$, and on it acts as $\sigma M_k+c_s+\sum_{j=1}^{N_f}h_x^{(j)}$, with $c_s$ the source's own constant and

$$
h_x:=C_1+g_bP_x+g_sP_{x_s}.
\tag{7.1}
$$

The field objects are identical with $c=+1$ and no pair terms among them, so the lowest field eigenvalue is $N_f\varepsilon_0(x)$, with $\varepsilon_0(x)$ the lowest eigenvalue of $h_x$ ((4.2) of 04-profile). At $g_b=g_s=0$ it is $-2t_f$, simple, with gap $t_f\omega_{\min}$. Under (W), $\lVert h_x-C_1\rVert<t_f\omega_{\min}/2$, so it stays simple and isolated and is analytic in $(g_b,g_s)$ (analytic perturbation theory). It is the eigenvalue connected to the body with the field in its lowest state. Hence, exactly for $\tau=0$ and both couplings,

$$
W_{k,\sigma}(x)-W_{k,\sigma}(x')=N_f\bigl[\varepsilon_0(x)-\varepsilon_0(x')\bigr],\qquad \varepsilon_0(x)=\min\operatorname{spec}h_x .
\tag{7.2}
$$

At $g_s=0$ this eigenvalue is $\sigma M_k+N_f\varepsilon_0(\sigma\gamma M_k)$, which is (6.7).

### Step 2. Expansion to first order in $g_s$, through $g_b^2$

A rotation of the ring maps $h_x$ to $h_{x'}$ when only one of the two point terms is present, so $\varepsilon_0(x)$ depends on $x$ only through the mixed coefficients $g_b^ag_s^c$, $a,c\ge1$. At first order in $g_s$ and through $g_b^2$ these are $g_bg_s$ and $g_b^2g_s$. We use Rayleigh–Schrödinger theory with $\phi_0=L^{-1/2}\sum_y\lvert y\rangle$, $V=g_bP_x+g_sP_{x_s}$ and reduced resolvent $R=Q(-2t_f-C_1)^{-1}Q=-G/t_f$. The standard formulas are $E^{(2)}=\langle VRV\rangle$ and $E^{(3)}=\langle VRVRV\rangle-\langle V\rangle\langle VR^2V\rangle$, and $\langle\phi_0\rvert P_yXP_{y'}\lvert\phi_0\rangle=X(y,y')/L$. Then:
- $g_bg_s$ comes from $2\langle P_xRP_{x_s}\rangle=-\frac{2}{Lt_f}G(x,x_s)$;
- $g_b^2g_s$ comes from the three orderings in $\langle VRVRV\rangle$, $\frac{1}{Lt_f^2}[2G_0G(x,x_s)+G(x,x_s)^2]$, and from $-\frac{1}{L^2t_f^2}[(G^2)(x,x)+2(G^2)(x,x_s)]$. Both $G_0$ and $(G^2)(x,x)$ are independent of $x$.

Multiplying by $N_f$:

$$
W_{k,\sigma}(x)=\text{const}-\frac{2ng_sg_b}{t_f}G(x,x_s)+\frac{ng_sg_b^2}{t_f^2}\Bigl[2G_0\,G(x,x_s)+G(x,x_s)^2-\frac2L(G^2)(x,x_s)\Bigr]+R_W,
\tag{7.3}
$$

with $R_W=O(g_s^2)+O(g_sg_b^3)$. The coefficients of $R_W$ are bounded by $N_f$ times powers of $1/(t_f\omega_{\min})$ (Cauchy estimates on the polydisc of analyticity, where $|\varepsilon_0+2t_f|\le|g_b|+|g_s|$). The $g_bg_s$ term is $M^{(2)}(x,x_s)$ of (4.4) of 04-profile with $g_1=g_b$, $g_2=g_s$.

### Step 3. Closed form on the ring

$G(\cdot,x_s)$ is the zero-sum solution of $\Lambda f=\delta_{x_s}-1/L$, and $(G^2)(\cdot,x_s)$ the zero-sum solution of $\Lambda f=G(\cdot,x_s)$ (since $\Lambda G=\mathbb 1-J/L$ and $GJ=0$, with $J$ the all-ones matrix). Write $\Delta^2f(r)=f(r+1)+f(r-1)-2f(r)$.
- As polynomials in $r$, $\Delta^2u=-2$ and $\Delta^2u^2=2L^2+2-12u$. These hold for $r=1,\dots,L-1$, where the periodic values $u(0)=u(L)=0$ agree with the polynomial.
- At $r=0$, $f(-1)=f(1)$.
- $\sum_rr(L-r)=L(L^2-1)/6$.

These give

$$
G(x,x_s)=G_0-\frac{u}{2L},\quad G_0=\frac{L^2-1}{12L},\qquad (G^2)(x,x_s)=c_L-\frac{u}{12L}-\frac{u^2}{24L},
\tag{7.4}
$$

with $c_L$ independent of $x$. Check at $r=0$: $2f(0)-2f(1)$ equals $1-1/L$ for $G$, and $\frac{L-1}{6L}+\frac{(L-1)^2}{12L}=G_0$ for $G^2$. Inserting (7.4) into (7.3) and collecting powers of $u$ (coefficient of $u$: $\frac{1}{6L^2}-\frac{2G_0}{L}=-\frac{L^2-2}{6L^2}$; of $u^2$: $\frac1{4L^2}+\frac1{12L^2}=\frac1{3L^2}$) gives

$$
W_{k,\sigma}(x)-W_{k,\sigma}(x')=\frac{ng_sg_b}{Lt_f}\Bigl[\Bigl(1-\frac{(L^2-2)\,g_b}{6Lt_f}\Bigr)(u-u')+\frac{g_b}{3Lt_f}\,(u^2-u'^2)\Bigr]+R_W,\qquad g_b=\sigma\gamma M_k,
\tag{7.5}
$$

where $u'=u(x')$. `code/v2/check_weight.py` confirms (7.4) to $6\times10^{-15}$, (7.3)$=$(7.5) and (7.5) against the exact (7.2) (section "Code").

### Step 4. Factorization of the weight

With $\gamma\,\delta N=-\frac{2n\gamma g_s}{t_f}G(\cdot,x_s)$ from (5.8), (7.5) reads

$$
W_{k,\sigma}(x)-W_{k,\sigma}(x')=\sigma M_k\,\gamma\bigl[\delta N(x)-\delta N(x')\bigr]+M_k^2\gamma^2\bigl[\Theta(x)-\Theta(x')\bigr]+R_W,\qquad \Theta:=\frac{ng_s}{3L^2t_f^2}\Bigl(u^2-\frac{L^2-2}{2}\,u\Bigr).
\tag{7.6}
$$

- *First order in $\gamma$.* The weight is one function of the place, $\gamma\,\delta N(x)$, times the factor $\sigma M_k$. This factor is the eigenvalue of the coupled operator $\sigma_3\otimes(m\mathbb 1_N+D)$ on the level and branch, i.e. the body's rest-and-clock cost. This is the weight of a test body in the made field.
- *Second order in $\gamma$.* The factor is $M_k^2$, the same for both branches, and $\Theta$ is not affine in $\delta N$: $\delta N$ is affine in $u$, $\Theta$ has the $u^2$ coefficient $\frac{ng_s}{3L^2t_f^2}\ne0$, and $1,u,u^2$ are linearly independent functions on the ring for $L\ge4$ (the distances $0,1,2$ give the distinct values $0$, $L-1$, $2(L-2)$ of $u$). In (7.5) the ratio of the $u^2$ to the $u$ coefficient is $\frac{g_b}{3Lt_f}+O(g_b^2)$, fixed by the orders $\gamma,\gamma^2$ alone. Two pairs $(k,\sigma)$ with different $g_b$ therefore have different place dependences at relative order $\gamma$. The two branches of every level always differ ($g_b=\pm\gamma M_k$, $M_k>0$, $\gamma\ne0$), and so do two levels with $M_k\ne M_{k'}$. Through second order in $\gamma$ the level enters only through $\sigma M_k$, but as a change of the shape of the place dependence, not as a factor.
- *Origin.* By Hellmann–Feynman, $\partial_{g_s}N_f\varepsilon_0|_{g_s=0}=N_f\lvert\phi_0(x_s;g_b,x)\rvert^2$, the field made by the body itself at the source's place. The $\gamma^2$ term is the nonlinearity of this made field in the body's coupling, i.e. the body's own mark.
- *Class and control (M9).* The second-order statement holds for every member of the class (it requires $g_s\ne0$, $\gamma\ne0$); there is no control in the class. Control for the first-order factor: a body coupled by $\sigma_3\otimes\mathbb 1_N$ in place of $\sigma_3\otimes(m\mathbb 1_N+D)$ has $g_b=\sigma\gamma$ and the factor $\sigma$, not $\sigma M_k$.

### Step 5. Regime R and the projected lattice acceleration

**Regime R (assumption of the package).**
- **R1 (isolated band).** At $\gamma=g_s=0$ the band of the start, $\sigma\omega_k(P)$ for all $P$ with the field in its lowest state, lies at distance at least $\delta_{k,\sigma}$ from the rest of the spectrum on level $k$, where
  $$\delta_{k,-}=\min(t_f\omega_{\min}-\eta_k,\,2M_k),\qquad \delta_{k,+}=\min(t_f\omega_{\min}-\eta_k,\,2M_k-4t_fN_f),\qquad \eta_k:=\sqrt{M_k^2+\tau^2}-M_k\le\frac{\tau^2}{2M_k}.$$
  The reason is that every excited field eigenstate has $t_f\omega_{\min}\le E_{\rm exc}\le4t_fN_f$ (lowest: one object moved to $q=\pm2\pi/L$; highest: every object at the top of its band, $\le2t_f$), and that $M_k\le\omega_k\le M_k+\eta_k$. Same-branch states with an excited field therefore lie at distance $\ge t_f\omega_{\min}-\eta_k$, and other-branch states at distance $\ge2M_k$ ($\sigma=-$) or $\ge2M_k-4t_fN_f$ ($\sigma=+$). $\delta_{k,\sigma}>0$ implies (6.4) for every $P$.
  - *Assumption:* $\delta_{k,\sigma}>0$, i.e. $\eta_k<t_f\omega_{\min}$ (sufficient: $\tau^2<2M_kt_f\omega_{\min}$), and for $\sigma=+$ also $2t_fN_f<M_k$.
  - *Assumption:* $N_f\bigl(|\gamma|(M_k+\tau)+|g_s|\bigr)<\delta_{k,\sigma}/2$. The left side bounds the norm of the coupling and source terms of (6.2) and of $g_s\Phi_{x_s}$, since $\lVert\Phi_x\rVert=N_f$.
  - *Consequence:* the Neumann series of the resolvent converges on a contour at distance $\delta_{k,\sigma}/2$ around the band. The spectral projector $\Pi$ of the $L$ eigenvalues of the band is therefore analytic in $\gamma,g_s$, and analytic in $\tau$ near $0$. $C$ commutes with $\Pi$, so a start in ${\rm ran}\,\Pi$ stays there, and no transitions out of the level occur.
- **Start (assumption, M9).** $\Psi=\sum_j\phi_j\lvert j\rangle\!\rangle$, where $\lvert j\rangle\!\rangle\in{\rm ran}\,\Pi$ is the image under Kato's transformation from $\tau=0$ of the held state $\lvert j,\sigma,k\rangle\otimes\lvert\Omega_j\rangle$ of Step 1, with $\Omega_j$ the lowest field state with the body at $j$.
  - The envelope $\phi$ is real, nonnegative and symmetric about a place $x_0\ne x_s$. It is supported on an arc of at most $L-4$ consecutive places that does not contain $x_s$, and it has $c_\phi:=\sum_j\phi_{j+1}\phi_{j-1}>0$. This class is nonempty for $L\ge7$. (7.7) below uses only that $\phi$ is real and the condition on its support.
  - The position $X$ of the body is counted from a cut that has two places outside the arc on each side. $s_\phi^2:=c_\phi^{-1}\sum_j\phi_{j+1}\phi_{j-1}(j-x_0)^2\ge0$, with $j-x_0$ counted along the arc.
- **R2 (sequential expansion in $\tau$).** The acceleration is computed at first order in $g_s$, as the coefficient of $\tau^2$ of its expansion at $\tau=0$. At this order the field follows the body: the dressed states carry the cloud $\Omega_j$ of the held body.
  - The remainder is $O(\tau^4\gamma g_s)$. It vanishes at $\gamma=0$, where the body is uncoupled, and it is not computed.
  - The part of $a$ that does not depend on $g_s$ is not part of the result. It vanishes through order $\tau^4$ for this support, because $H^{(0)}$ (below) is a constant at $g_s=0$ and $\langle[K,[K,\hat x]]\rangle=0$. Beyond that order, the dressed tails of the envelope reach the cut.
  - The energy denominators of the expansion are the branch gap $\approx2M_k$ and the isolation gap $\delta_{k,\sigma}$. The $\tau^2$ term therefore describes the acceleration only when $\tau\ll M_k$ and $\tau^2\ll M_k\delta_{k,\sigma}$. Neither condition implies the other.

**Acceleration.** By (1.9), $a_{k,\sigma}(0):=\frac{\mathrm d^2}{\mathrm d\lambda^2}\langle\Psi\vert X\vert\Psi\rangle\big|_{\lambda=0}=-\langle\Psi\vert[C,[C,X]]\vert\Psi\rangle$. Since $[C,\Pi]=0$, only $\Pi C\Pi$ and $\Pi X\Pi$ enter. In the orthonormal basis $\lvert j\rangle\!\rangle$:
- $\Pi C\Pi=H^{(0)}+\tau^2H^{(2)}+O(\tau^4)$, with $H^{(0)}=\text{const}+W_{k,\sigma}(\hat x)$ by Step 1. There is no term of first order, because $A$ (and the extra term of (c)) changes the branch.
- $\Pi X\Pi=\hat x+\tau^2X^{(2)}+O(\tau^4)$, with $\hat x\lvert j\rangle\!\rangle=j\lvert j\rangle\!\rangle$. There is no term of first order, because $X$ does not change the branch.
- $H^{(2)}$ and $X^{(2)}$ connect $j$ only to $j$ and $j\pm2$, because each factor of the motion term (and of the extra term of (c)) moves the body by one place.

Since $[H^{(0)},\hat x]=0$, the $\tau^2$ coefficient is $-\langle[W,[H^{(2)},\hat x]]\rangle-\langle[W,[W,X^{(2)}]]\rangle$.
- The second term has matrix elements $(W(j)-W(j'))^2X^{(2)}_{jj'}=O(g_s^2)$.
- In the first term, the $O(g_s)$ part of $\tau^2H^{(2)}$ gives $O(g_s^2)$ for the same reason. What remains is the source-free part $K$. It commutes with the joint translation $\mathcal T\lvert j\rangle\!\rangle=\lvert j+1\rangle\!\rangle$ and has displacements $0,\pm2$ only. With $E(P)=E(-P)$ (reflection of body and field about a place combined with $\sigma_3$, a symmetry of (6.2)), $K=\text{const}+\frac{E''_{k,\sigma}(0)}4(1-\cos2\hat P)$, with $\cos2\hat P:=\frac12(\mathcal T^2+\mathcal T^{-2})$ and $E''$ taken at order $\tau^2$.
- $\phi$ vanishes on the two places on each side of the cut. The hops by two places across the cut therefore do not contribute, and $[\hat x,\mathcal T^{\pm2}]=\pm2\mathcal T^{\pm2}$ may be used.
- The velocity is $i[K,\hat x]=\frac{iE''}4(\mathcal T^2-\mathcal T^{-2})$. Its expectation vanishes for real $\phi$: the start is at rest.
- With $[W,\mathcal T^{\pm2}]=\mathcal T^{\pm2}\bigl(W(\hat x\pm2)-W(\hat x)\bigr)$, and shifting the summation index by one place:

$$
a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}\,W'_{k,\sigma}(j)+O(g_s^2)+O(\tau^4\gamma g_s).
\tag{7.7}
$$

The central slope is a result of this step, not a choice: it comes from the hops by two places. The prefactor $c_\phi=\langle\Psi\vert\cos2\hat P\vert\Psi\rangle\in(0,1]$ is the momentum-width factor of the envelope. `code/v2/check_weight.py` confirms (7.7) with $E''=1$ against $-\langle[H,[H,X]]\rangle$ of $H=K+W$ on the ring (section "Code").

### Step 6. Acceleration through relative order $\gamma$

Through order $\gamma$, (6.10) gives $E''_{k,\sigma}(0)=\sigma\frac{\tau^2}{M_k}(1+\epsilon\gamma n)+O(\gamma^2)$, with $\epsilon=-1$ for (a) and $\epsilon=+1$ for (c) ($\Delta''=O(\gamma^2)$). At this order it is exactly of order $\tau^2$. From (7.6), $W'(j)=\sigma M_k\gamma\,\delta N'(j)+M_k^2\gamma^2\Theta'(j)+O(g_s\gamma^3)+O(g_s^2)$. For a place $j$ with $r_j:=(j-x_s)\bmod L\in\{1,\dots,L-1\}$:
- $u'(j)=L-2r_j$;
- $(u^2)'(j)=u'(j)\,[u(j+1)+u(j-1)]=(L-2r_j)(2u(j)-2)$;
- $L^2+2-4u(j)=(L-2r_j)^2+2$.

Hence $\delta N'(j)=\frac{ng_s(L-2r_j)}{Lt_f}$ and $\Theta'(j)=-\frac{ng_s(L-2r_j)[(L-2r_j)^2+2]}{6L^2t_f^2}$.

Every $j$ with $\phi_{j+1}\phi_{j-1}\ne0$ lies on the arc, so $r_j\ne0$ and $L-2r_j=(L-2r_0)-2(j-x_0)$, with $r_0:=(x_0-x_s)\bmod L$. The weights $\phi_{j+1}\phi_{j-1}$ are symmetric about $x_0$, so their odd moments vanish. This gives
- $\sum_j\phi_{j+1}\phi_{j-1}(L-2r_j)=c_\phi(L-2r_0)$;
- $\sum_j\phi_{j+1}\phi_{j-1}(L-2r_j)^3=c_\phi(L-2r_0)[(L-2r_0)^2+12s_\phi^2]$.

Inserting into (7.7), the factors $\sigma^2=1$ and $M_k/M_k$ cancel in the leading term:

$$
a_{k,\sigma}(0)=-\frac{\tau^2\gamma\,ng_s\,(L-2r_0)}{Lt_f}\,c_\phi\Bigl[1+\epsilon\gamma n-\sigma\,\frac{\gamma M_k}{t_f}\,\frac{(L-2r_0)^2+2+12s_\phi^2}{6L}\Bigr]+O(\tau^2g_s\gamma^3)+O(g_s^2)+O(\tau^4\gamma g_s).
\tag{7.8}
$$

- *Leading term.* $-\tau^2\gamma\,c_\phi\,\delta N'(x_0)$ is the same for all levels and both branches, for a common envelope.
- *Relative order $\gamma$.* The universal part is $\epsilon\gamma n$. The part that depends on level and branch is

$$
\Delta_{k,\sigma}(x_0)=-\sigma\,\frac{\gamma M_k}{t_f}\,\frac{(L-2r_0)^2+2+12s_\phi^2}{6L}.
\tag{7.9}
$$

  - It is odd in the branch and proportional to the half-gap $M_k$. Its geometric factor is $\ge1/(3L)$, because $s_\phi^2\ge0$. It is therefore nonzero for every start of the class when $\gamma\ne0$.
  - The packet term $12s_\phi^2$ is fixed by the start. It exceeds the lattice term $2$ as soon as $s_\phi^2>1/6$: for the triangular envelope of half-width $2$, $12s_\phi^2=7.2$ (`check_weight.py`).
  - For $\gamma g_s\ne0$ and off the antipode $L=2r_0$ (where $a=0$ by reflection symmetry), the fall therefore depends on level and branch at relative order $\gamma$, with parameter $\gamma M_k/t_f$.
  - In the class there is no control. Control for the universal leading term: the coupling $\sigma_3\otimes\mathbb 1_N$ of Step 4 gives $a=-\tau^2\gamma\,c_\phi\,\delta N'(x_0)/M_k$, level-dependent already at leading order.

## Result

- **(7.2)** [formula] Class: the held body ($\tau=0$), couplings (a) and (c), $N_f\ge1$, $L\ge5$, every level and branch, lowest adapted field state, (W). The weight difference is exactly $N_f[\varepsilon_0(x)-\varepsilon_0(x')]$, with $\varepsilon_0(x)$ the lowest eigenvalue of one field object with the two held point terms $g_b=\sigma\gamma M_k$ at $x$ and $g_s$ at $x_s$, (7.1).
- **(7.3)** [formula] Same class. At first order in $g_s$ and through second order in $\gamma$, the weight is $-\frac{2ng_sg_b}{t_f}\Lambda^+(x,x_s)+\frac{ng_sg_b^2}{t_f^2}[2G_0\Lambda^+(x,x_s)+\Lambda^+(x,x_s)^2-\frac2L(\Lambda^{+2})(x,x_s)]$ up to an $x$-independent constant, with remainder $O(g_s^2)+O(g_sg_b^3)$ at fixed $L,N_f,t_f$.
- **(7.5)** [formula] Same class and order. On the ring, $W_{k,\sigma}(x)-W_{k,\sigma}(x')=\frac{ng_sg_b}{Lt_f}[(1-\frac{(L^2-2)g_b}{6Lt_f})(u-u')+\frac{g_b}{3Lt_f}(u^2-u'^2)]$, with $u=r(L-r)$; (7.4) gives $\Lambda^+$ and $\Lambda^{+2}$ explicitly.
- **(7.6)** [theorem] Same class and order. The weight is $\sigma M_k\gamma\,\delta N(x)+M_k^2\gamma^2\Theta(x)$.
  - At first order in $\gamma$ it is proportional, for all levels and both branches, to the one function $\gamma\,\delta N(x)$ of (5.8), with the factor $\sigma M_k$ (the eigenvalue of $\sigma_3\otimes(m\mathbb 1_N+D)$, the rest-and-clock cost).
  - Through second order, for $\gamma g_s\ne0$, it is not proportional to one function times a factor. The shape of its place dependence changes at relative order $\gamma$ between the two branches of every level, and between levels with different $M_k$.
  - No member of the class is a control.
- **(7.7)** [formula] Conditions: regime R, both couplings.
  - R1: the band of level $k$ and branch $\sigma$ is isolated, $\delta_{k,\sigma}>0$, with $N_f(|\gamma|(M_k+\tau)+|g_s|)<\delta_{k,\sigma}/2$.
  - The start lies in the band, with a real envelope $\phi$ supported on an arc of at most $L-4$ places that does not contain $x_s$.
  - R2: the acceleration is the coefficient of $\tau^2$, at first order in $g_s$.

  Under these conditions, the drift acceleration from rest at $P=0$ is $a_{k,\sigma}(0)=-E''_{k,\sigma}(0)\sum_j\phi_{j+1}\phi_{j-1}W'_{k,\sigma}(j)$, with the central slope. The remainder is $O(g_s^2)+O(\tau^4\gamma g_s)$.
- **(7.8)** [formula] Conditions: as for (7.7); every start of the class (envelope nonnegative and symmetric about $x_0\ne x_s$, $c_\phi>0$; nonempty for $L\ge7$); first order in $g_s$; through relative order $\gamma$. Then $a_{k,\sigma}(0)=-\frac{\tau^2\gamma ng_s(L-2r_0)}{Lt_f}c_\phi[1+\epsilon\gamma n-\sigma\frac{\gamma M_k}{t_f}\frac{(L-2r_0)^2+2+12s_\phi^2}{6L}]$.
  - $\epsilon=-1$ for (a) and $+1$ for (c).
  - $c_\phi=\langle\cos2\hat P\rangle\in(0,1]$, and $s_\phi^2$ is the variance of the weights $\phi_{j+1}\phi_{j-1}$.
  - The remainder is $O(\tau^2g_s\gamma^3)+O(g_s^2)+O(\tau^4\gamma g_s)$. The $\tau^2$ term describes the acceleration only for $\tau\ll M_k$ and $\tau^2\ll M_k\delta_{k,\sigma}$.
- **(7.9)** [negative] Same conditions.
  - The fall of a body with its own mark is the same for all levels and both branches at leading order: $-\tau^2\gamma\,c_\phi\,\delta N'(x_0)$, for a common envelope.
  - It is not the same at relative order $\gamma$. A level- and branch-dependent term $\Delta_{k,\sigma}=-\sigma\frac{\gamma M_k}{t_f}\frac{(L-2r_0)^2+2+12s_\phi^2}{6L}$ appears. It is of first order in the parameter $\gamma M_k/t_f$, odd in $\sigma$ and proportional to $M_k$. Its geometric factor depends on the start through $s_\phi^2$ and is $\ge1/(3L)$.
  - For $\gamma g_s\ne0$, the level and branch dependence of the fall is present for every member of the class and every start of the class with $L\ne2r_0$. At the antipode $L=2r_0$, $a=0$.

## Consistency checks

1. **Dimensions.** $\tau,t_f,g_s,M_k$ have the unit of the contract, $1/\lambda$; $\gamma$ and $n$ are pure numbers. The prefactor of (7.8) is $\tau^2\times$(pure number), in position labels per $\lambda^2$ as in (5.7), and $\gamma M_k/t_f$ is a pure number. Passed.
2. **Test-body limit against (5.7) with (5.8).** Dropping the own-mark term $\Theta$ in (7.8) leaves $-\tau^2\gamma\,c_\phi\,\delta N'(x_0)(1+\epsilon\gamma n)$. (5.7) with $N_\varphi=n+\delta N$, at first order in $g_s$, gives $-\tau^2\gamma\delta N'/\nu=-\tau^2\gamma\delta N'(1-\gamma n)+O(\gamma^3)$ for (a), and $-\tau^2\nu^2\gamma\delta N'/\nu=-\tau^2\gamma\delta N'(1+\gamma n)$ for (c). The two agree through relative order $\gamma$, up to the common momentum-width factor $c_\phi$, which (5.7) as quoted does not show. Passed.
3. **Limits.** At $\tau\to0$, (7.8) vanishes, as it must for the flat dressed level (6.7). At $g_s=0$ it vanishes by translation invariance. At the antipode $L=2r_0$ it vanishes by the reflection symmetry of $W$ about $x_s$. Passed.

## Open issues

- Regime R is an assumption. The order $\tau^4$ of the acceleration (moving cloud, admixture of the other branch) is not computed.
- The isolation and smallness conditions of R1 are sufficient conditions (norm bounds), not necessary ones.
- (7.8) is derived for envelopes symmetric about a place. For other real envelopes (7.7) still holds, and the centre and the third moment of the weights $\phi_{j+1}\phi_{j-1}$ enter.
- Field states other than the lowest adapted one, and the second order in $g_s$, are not treated (scope).

## Code

- `code/v2/check_weight.py` (check). For $N_f=1$, $t_f=1$, $x_s=0$, on rings $L=8$ and $L=13$, it compares $W(x)-W(x_s)$ from the exact lowest eigenvalue of (7.1) with (7.3), (7.5) and with the first-order term (4.4) alone. Output: `code/v2/output/check_weight.txt`.
  - (7.4) holds to $\le6\times10^{-15}$, and (7.3) and (7.5) agree to $\le6\times10^{-21}$.
  - Grid: $M_k\in\{1,2.5\}$, $\sigma=\pm$, $\gamma\in\{0.002,0.01\}$, $g_s\in\{0.001,0.004\}$. The maximal deviation is $3\times10^{-10}$ to $1\times10^{-7}$, i.e. $5\times10^{-4}$ to $4\times10^{-3}$ of the weight, dominated by the $g_s^2$ remainder.
  - Scaling: the part of the deviation odd in $g_s$ scales as $g_b^{2.99}$, against $g_b^{1.99}$ for (4.4) alone. The part even in $g_s$ scales as $g_s^{2.000}$.
  - The local relative slope correction $[W'(x_0)/W'_{\rm lead}(x_0)-1]/g_b$ matches $-[(L-2r_0)^2+2]/(6Lt_f)$, the $s_\phi=0$ part of (7.9), for every $r_0$, to $\le1.2\times10^{-4}$. The worst case is where $|L-2r_0|=1$ amplifies rounding.
  - Packets, for (7.7) and (7.8). It computes $-\langle[H,[H,X]]\rangle$ for $H=\frac14(1-\cos2\hat P)+W$ ($E''=1$), with $W$ the exact weight, for triangular envelopes of half-width $h=1$ ($L=8$) and $h=1,\dots,4$ ($L=13$), at every allowed $x_0$, with the cut three places beyond the arc.
    - $\langle[K,[K,X]]\rangle=0$ exactly.
    - The leading ratio, which includes $c_\phi$ ($0.167$ to $0.800$), is $1$ to $\le3\times10^{-6}$.
    - The geometric factor matches $-[(L-2r_0)^2+2+12s_\phi^2]/(6Lt_f)$, with $12s_\phi^2$ from $0$ to $26.1$, to $\le1.2\times10^{-4}$.

## Methods used

- Block decomposition by conserved labels; non-interacting identical objects
- Rayleigh–Schrödinger perturbation theory to third order; analytic perturbation theory with Cauchy estimates
- Lattice Green function and its square on the ring (difference equations)
- Hellmann–Feynman theorem
- Isolation of a spectral band by norm bounds and a resolvent contour; Kato's transformation to the held basis
- Heisenberg double commutator for the projected lattice contract; moments of a symmetric envelope
- Numerical exact diagonalization and log-log scaling fits (check)

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