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

# The field made by a held source as a common local scale for a test body

- **Subproject:** 06-ilang-source
- **Package:** 05-made-field-scale
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-19

## Changes from previous version

- Step 1: the bound (5.2) integrates the field fluctuation along the actual trajectory, $\sigma_\Phi(s\mu)$ with $s=\operatorname{sgn}\lambda$ (I5). New paragraph "Reduced view" with (5.15): the first-order fluctuation term vanishes after the field trace, and the view of body and medium follows $\bar C_{\varphi(\lambda)}$ up to a second-order remainder (I1).
- Step 2: explicit extraction rule for the on-site blocks from $\bar C_\varphi$ (on-site operator on branch ⊗ medium, Pauli decomposition, field-independent scalar background) and its invariance under place-dependent relistings between body term and record term (I3).
- S3, S6, Step 4: labels and closing bond of $\ell$; lattice slope and curvature defined; (5.7) carries the curvature term between the slope and the forward difference used by (4.24) (I2, I6).
- Step 5: sign of the ring check corrected ($\Lambda G$ is minus the second difference); weak-source condition (W1) and the remainder $R_N$ stated (I4, I6).
- Step 6: exact forms of the clock ratio and the acceleration; error budget of (5.11)–(5.14) (lapse expansion (W2), remainder of (5.8)) and the conditions under which the signs follow, without (W2) (I4). Range: ring curvature $+1/L$, curvature correction of first order in $g_s$, directional condition $w\ll\lvert\varrho_\parallel\rvert$ on the tori (I6).
- Step 6: new paragraphs "Graph separation and recorded distance" with (5.16): (5.13), (5.14) are in graph coordinates; recorded and graph separation coincide on $\ell$ iff the label separation is at most $L/2$; reversal of "towards" across the closing bond; no recorded transverse separation on the tori (I2, note of 2026-10-19).
- Result, Consistency checks, Open issues, Methods used: updated accordingly.

## Response to verification

- I1: Accepted and fixed in Step 1, paragraph "Reduced view", (5.15). (5.2) is kept as the first-order error of the joint state (the test body's change of the field); the view of body and medium has no first-order error, and its remainder is second order in $\gamma$, with its range in $\lambda$.
- I2: Accepted and fixed in S3 and Step 6, paragraphs "Graph separation and recorded distance", "Ring", "Tori", (5.16): (5.13), (5.14) are stated in graph coordinates; the recorded geometry on $\ell$, the closing bond and the missing transverse record are stated, together with where "towards the source" holds in the recorded geometry.
- I3: Accepted and fixed in Step 2, paragraph "Extraction rule" (invariance under $C_{\rm D}\to C_{\rm D}+\sum_xf_x\Pi_x$, $K_x\to K_x-f_x\mathbb 1_c$ shown).
- I4: Accepted and fixed in Step 5 ((W1), $R_N$) and Step 6 ((W2), error terms of (5.11), (5.12), sign conditions).
- I5: Accepted and fixed in Step 1, (5.2).
- I6: Accepted and fixed in S6, Step 4 ((5.7)), Step 5 (check) and Step 6, paragraph "Range of the acceleration".

## Setup and assumptions

**Inputs** (as quoted in question.md).
- (2.1) of 02-what-a-body-presents@v3: $C_{bf}=\gamma\sum_xQ_x\otimes\Phi_x$, $Q^{(a)}_x=\Pi_xO$, $Q^{(c)}_x=\Pi_xO+\tfrac12(A_x+A_{x-1})$, with $\Pi_x=\lvert x\rangle\langle x\rvert\otimes\mathbb 1_2\otimes\mathbb 1_N$, $O=\sigma_3\otimes(m\mathbb 1_N+D)$, $A_x$ the bond $(x,x+1)$ of the motion term, $A=\sum_xA_x$, $B=\sum_x\Pi_xO$.
- (3.6), (3.9) of 03-held-sources@v4: $\chi_n(x,y)=\sum_{m:\,e_m\neq e_n}2\operatorname{Re}\bigl(\langle\varphi_n\vert\Phi_x\vert\varphi_m\rangle\langle\varphi_m\vert\Phi_y\vert\varphi_n\rangle\bigr)/(e_m-e_n)$ and $\langle\varphi_n(g)\vert\Phi_y\vert\varphi_n(g)\rangle=\bar\Phi_n(y)-\sum_ig_i\chi_n(y,x_i)+O(\lVert\Phi_y\rVert v^2/\Delta_n^2)$.
- (4.4) of 04-profile@v4: $M^{(2)}(x_1,x_2)=-\frac{2ng_1g_2}{t_f}\Lambda^+(x_1,x_2)$, $\Lambda^+(x,y)=\lvert X_f\rvert^{-1}\sum_{\mathbf k\neq0}e^{i\mathbf k\cdot(x-y)}/\omega(\mathbf k)$, $\omega(\mathbf k)=\sum_{\mu=1}^{D}4\sin^2\frac{k_\mu}2$.
- (17.11) of 05-ilang-free-fall/17-typical-or-forced@v3: for linear level profiles $M_{x,k}=M_k+G_kx$, $V_{x,k}=-\mathcal F_kx$: R3 $\iff G_k=\gamma'M_k,\ \mathcal F_k=\phi M_k\ \forall k\iff\boldsymbol g\in\mathcal U_4^{\pm}$ (fall from rest level independent on both branches). Here $\gamma'$ is the $\gamma$ of (17.11), not the coupling constant $\gamma$.
- (4.16), (4.24) of 05-ilang-free-fall/04-branch-fall@v3: $a_{k,\sigma}(0)=\frac{\tau_k^2}{M_k}(\sigma\mathcal F_k-G_k)$; $\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\big\vert_{\rm drift}=d_0\,a_{k,+}(0)\,\hat e=-\frac{v_{\max}^2}{d_0}\frac{r(x+1)-r(x)}{r(x)}\hat e$, with $v_{\max}=\tau d_0$, $d_0=\kappa\sigma_X$, $r(x)$ the clock rate at $x$.

**Assumptions** (the description chosen for study, M8, M9).
- **S1 Field.** $N_f\ge1$ identical field objects ($c=+1$), no pair terms among them; places $X_f$: a ring of $L\ge5$ places ($D=1$), an $L\times L$ torus ($D=2$) or an $L^3$ torus ($D=3$); own term $C_1=-t_f\sum_{\{x,y\}}(\lvert x\rangle\langle y\rvert+\lvert y\rangle\langle x\rvert)$, $t_f>0$; $\Phi_x$ the number of field objects at $x$; $n=N_f/\lvert X_f\rvert$. With the source the field operator is $C_f^s:=\sum_{j=1}^{N_f}C_1^{(j)}+g_s\Phi_{x_s}$ on the symmetric subspace $\mathcal H_f$.
- **S2 Source.** The held source is in a common eigenvector of its own term and of its coupled operator (eigenvalue $g_s$). $C$ then keeps the source in that vector, and the source enters only through $g_s\Phi_{x_s}$ and a constant; it is suppressed below.
- **S3 Test body.** Model D with positions $x\in\ell$, a line of $X_f$ (the whole ring for $D=1$; on the tori a line along a lattice direction $\hat e_\ell$, a cycle of $L$ places), labelled $x=0,\dots,L-1$ in the record $K_x=\kappa\,x\,X$ of a medium $c$; the bonds of the motion term are those of $\ell$, the closing bond $(L-1,0)$ included. $C_{bc}$ is the part of $C$ not involving the field: $C_{\rm D}=A+B$, the record term and the medium's own term $C_c$. Coupling (2.1), form (a) or (c). $d_k=2\pi k/(N\lambda_0)$, $M_k=m+d_k$, $\lVert O\rVert=M_{N-1}$.
- **S4 Start.** $\Psi(0)=\xi\otimes\varphi$ with any body–medium state $\xi\in\mathcal H_{bc}=\mathcal H_b\otimes\mathcal H_c$ and a field state $\varphi$; $\varphi(\lambda):=e^{-iC_f^s\lambda}\varphi$. Items 2–3 use a stationary $\varphi$ (an eigenvector of $C_f^s$), item 4 the lowest adapted state (Step 5).
- **S5 Positivity** (items 3–4). $\nu(x):=1+\gamma N_\varphi(x)>0$ on $\ell$, and $1+\gamma n>0$.
- **S6 Regime of item 3.** The conditions of (4.16) of 05-ilang-free-fall/04-branch-fall (branch start at rest, first order in the gradients, record background with one step $d_0\hat e$ per bond), so the support of the start (width $w$ around $x_0$) does not contain the closing bond of $\ell$ (Step 6). Local-slope approximation: with the lattice slope $N'(x):=\tfrac12\bigl(N_\varphi(x+1)-N_\varphi(x-1)\bigr)$ and curvature $N''(x):=N_\varphi(x+1)+N_\varphi(x-1)-2N_\varphi(x)$, $N_\varphi(x)=N_\varphi(x_0)+N'(x_0)(x-x_0)$ on the support up to the curvature term, with $\lvert N''\rvert w\ll\lvert N'(x_0)\rvert$ on the support.

Notation: $N_\lambda(x):=\langle\varphi(\lambda)\vert\Phi_x\vert\varphi(\lambda)\rangle$, $\delta\Phi_x:=\Phi_x-N_\lambda(x)$, $\sigma_\Phi(\lambda):=\max_{x\in\ell}\bigl(\operatorname{Var}_{\varphi(\lambda)}\Phi_x\bigr)^{1/2}$. For stationary $\varphi$, $N_\lambda=N_\varphi$ and $\sigma_\Phi$ are constant.

## Derivation

### Step 1. The operator by which the test body evolves

By S2, on $\mathcal H_{bc}\otimes\mathcal H_f$ and up to a constant (a common phase, (1.2)), $C=C_{bc}\otimes\mathbb 1+\mathbb 1\otimes C_f^s+\gamma\sum_{x\in\ell}Q_x\otimes\Phi_x$. For a unit vector $\varphi$ define the partial expectation $\bar C_\varphi$ on $\mathcal H_{bc}$ by $\langle\xi'\vert\bar C_\varphi\vert\xi\rangle:=\langle\xi'\otimes\varphi\vert C\vert\xi\otimes\varphi\rangle$. Inserting $C$:

$$
\bar C_\varphi=C_{bc}+\gamma\sum_{x\in\ell}N_\varphi(x)\,Q_x+E_\varphi\mathbb 1,\qquad N_\varphi(x)=\langle\varphi\vert\Phi_x\vert\varphi\rangle,\quad E_\varphi=\langle\varphi\vert C_f^s\vert\varphi\rangle .
\tag{5.1}
$$

Let $\xi(\lambda)$ solve $i\,\mathrm d\xi/\mathrm d\lambda=(\bar C_{\varphi(\lambda)}-E_\varphi)\xi$ ($E_{\varphi(\lambda)}=E_\varphi$ is conserved) and $\Psi_{\rm eff}:=\xi(\lambda)\otimes\varphi(\lambda)$. The product rule gives $i\dot\Psi_{\rm eff}=C\Psi_{\rm eff}-\gamma W_\lambda\Psi_{\rm eff}$ with $W_\lambda:=\sum_xQ_x\otimes\delta\Phi_x$. Hence $D:=\Psi-\Psi_{\rm eff}$ obeys $i\dot D=CD+\gamma W_\lambda\Psi_{\rm eff}$, $D(0)=0$, and Duhamel's formula with the unitarity of $e^{-iC\lambda}$ gives $\lVert D(\lambda)\rVert\le\lvert\gamma\rvert\int_0^{\lvert\lambda\rvert}\lVert W_{s\mu}\Psi_{\rm eff}(s\mu)\rVert\,\mathrm d\mu$ along the actual trajectory, $s:=\operatorname{sgn}\lambda$. Write $W\Psi_{\rm eff}=\sum_xv_x$, $v_x=Q_x\xi\otimes\delta\Phi_x\varphi$, $\lVert\delta\Phi_x\varphi\rVert^2=\operatorname{Var}\Phi_x$.
- (a): $Q_xQ_y=\delta_{xy}\Pi_xO^2$, so $\lVert\sum_xv_x\rVert^2=\sum_x\langle\xi\vert\Pi_xO^2\vert\xi\rangle\operatorname{Var}\Phi_x\le\lVert O\rVert^2\sigma_\Phi^2$.
- (c): $Q_x\xi$ lies on the places $x-1,x,x+1$, so $\langle v_x,v_y\rangle=0$ for $\lvert x-y\rvert\ge3$ and $\lVert\sum_xv_x\rVert^2\le5\sum_x\lVert v_x\rVert^2$. With $\lVert A_x\rVert=\tau/2$, $\lVert Q_x\xi\rVert\le(\lVert O\rVert+\frac\tau2)\lVert(\Pi_{x-1}+\Pi_x+\Pi_{x+1})\xi\rVert$, whose squares sum to $3$.

These bounds hold for every unit $\xi$, so $\lVert W_\lambda(\chi\otimes\varphi(\lambda))\rVert\le c_Q\sigma_\Phi(\lambda)$ for every unit $\chi\in\mathcal H_{bc}$. Therefore

$$
\bigl\lVert\Psi(\lambda)-\xi(\lambda)\otimes\varphi(\lambda)\bigr\rVert\le\delta(\lambda):=\lvert\gamma\rvert\,c_Q\int_0^{\lvert\lambda\rvert}\sigma_\Phi(s\mu)\,\mathrm d\mu,\qquad c_{Q^{(a)}}=\lVert O\rVert,\quad c_{Q^{(c)}}=\sqrt{15}\,\bigl(\lVert O\rVert+\tfrac\tau2\bigr).
\tag{5.2}
$$

For stationary $\varphi$, $\xi(\lambda)=e^{-i(\bar C_\varphi-E_\varphi)\lambda}\xi$ and $\delta(\lambda)=\lvert\gamma\rvert c_Q\sigma_\Phi\lvert\lambda\rvert$. The neglected term $\gamma W$ is the change of the field by the test body (A15). (5.2) bounds the joint state; its error is first order in $\gamma$, like the retained term $\gamma\sum_xN_\varphi(x)Q_x$, so (5.2) alone does not fix the body's evolution at first order. The view of body and medium does.

*Reduced view.* Let $\rho(\lambda):=\operatorname{Tr}_f\lvert\Psi(\lambda)\rangle\langle\Psi(\lambda)\rvert$, the view (1.5) of body and medium, and $H_\lambda:=\bar C_{\varphi(\lambda)}-E_\varphi$. In $\operatorname{Tr}_f$ of $i\,\mathrm d\lvert\Psi\rangle\langle\Psi\rvert/\mathrm d\lambda=[C,\lvert\Psi\rangle\langle\Psi\rvert]$ the field's own term drops ($\operatorname{Tr}_f[\mathbb 1\otimes X,Y]=0$ for field operators $X$), and $\Phi_x=N_\lambda(x)+\delta\Phi_x$ gives $i\,\mathrm d\rho/\mathrm d\lambda=[H_\lambda,\rho]+\gamma(G-G^\dagger)$ with $G:=\operatorname{Tr}_f\bigl(W_\lambda\lvert\Psi\rangle\langle\Psi\rvert\bigr)$. $\lvert\xi\rangle\langle\xi\rvert$ obeys the same equation without $G$. With $U$ the unitary evolution of $H_\lambda$, $\Delta:=\rho-\lvert\xi\rangle\langle\xi\rvert=-i\gamma\int_0^\lambda U(\lambda,\mu)(G-G^\dagger)(\mu)U(\lambda,\mu)^\dagger\,\mathrm d\mu$, so $\lVert\Delta(\lambda)\rVert_1\le2\lvert\gamma\rvert\int_0^{\lvert\lambda\rvert}\lVert G(s\mu)\rVert_1\,\mathrm d\mu$. Split $\lvert\Psi\rangle\langle\Psi\rvert=\lvert\Psi\rangle\langle\Psi_{\rm eff}\rvert+\lvert\Psi_{\rm eff}\rangle\langle D\rvert+\lvert D\rangle\langle D\rvert$; the partial trace does not increase the trace norm.
- $\operatorname{Tr}_f\bigl(W\lvert\Psi\rangle\langle\xi\otimes\varphi\rvert\bigr)=\lvert(\mathbb 1\otimes\langle\varphi\rvert)W\Psi\rangle\langle\xi\rvert$, and $(\mathbb 1\otimes\langle\varphi\rvert)W\Psi_{\rm eff}=\sum_xQ_x\xi\,\langle\varphi\vert\delta\Phi_x\vert\varphi\rangle=0$: the first-order term vanishes after the field trace. What remains is $\lvert(\mathbb 1\otimes\langle\varphi\rvert)WD\rangle\langle\xi\rvert$, of trace norm $\sup_{\lVert\chi\rVert=1}\lvert\langle W(\chi\otimes\varphi)\vert D\rangle\rvert\le c_Q\sigma_\Phi\lVert D\rVert$.
- $\bigl\lVert\operatorname{Tr}_f\bigl(W\lvert\Psi_{\rm eff}\rangle\langle D\rvert\bigr)\bigr\rVert_1\le\lVert W\Psi_{\rm eff}\rVert\,\lVert D\rVert\le c_Q\sigma_\Phi\lVert D\rVert$.
- $\bigl\lVert\operatorname{Tr}_f\bigl(W\lvert D\rangle\langle D\rvert\bigr)\bigr\rVert_1\le\lVert W\rVert\,\lVert D\rVert^2\le c_W\phi_{\max}\lVert D\rVert^2$, with $\phi_{\max}(\lambda):=\max_{x\in\ell}\lVert\Phi_x-N_\lambda(x)\rVert\le N_f$, $c_{W^{(a)}}=\lVert O\rVert$ ($W^{(a)}$ is block diagonal in the places) and $c_{W^{(c)}}=\lVert O\rVert+\tfrac{3\tau}2$ (the bond terms $A_x\otimes\tfrac12(\delta\Phi_x+\delta\Phi_{x+1})$ fall into at most three classes of disjoint bonds, each of norm $\le\tfrac\tau2\phi_{\max}$).

With $\lVert D(s\mu)\rVert\le\delta(s\mu)$ and $\mathrm d\delta(s\mu)/\mathrm d\mu=\lvert\gamma\rvert c_Q\sigma_\Phi(s\mu)$, the first two terms integrate to $2\delta(\lambda)^2$:

$$
\bigl\lVert\rho(\lambda)-\lvert\xi(\lambda)\rangle\langle\xi(\lambda)\rvert\bigr\rVert_1\le2\delta(\lambda)^2+2\lvert\gamma\rvert c_W\int_0^{\lvert\lambda\rvert}\phi_{\max}(s\mu)\,\delta(s\mu)^2\,\mathrm d\mu\le2\delta(\lambda)^2\bigl(1+\lvert\gamma\rvert c_W\hat\phi(\lambda)\lvert\lambda\rvert\bigr),\qquad\hat\phi(\lambda):=\sup_{0\le\mu\le\lvert\lambda\rvert}\phi_{\max}(s\mu).
\tag{5.15}
$$

So the view of body and medium follows $\bar C_{\varphi(\lambda)}$ with no first-order error; the remainder is second order in $\gamma$, and every reading probability of body and medium differs by at most half of it. It is small for $\delta(\lambda)\ll1$ and $\lvert\gamma\rvert c_W\hat\phi\lvert\lambda\rvert\,\delta(\lambda)^2\ll1$; for stationary $\varphi$ this is $\lvert\lambda\rvert\ll(\lvert\gamma\rvert c_Q\sigma_\Phi)^{-1}$ and $\lvert\gamma\rvert^3c_Wc_Q^2\phi_{\max}\sigma_\Phi^2\lvert\lambda\rvert^3\ll1$. The second-order terms themselves (among them the test body's own mark, 06-own-mark) are not computed.

*M6.* $\bar C_\varphi$ and the view $\rho$ are computed from $C$ and the state alone. It is therefore unchanged by every relisting that leaves $C$ unchanged, e.g. $\Phi_x\to\Phi_x-c_x$ together with $C_{\rm D}\to C_{\rm D}+\gamma\sum_xc_xQ_x$, place-dependent $c_x$ included. It does not depend on the labels of the field objects ($\Phi_x$ is symmetric) or on the phase of $\varphi$, and its multiple of $\mathbb 1$ is a common phase. The split of $\bar C_\varphi$ into $C_{bc}$ and $\gamma\sum_xN_\varphi(x)Q_x$ is not fixed: the separate values of the bare blocks, of $\gamma$ and of $N_\varphi(x)$ depend on the listing.

### Step 2. Level blocks

*Extraction rule.* The position, branch and clock places of the body belong to its type (A2), and the clock levels $\lvert f_k\rangle$ are built from the clock places. Every term of $\bar C_\varphi$ acts on the clock label through $\mathbb 1_N$ or $D$, so $\bar C_\varphi$ is block diagonal in the levels. Its level-$k$ block between the places $x'$ and $x$, an operator on branch ⊗ medium, is therefore fixed by $C$ and $\varphi$: for $x'=x$ the on-site operator $\mathfrak B_{x,k}$, for $x'=x+1$ the hopping. The Pauli decomposition on the branch label, $\mathfrak B_{x,k}=\sigma_3\otimes\hat M_{x,k}+\mathbb 1_2\otimes\hat V_{x,k}+\sigma_1\otimes(\cdot)+\sigma_2\otimes(\cdot)$, is unique. From (5.1), since $A$ is off-diagonal in the places, $\hat M_{x,k}=\nu(x)M_k\mathbb 1_c$, $\hat V_{x,k}=K_x+C_c+E_\varphi$, and there is no $\sigma_1,\sigma_2$ part. No field coupling has a $\mathbb 1_2$ part on the branch label, so $\hat V_{x,k}$ does not depend on $k$, and depends on $\varphi$ only through the constant $E_\varphi$. Hence $\hat V^{(0)}_x:=\hat V_{x,k}-E_\varphi$ is a field-independent operator on the medium, fixed by $C$ up to a constant: the record background, $K_x+C_c$ in the listing of the setting, in which (4.16), (4.24) and R3 are stated. The effective on-site block is $B_{x,k}:=\mathfrak B_{x,k}-\mathbb 1_2\otimes\hat V^{(0)}_x$, a multiple of $\mathbb 1_c$ and thus a $2\times2$ block. The relisting $C_{\rm D}\to C_{\rm D}+\sum_xf_x\Pi_x$, $K_x\to K_x-f_x\mathbb 1_c$ with real place-dependent $f_x$ leaves $C$, $\bar C_\varphi$, $\mathfrak B_{x,k}$ and $\hat V^{(0)}_x$ unchanged, hence $B_{x,k}$; it only renames parts of $\hat V^{(0)}_x$. So does $\Phi_x\to\Phi_x-c_x$ with $C_{\rm D}\to C_{\rm D}+\gamma\sum_xc_xQ_x$, which changes $C_{bc}$ but adds no $\mathbb 1_2$ part.

*Blocks.* Let $\hat\nu:=\sum_x\nu(x)\Pi_x$ with $\nu(x)=1+\gamma N_\varphi(x)$. On level $k$, $O=M_k\sigma_3$. For (a), (5.1) gives $\gamma\sum_xN_\varphi(x)\Pi_xO=(\hat\nu-1)B$. For (c) it gives in addition $\gamma\sum_xN_\varphi(x)\tfrac12(A_x+A_{x-1})=\gamma\sum_x\tfrac12(N_\varphi(x)+N_\varphi(x+1))A_x$. Since $A_x$ connects only $x$ and $x+1$, $\tfrac12\{\hat\nu,A_x\}=\tfrac12(\nu(x)+\nu(x+1))A_x$. With the extraction rule, the level-$k$ blocks are

$$
\begin{aligned}
&B_{x,k}=M_{x,k}\sigma_3+V_{x,k}\mathbb 1_2,\qquad M_{x,k}=\nu(x)\,M_k,\qquad V_{x,k}=E_\varphi,\\
&\text{hopping on }(x,x+1):\ \tfrac{\tau_{x,x+1}}2(-i\sigma_1),\qquad \tau^{(a)}_{x,x+1}=\tau,\qquad \tau^{(c)}_{x,x+1}=\tau\,\tfrac12\bigl(\nu(x)+\nu(x+1)\bigr),
\end{aligned}
\tag{5.3}
$$

i.e. the Model D part of $\bar C_\varphi$ is $\hat\nu B+A$ for (a) and $\tfrac12\{\hat\nu,C_{\rm D}\}$ for (c). Fixed by $C$ and $\varphi$ (extraction rule): $B_{x,k}$ and the hopping amplitudes, i.e. $M_{x,k}$, the hopping amplitudes and $V_{x,k}$ up to the common constant. Hence the level-independent ratios $\nu(y)/\nu(x)=M_{y,k}/M_{x,k}$ are fixed. Listing-dependent: the factorization $M_{x,k}=\nu(x)M_k$ (a constant relisting $c_x=c$ multiplies $M_k$, and $\tau$ for (c), by $1+\gamma c$ and divides $\nu$ by it) and the place-independent constant $E_\varphi$ in $V_{x,k}$.

### Step 3. R3 place by place, the hopping, and a control

From (5.3), for both couplings and every $\varphi$ (at every $\lambda$ if $\varphi$ is not stationary):

$$
B_{x+1,k}-B_{x,k}=M_k\,\Gamma(x),\qquad \Gamma(x)=\gamma\bigl(N_\varphi(x+1)-N_\varphi(x)\bigr)\sigma_3 ;\qquad\text{for }\nu(x)\neq0:\ B_{x+1,k}-B_{x,k}=M_{x,k}\,\frac{\nu(x+1)-\nu(x)}{\nu(x)}\,\sigma_3 .
\tag{5.4}
$$

$\Gamma(x)$ does not depend on $k$, so R3 holds at every place. It has no $\mathbb 1_2$ part ($\phi=0$ in the parametrization of (17.11) of 05-ilang-free-fall/17-typical-or-forced). The second form uses only fixed quantities.

*Hopping.* (a): the hopping is $\tau$ at every bond, while the on-site blocks change by $\nu(x+1)/\nu(x)$. The two factors differ at every place with $\gamma(N_\varphi(x+1)-N_\varphi(x))\neq0$. (c): by (5.3), every matrix element of the Model D part of $\bar C_\varphi$ between places $x,x'$ ($x'=x$ or a neighbour) equals that of $C_{\rm D}$ times $\tfrac12(\nu(x)+\nu(x'))$. The hopping of every level thus carries the same local factor as the on-site blocks, evaluated as the mean over the two ends of the bond.

*Control.* Take the pair term of the same form with $Q^{(r)}_x=\Pi_x\,\sigma_3\otimes m\mathbb 1_N$ (rest term only). Then $B^{(r)}_{x,k}=(M_k+\gamma mN_\varphi(x))\sigma_3+E_\varphi\mathbb 1_2$. R3 at $x$ would require $M_k\Gamma=\gamma m\Delta N\sigma_3$ for $k=0,1$, $\Delta N:=N_\varphi(x+1)-N_\varphi(x)$. With $M_0=m$ this forces $\Gamma=\gamma\Delta N\sigma_3$ and then $d_1\gamma\Delta N=0$, where $d_1=2\pi/(N\lambda_0)>0$. Under (4.16) of 05-ilang-free-fall/04-branch-fall ($G_k=\gamma mN'$, $\mathcal F_k=0$, $\tau_k=\tau$):

$$
B^{(r)}_{x+1,k}-B^{(r)}_{x,k}=\gamma m\,\Delta N\,\sigma_3,\qquad a^{(r)}_{k,\sigma}(0)=-\tau^2\,\frac{\gamma m\,N'(x_0)}{M_k+\gamma mN_\varphi(x_0)} .
\tag{5.5}
$$

So R3 fails at every place with $\gamma\Delta N\neq0$, and there the fall from rest depends on the level.

### Step 4. Clock rate and fall in a stationary field state

The on-site eigenvalues are $E_\varphi\pm\nu(x)M_k$. By S5 the upper one is $E_\varphi+\nu(x)M_k$, and its spacing between levels $k,l$ is $\nu(x)(d_l-d_k)$. The clock-rate ratio of two places is therefore

$$
\frac{r(y)}{r(x)}=\frac{\nu(y)}{\nu(x)}=\frac{M_{y,k}}{M_{x,k}},
\tag{5.6}
$$

the same for every pair of levels. It is exact in $\bar C_\varphi$ (no gradient expansion), and the lower branch gives the same ratio.

*Fall.* Under S6 the profiles across the start are those of 05-ilang-free-fall with base half-gap $\nu(x_0)M_k$, $G_k=\gamma N'(x_0)M_k$, $\mathcal F_k=0$, and hopping $\tau_k=\tau_{\rm loc}(x_0)$: $\tau$ for (a), $\tau\nu(x_0)$ for (c) up to first order in the gradient. The variation of the hopping across the start (c) does not enter at first order. In the band energy of a branch, $E_\sigma(x,p)=E_\varphi+\sigma\sqrt{M_{x,k}^2+\tau_{\rm loc}(x)^2\sin^2p}$, the hopping multiplies $\sin^2p$, whose $x$-derivative vanishes at rest. With (4.16) of 05-ilang-free-fall/04-branch-fall and (5.6):

$$
a_{k,\sigma}(0)=-\tau_{\rm loc}(x_0)^2\,\frac{\gamma N'(x_0)}{\nu(x_0)}=-\tau_{\rm loc}(x_0)^2\Bigl[\frac{r(x_0+1)-r(x_0)}{r(x_0)}-\frac{\gamma N''(x_0)}{2\nu(x_0)}\Bigr],\qquad v_{\max}(x_0):=\tau_{\rm loc}(x_0)\,d_0 .
\tag{5.7}
$$

The second equality is exact: by (5.6), $(r(x_0+1)-r(x_0))/r(x_0)=\gamma\bigl(N_\varphi(x_0+1)-N_\varphi(x_0)\bigr)/\nu(x_0)$, and $N_\varphi(x_0+1)-N_\varphi(x_0)=N'(x_0)+\tfrac12N''(x_0)$. $a_{k,\sigma}$ is the same for all levels and both branches, as (17.11) of 05-ilang-free-fall/17-typical-or-forced requires for R3 with $\phi=0$. Multiplying by $d_0$: (4.24) holds with the maximal speed $v_{\max}(x_0)$ of the effective contract at the start, up to the curvature term, i.e. to relative error $\lvert N''(x_0)\rvert/(2\lvert N'(x_0)\rvert)$, which is $\ll1/w$ by S6. For (a), $v_{\max}(x_0)=\tau d_0$ at every place. For (c), $v_{\max}(x_0)=\tau\nu(x_0)d_0$ changes from place to place by the factor of $r$; with the value at a fixed reference place $y$ instead, (4.24) acquires the factor $(r(x_0)/r(y))^2$. $\tau_{\rm loc}$ and $\gamma N'/\nu=(M_{x_0+1,k}-M_{x_0-1,k})/(2M_{x_0,k})$ are fixed by $C$ and $\varphi$.

### Step 5. The made field in the lowest adapted state

Without pair terms among the field objects, $C_f^s=\sum_jh^{(j)}$ with $h=C_1+g_s\lvert x_s\rangle\langle x_s\rvert$. $h$ has off-diagonal entries $-t_f\le0$ on a connected graph, so by Perron–Frobenius its lowest eigenvector $u_g$ is unique and positive, for every real $g_s$. The lowest state of $C_f^s$ is then $u_g^{\otimes N_f}$, which is symmetric and unique. In it the $N_f$ objects are independent: $N(y)=N_fu_g(y)^2$ and $\operatorname{Var}\Phi_y=N_fu_g(y)^2(1-u_g(y)^2)\le N(y)$. In (5.2), $\sigma_\Phi^2\le\max_yN(y)$.

Apply (3.9) to one field object ($\Phi_y\to\lvert y\rangle\langle y\rvert$, $v=\lvert g_s\rvert$). The eigenvectors of $C_1$ are the plane waves $\lvert X_f\rvert^{-1/2}e^{i\mathbf k\cdot x}$ with eigenvalues $-2Dt_f+t_f\omega(\mathbf k)$; the lowest is uniform, with gap $t_f\omega_{\min}$, $\omega_{\min}=4\sin^2(\pi/L)$. In (3.6) each term is $\lvert X_f\rvert^{-2}e^{i\mathbf k\cdot(y-x_s)}/(t_f\omega(\mathbf k))$, so $\chi_0=\frac{2}{\lvert X_f\rvert t_f}\Lambda^+$. Multiplying by $N_f$:

$$
N(y)=n-\frac{2n\,g_s}{t_f}\,\Lambda^+(y,x_s)+O\Bigl(\frac{N_fg_s^2}{t_f^2\omega_{\min}^2}\Bigr),\qquad \delta N(y):=N(y)-n,
\tag{5.8}
$$

with the kernel $\frac{2n}{t_f}\Lambda^+$ of (4.4). (3.9) is used with $v=\lvert g_s\rvert$ and the gap $\Delta_0=t_f\omega_{\min}$, under **(W1)** $\lvert g_s\rvert\ll t_f\omega_{\min}$, with $\omega_{\min}\approx4\pi^2/L^2$ for $L\gg1$: at fixed $g_s$ it fails as $L$ grows, and the large-$L$ forms below are used at finite $L$ within (W1). $R_N\lesssim N_fg_s^2/(t_f\omega_{\min})^2$ denotes the remainder of (5.8) at one place. On the ring, with $\varrho\in[0,L/2]$ the ring distance from $x_s$,

$$
\Lambda^+(x,x_s)=\frac{L^2-1}{12L}-\frac{\varrho(L-\varrho)}{2L}.
\tag{5.9}
$$

Check: for $G:=$ (5.9), $(\Lambda G)(x)=2G(x)-G(x+1)-G(x-1)$ is $-1/L$ for $\varrho\neq0$ (the second difference of $-\varrho(L-\varrho)/(2L)$ is $+1/L$) and $1-1/L$ at $\varrho=0$, so $\Lambda G=\delta_{x_s}-1/L$; the constant makes $\sum_xG=0$, which fixes $\Lambda^+\delta_{x_s}$. On the tori, with $\vec\varrho=x-x_s$ (nearest image) and $\varrho=\lvert\vec\varrho\rvert$, the standard asymptotics of the lattice Green function give, for $1\ll\varrho\ll L$,

$$
\Lambda^+(x,x_s)=c_L-\frac{\ln\varrho}{2\pi}+O(\varrho^{-2})+O\Bigl(\frac{\varrho^2}{L^2}\Bigr)\ (D=2),\qquad \Lambda^+(x,x_s)=\frac{1}{4\pi\varrho}-c_L'+O(\varrho^{-3})+O\Bigl(\frac{\varrho^2}{L^3}\Bigr)\ (D=3),
\tag{5.10}
$$

with constants $c_L,c'_L$ and gradients given by the term-by-term derivatives.

### Step 6. Dependence on the place relative to the source

By (5.6) and (5.7), exactly, $r(y)/r(x)-1=\gamma(N(y)-N(x))/\nu(x)$ and $a_{k,\sigma}(0)=-\tau_{\rm loc}(x_0)^2\gamma N'(x_0)/\nu(x_0)$. Let $\bar\gamma:=\gamma/(1+\gamma n)$, which has the sign of $\gamma$ by S5, and $\tau_0:=\tau$ (a), $\tau_0:=\tau(1+\gamma n)$ (c). Then $\nu(x)=(1+\gamma n)(1+\bar\gamma\,\delta N(x))$, $\tau_{\rm loc}(x_0)=\tau_0(1+\bar\gamma\,\delta N(x_0))$ for (c), and exactly

$$
\frac{r(y)}{r(x)}-1=\frac{\bar\gamma\bigl(\delta N(y)-\delta N(x)\bigr)}{1+\bar\gamma\,\delta N(x)},\qquad a_{k,\sigma}(0)=-\tau_0^2\,\bar\gamma\,N'(x_0)\times\begin{cases}\bigl(1+\bar\gamma\,\delta N(x_0)\bigr)^{-1}&\text{(a)},\\ 1+\bar\gamma\,\delta N(x_0)&\text{(c)}.\end{cases}
$$

Let $\varepsilon_\nu:=\lvert\bar\gamma\rvert\max_{y\in\ell}\lvert\delta N(y)\rvert$. The expansion needs **(W2)** $\varepsilon_\nu\ll1$, which for $\delta N\not\equiv0$ fails as $1+\gamma n\to0$, where $\bar\gamma$ diverges; on the ring, by (5.9), $\varepsilon_\nu\le\lvert\bar\gamma\rvert\bigl(\frac{n\lvert g_s\rvert}{t_f}\frac{L^2-1}{6L}+R_N\bigr)$. Expanding in $\varepsilon_\nu$ and inserting (5.8), with $\partial_\ell$ the central difference along $\ell$ (as $N'$ in S6):

$$
\frac{r(y)}{r(x)}=1+\frac{2n\bar\gamma g_s}{t_f}\bigl[\Lambda^+(x,x_s)-\Lambda^+(y,x_s)\bigr]\bigl(1+O(\varepsilon_\nu)\bigr)+O\bigl(\lvert\bar\gamma\rvert R_N\bigr),
\tag{5.11}
$$

$$
a_{k,\sigma}(0)\big\vert_{x_0}=\frac{2n\bar\gamma g_s}{t_f}\,\tau_0^2\;\partial_\ell\Lambda^+(x_0,x_s)\bigl(1+O(\varepsilon_\nu)\bigr)+O\bigl(\tau_0^2\lvert\bar\gamma\rvert R_N\bigr).
\tag{5.12}
$$

The leading terms are of first order in $g_s$; the errors are the lapse expansion (relative, $O(\varepsilon_\nu)$) and the remainder of (5.8) (absolute). With (5.9) on the ring ($\varrho_x$ the ring distance of $x$ from $x_s$, a graph separation; $a$ counted positive away from the source along the shorter arc), the leading terms are

$$
\frac{r(y)}{r(x)}=1+\frac{n\bar\gamma g_s}{t_fL}\bigl[\varrho_y(L-\varrho_y)-\varrho_x(L-\varrho_x)\bigr],\qquad a_{k,\sigma}(0)=-\frac{n\bar\gamma g_s}{t_f}\,\tau_0^2\Bigl(1-\frac{2\varrho}{L}\Bigr).
\tag{5.13}
$$

With (5.10) on the tori, to leading order in $1\ll\varrho\ll L$ ($\vec\varrho=x-x_s$ the nearest-image coordinate difference, $\varrho$ its Euclidean norm, not the $\ell^1$ graph distance; $\varrho_\parallel:=\hat e_\ell\cdot\vec\varrho$):

$$
\frac{r(y)}{r(x)}=1+\frac{n\bar\gamma g_s}{\pi t_f}\ln\frac{\varrho_y}{\varrho_x},\quad a=-\frac{n\bar\gamma g_s\tau_0^2}{\pi t_f}\frac{\hat e_\ell\cdot\vec\varrho}{\varrho^2}\ (D=2);\qquad \frac{r(y)}{r(x)}=1+\frac{n\bar\gamma g_s}{2\pi t_f}\Bigl(\frac1{\varrho_x}-\frac1{\varrho_y}\Bigr),\quad a=-\frac{n\bar\gamma g_s\tau_0^2}{2\pi t_f}\frac{\hat e_\ell\cdot\vec\varrho}{\varrho^3}\ (D=3).
\tag{5.14}
$$

*Range of the acceleration (S6).* On the ring, for $x\neq x_s$, $\Lambda^+(\cdot,x_s)$ is quadratic along $\ell$ with second difference $+1/L$, and its central-difference slope in the direction away from $x_s$ is exactly $-(\tfrac12-\varrho/L)$; at $x_s$ it has a kink (second difference $-(1-1/L)$). So (5.13) is the local-slope value of the first-order profile without further error. S6 requires $w\ll\varrho$ and $\lvert N''\rvert w\ll\lvert N'\rvert$, i.e. $w\ll L/2-\varrho$. The curvature term of (5.7) is a correction of first order in $g_s$ to (4.24), of relative size $1/(L-2\varrho)$. On the tori the leading forms give a slope along $\ell$ proportional to $\varrho_\parallel/\varrho^D$ and $\lvert\partial_\ell^2N\rvert/\lvert\partial_\ell N\rvert=\lvert\varrho^2-D\varrho_\parallel^2\rvert/(\varrho^2\lvert\varrho_\parallel\rvert)\le(D-1)/\lvert\varrho_\parallel\rvert$. The directional condition of S6 thus holds for $w\ll\lvert\varrho_\parallel\rvert$ (which implies $w\ll\varrho$), and fails near the foot point $\varrho_\parallel=0$ of $x_s$ on $\ell$, where the slope along $\ell$ vanishes.

*Signs (graph coordinates).* By the exact forms above and S5, $\operatorname{sgn}\bigl(r(y)/r(x)-1\bigr)=\operatorname{sgn}\bigl(\gamma(N(y)-N(x))\bigr)$ and $\operatorname{sgn}a_{k,\sigma}(0)=-\operatorname{sgn}\bigl(\gamma N'(x_0)\bigr)$; (W2) is not needed. These are the signs of the leading terms of (5.11), (5.12) when $R_N<\frac{n\lvert g_s\rvert}{t_f}\lvert\Lambda^+(x,x_s)-\Lambda^+(y,x_s)\rvert$, resp. $R_N<\frac{2n\lvert g_s\rvert}{t_f}\lvert\partial_\ell\Lambda^+(x_0,x_s)\rvert$. With $R_N\lesssim N_fg_s^2/(t_f\omega_{\min})^2$ and $N_f=n\lvert X_f\rvert$, sufficient conditions are $\lvert g_s\rvert\ll t_f\omega_{\min}^2\lvert\Lambda^+(x,x_s)-\Lambda^+(y,x_s)\rvert/\lvert X_f\rvert$, resp. $\lvert g_s\rvert\ll t_f\omega_{\min}^2\lvert\partial_\ell\Lambda^+(x_0,x_s)\rvert/\lvert X_f\rvert$; they imply (W1), since $\lvert\Lambda^+(x,y)\rvert\le\lVert\Lambda^+\rVert=1/\omega_{\min}$. $\Lambda^+(\cdot,x_s)$ decreases with $\varrho$: exactly on the ring for $0\le\varrho\le L/2$ by (5.9), and in the leading forms (5.10) on the tori. Hence, under these conditions, for $g_s\gamma>0$ the clock runs slower at smaller $\varrho$ and the drift acceleration points to decreasing $\varrho$: on the ring along the shorter arc towards $x_s$, on the tori along $\ell$ towards the foot point. For $g_s\gamma<0$ both reverse.

*Graph separation and recorded distance.* $\varrho$, $\vec\varrho$ and "towards" above are positions in the field graph. The record $K_x=\kappa\,x\,X$ fixes recorded positions only on $\ell$: in the background of (4.24) the recorded position moves by $d_0\hat e$ per label step on every bond of $\ell$ except the closing bond $(L-1,0)$, whose record increment is $-\kappa(L-1)X$ instead of $\kappa X$; across it the background of (4.16) and (4.24) does not hold (S6). The setting fixes no record of the source; for $x_s\in\ell$ its recorded position is taken as that of the label $x_s$. For labels of $\ell$ with label separation $s\in\{0,\dots,L-1\}$ the recorded distance is $d_0s$, while their graph separation along $\ell$ is $\min(s,L-s)$; the two coincide iff $s\le L/2$.

*Ring.* With $s:=\lvert x_0-x_s\rvert$, $\varrho=\min(s,L-s)$ and $\varrho(L-\varrho)=s(L-s)$, the leading terms of (5.13) read in recorded terms

$$
\frac{r(y)}{r(x)}=1+\frac{n\bar\gamma g_s}{t_fL}\bigl[s_y(L-s_y)-s_x(L-s_x)\bigr],\qquad a^{\rm rec}_{k,\sigma}(0)=\frac{n\bar\gamma g_s}{t_f}\,\tau_0^2\Bigl(1-\frac{2s}{L}\Bigr),
\tag{5.16}
$$

with $a^{\rm rec}$ counted positive towards the recorded position of $x_s$, the errors of (5.11), (5.12), and starts whose support contains neither $x_s$ nor the closing bond. For $s<L/2$ the recorded and the graph statements agree. For $s>L/2$ the shorter arc crosses the closing bond: for $g_s\gamma>0$ the fall is towards the source in the graph but away from it in the recorded geometry, and along $\ell$ the clock is slower at larger recorded distance from $x_s$.

*Tori.* With $\vec\varrho_\perp:=\vec\varrho-\varrho_\parallel\hat e_\ell$: the record fixes the recorded distance $d_0s_\parallel$ from $x_0$ to the foot point of $x_s$ on $\ell$, $s_\parallel$ the label separation ($s_\parallel=\lvert\varrho_\parallel\rvert$ iff $s_\parallel\le L/2$). The transverse separation $\vec\varrho_\perp$, and with it $\varrho$, has no recorded counterpart in this setting, so the size of (5.14) depends on graph quantities that the record does not fix. In recorded terms (5.14) says, for $g_s\gamma>0$ and $s_\parallel<L/2$: along $\ell$ the fall points to the foot point and the clock is slower nearer to it; for $s_\parallel>L/2$ both are reversed in the recorded geometry, as on the ring.

## Result

- **(5.1), (5.2), (5.15)** [theorem]. Class: couplings (a) and (c); every field state $\varphi$; product starts $\xi\otimes\varphi$ with every body–medium state $\xi$ (S1–S4); every real $\lambda$, $s=\operatorname{sgn}\lambda$. The view of body and medium equals $\lvert\xi(\lambda)\rangle\langle\xi(\lambda)\rvert$, with $\xi$ evolving by the partial expectation $\bar C_{\varphi(\lambda)}$ of the total contract in the field state (5.1), up to a trace-norm remainder of at most $2\delta(\lambda)^2\bigl(1+\lvert\gamma\rvert c_W\hat\phi(\lambda)\lvert\lambda\rvert\bigr)$ (5.15). The first-order term vanishes after the field trace, so this remainder is second order in $\gamma$; it is small for $\delta\ll1$ and $\lvert\gamma\rvert c_W\hat\phi\lvert\lambda\rvert\delta^2\ll1$. The joint state equals $\xi(\lambda)\otimes\varphi(\lambda)$ up to a common phase and a remainder of norm at most $\delta(\lambda)=\lvert\gamma\rvert c_Q\int_0^{\lvert\lambda\rvert}\sigma_\Phi(s\mu)\,\mathrm d\mu$ (5.2), first order in $\gamma$: the test body's change of the field. For stationary $\varphi$ the operator is $\lambda$-independent and $\delta=\lvert\gamma\rvert c_Q\sigma_\Phi\lvert\lambda\rvert$; in the lowest adapted state $\sigma_\Phi^2\le\max_yN(y)$ and $\hat\phi\le N_f$. $\bar C_\varphi$ and the view are fixed by $C$ and the state; the split of $\bar C_\varphi$ into own and field-made parts is not.
- **(5.3)** [formula]. Same class. Extraction rule (Step 2): $B_{x,k}$ is the level-$k$ on-site operator of $\bar C_\varphi$ minus its field-independent scalar background $\hat V^{(0)}_x$, which is $K_x+C_c$ in the listing of the setting. The level-$k$ blocks are: on-site $\nu(x)M_k\sigma_3+E_\varphi\mathbb 1_2$; hopping $\tau$ for (a) and $\tau\frac12(\nu(x)+\nu(x+1))$ for (c). Fixed by $C$ and $\varphi$: $B_{x,k}$, also under place-dependent relistings between body term and record term, and the hopping amplitudes; hence $M_{x,k}$, $V_{x,k}$ up to a common constant, and the ratios $M_{y,k}/M_{x,k}=\nu(y)/\nu(x)$. Listing-dependent: the factorization $M_{x,k}=\nu(x)M_k$ and the place-independent constant in $V_{x,k}$.
- **(5.4)** [theorem]. Class: (a) and (c), every field state, every place (at every $\lambda$). R3 holds with $\Gamma(x)=\gamma(N_\varphi(x+1)-N_\varphi(x))\sigma_3$, without a $\mathbb 1_2$ part. In fixed quantities, $B_{x+1,k}-B_{x,k}=M_{x,k}\frac{\nu(x+1)-\nu(x)}{\nu(x)}\sigma_3$ for $\nu(x)\neq0$. The hopping changes by the same factor as the on-site blocks for (c), with the bond carrying the mean $\frac12(\nu(x)+\nu(x+1))$ of the factors of its ends. For (a) it does not: the hopping stays $\tau$ at every place, and the factors differ wherever $\gamma(N_\varphi(x+1)-N_\varphi(x))\neq0$. No field state violates R3 for (a) or (c); the control is (5.5).
- **(5.5)** [negative]. Control: the pair term $\gamma\sum_x\Pi_x\sigma_3\otimes m\mathbb 1_N\otimes\Phi_x$ (rest term only). R3 fails at every place with $\gamma(N_\varphi(x+1)-N_\varphi(x))\neq0$, and there the fall from rest (under S6) depends on the level.
- **(5.6)** [formula]. Class: (a), (c), stationary $\varphi$, S5. The clock-rate ratio of two places is $r(y)/r(x)=\nu(y)/\nu(x)=M_{y,k}/M_{x,k}$, the same for every pair of levels and for both branches. It is exact in $\bar C_\varphi$.
- **(5.7)** [theorem]. Class: (a), (c), stationary $\varphi$, branch starts at rest, S5–S6; first order in the gradients. $a_{k,\sigma}(0)=-\tau_{\rm loc}(x_0)^2\,\gamma N'(x_0)/\nu(x_0)$ with the lattice slope $N'$, the same for all levels and both branches. (4.24) of 05-ilang-free-fall/04-branch-fall holds with $v_{\max}(x_0)=\tau_{\rm loc}(x_0)d_0$ up to the curvature term of (5.7), of relative size $\lvert N''(x_0)\rvert/(2\lvert N'(x_0)\rvert)$. For (a) $v_{\max}(x_0)$ is $\tau d_0$ at every place. For (c) it is $\tau\nu(x_0)d_0$, which changes from place to place by the factor of $r$; with a place-independent $v_{\max}$ taken at a reference place $y$, (4.24) carries the factor $(r(x_0)/r(y))^2$.
- **(5.8), (5.9)** [formula]. The lowest adapted state is $u_g^{\otimes N_f}$, unique for every real $g_s$. Under (W1) $\lvert g_s\rvert\ll t_f\omega_{\min}$, in it $N(y)=n-\frac{2ng_s}{t_f}\Lambda^+(y,x_s)$ up to $R_N\lesssim N_fg_s^2/(t_f\omega_{\min})^2$, with $\Lambda^+$ on the ring given exactly by (5.9).
- **(5.11), (5.12)** [formula]. Class: (a), (c), lowest adapted state, S5, (W1) (and S6 for the acceleration). Clock-rate ratio $1+\frac{2n\bar\gamma g_s}{t_f}[\Lambda^+(x,x_s)-\Lambda^+(y,x_s)]$; acceleration $\frac{2n\bar\gamma g_s}{t_f}\tau_0^2\partial_\ell\Lambda^+(x_0,x_s)$, level and branch independent; first order in $g_s$, with $\bar\gamma=\gamma/(1+\gamma n)$ and $\tau_0$ the source-free hopping of $\bar C_\varphi$, up to a relative error $O(\varepsilon_\nu)$, small under (W2) $\varepsilon_\nu\ll1$, and an absolute error $O(\lvert\bar\gamma\rvert R_N)$ ($O(\tau_0^2\lvert\bar\gamma\rvert R_N)$ for the acceleration).
- **(5.13), (5.14)** [formula, criterion]. Graph coordinates, with the errors of (5.11), (5.12). Ring: (5.13), exact in $L$ for the first-order profile; the acceleration is its local-slope value, for $w\ll\varrho$ and $w\ll L/2-\varrho$. Square and cubic tori, leading forms for $1\ll\varrho\ll L$ ($\varrho$ the Euclidean norm of the coordinate difference): (5.14), with the logarithm ($D=2$) and $1/\varrho$ ($D=3$) in the clock ratio, and the acceleration along $\ell$ the projection on $\hat e_\ell$ of a radial vector of size $\propto\varrho^{-1}$ ($D=2$), $\propto\varrho^{-2}$ ($D=3$), for $w\ll\lvert\varrho_\parallel\rvert$. Signs, with S5 and without (W2): the clock-rate difference has the sign of $\gamma(N(y)-N(x))$ and the acceleration that of $-\gamma N'(x_0)$. When $R_N$ is below the first-order differences (sufficient: $\lvert g_s\rvert\ll t_f\omega_{\min}^2\lvert\Lambda^+(x,x_s)-\Lambda^+(y,x_s)\rvert/\lvert X_f\rvert$, resp. the same with $\lvert\partial_\ell\Lambda^+(x_0,x_s)\rvert$): for $g_s\gamma>0$ the clock is slower at smaller graph separation $\varrho$ and the fall points to decreasing $\varrho$ (ring: along the shorter arc; tori: along $\ell$ towards the foot point of $x_s$); for $g_s\gamma<0$ the clock is faster nearer the source and the fall points away from it.
- **(5.16)** [criterion]. Recorded geometry. On $\ell$ the recorded distance $d_0s$ (label separation $s$) and the graph separation $\min(s,L-s)$ coincide iff $s\le L/2$. On the ring, in terms of $s=\lvert x_0-x_s\rvert$, the leading terms are (5.16), for starts whose support contains neither $x_s$ nor the closing bond $(L-1,0)$. Under the sign conditions of (5.13) and for $g_s\gamma>0$, the fall is towards the recorded position of $x_s$ for $s<L/2$ and away from it for $s>L/2$, where the shorter arc crosses the closing bond. On the tori the record fixes only the recorded distance $d_0s_\parallel$ along $\ell$ to the foot point of $x_s$; the transverse separation $\vec\varrho_\perp$ has no recorded counterpart in the setting, so the size of (5.14) is a function of graph coordinates.

## Consistency checks

- **Dimensions.** $\gamma$ and $n,\Lambda^+$ are numbers; $g_s,t_f,\tau,M_k$ have the dimension of $C$, i.e. $1/\lambda$. Then $n\bar\gamma g_s/t_f$, $\delta=\lvert\gamma\rvert c_Q\sigma_\Phi\lvert\lambda\rvert$ and $\lvert\gamma\rvert c_W\phi_{\max}\lvert\lambda\rvert$ are dimensionless, and $a\propto\tau_0^2$ has the dimension of label$/\lambda^2$. Passed.
- **$g_s=0$ or $\gamma=0$.** For $g_s=0$, (5.8) gives $N=n$ uniform, hence $\Gamma=0$, $r(y)/r(x)=1$ and $a=0$ in (5.11)–(5.14) and (5.16). For $\gamma=0$, $\bar C_\varphi=C_{bc}+E_\varphi$ and the bounds (5.2), (5.15) vanish. Passed.
- **Sharp field.** Take $t_f=0$ and $\varphi$ a joint eigenvector of all $\Phi_x$. Then $\varphi$ is stationary, $\sigma_\Phi=0$, (5.2) and (5.15) are exact, and $C(\xi\otimes\varphi)=(\bar C_\varphi\xi)\otimes\varphi$ directly: the given lapse field of 05-ilang-free-fall with on-site factor $1+\gamma N(x)$. Passed.

## Open issues

- The bounds (5.2) and (5.15) grow with $\lvert\lambda\rvert$, and (5.15) contains $\phi_{\max}\le N_f$, which is crude for many field objects. A bound that uses the field gap $t_f\omega_{\min}$ (a non-growing first-order joint-state remainder plus a second-order shift, which is the test body's own mark of 06-own-mark) is not derived.
- The remainder $R_N$ of (5.8), taken from (3.9) per field object, is crude; the sign conditions it gives shrink fast with $L$. A sharp estimate of the second-order profile is not derived.
- Non-product starts, and non-stationary field states for items 3–4 (a $\lambda$-dependent profile), are not treated.
- Regions with $\nu(x)\le0$ or without (W2), the neighbourhoods $\varrho\lesssim w$ of the source, the ring antipode, $\varrho\sim L$ and the foot point $\lvert\varrho_\parallel\rvert\lesssim w$ on the tori, and starts across the closing bond of $\ell$ are outside the formulas.
- The setting fixes no record of the source and no recorded transverse separation on the tori; a recorded distance to a source off $\ell$ would need a record beyond the single line $\ell$, which is not part of the setting.

## Methods used

- partial expectation (mean field) over the field state
- Duhamel formula, unitarity
- partial trace and trace-norm estimate of the reduced view
- Pauli decomposition of on-site operators
- Rayleigh–Schrödinger perturbation theory (via (3.9))
- Perron–Frobenius theorem, product states of independent identical objects
- plane-wave diagonalization, graph Laplacian pseudo-inverse, lattice Green function asymptotics
- semiclassical band dynamics, local-slope (Taylor) expansion

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