Version 1 · earlier version; the current one is v2 · External review, round 1: major errors

# Causal order of single-cell readings on a contract chain

- **Subproject:** 04-ilang-time
- **Package:** 07-causal-order
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-10

## Setup and assumptions

- **Model** (setting of the question). Cells $1,\dots,n$ ($n\ge2$), each the only instance of its type, with place sets $H_i$, $d_i\ge2$, and $C=\sum_{i=1}^nD_i+\sum_{i=1}^{n-1}J_i$, where $D_i$ acts on cell $i$, $J_i$ on cells $i,i+1$, and $\lVert J\rVert:=\max_i\lVert J_i\rVert$. No two objects share a type, so (1.11) imposes nothing: $\mathcal H_{\rm adm}=\mathcal H$. Cells are distinguished by their types, not by labels (M6).
- **Readings.** $R_i$ on cell $i$ at $\lambda_1$ has outcomes $s$ and projectors $\Pi_s:=\Pi^{(i)}_s$ ($m$ classes). $R_j$ on cell $j$ at $\lambda_2$ has outcomes $t$ and projectors $\Pi'_t:=\Pi^{(j)}_t$. Here $t$ is the question's $r'$, renamed so that it does not clash with the distance $r:=\lvert i-j\rvert$. Readings are partitions of places (A4), so all reading projectors are diagonal in the joint place basis. The delay is $\delta:=\lambda_2-\lambda_1$.
- **Start** (A8, A9). The start is the state $\Psi_0$ at some $\lambda_0\le\min(\lambda_1,\lambda_2)$. Readings act in increasing order of $\lambda$, and (1.9) acts between them. For $\lambda_1\le\lambda_2$ let $\Psi_1:=e^{-\mathrm iC(\lambda_1-\lambda_0)}\Psi_0$. This map is a bijection of unit vectors, so "for every start" means "for every unit vector $\Psi_1$". Outcomes of probability $0$ contribute nothing to outcome averages.
- **Notation.** $\mathrm i$ is the imaginary unit; the letter $i$ always denotes a cell. $\mathrm{ad}_XY:=[X,Y]$, $X(\delta):=e^{\mathrm iC\delta}Xe^{-\mathrm iC\delta}$ and $\mathcal M_i(X):=\sum_s\Pi_sX\Pi_s$. The support of an operator is the set of cells on which it acts nontrivially. Operators with disjoint supports commute, and $\operatorname{supp}[X,Y]\subseteq\operatorname{supp}X\cup\operatorname{supp}Y$.
- **M6.** Every quantity below depends on $\Psi_1$ only through $\lvert\Psi_1\rangle\langle\Psi_1\rvert$. It depends on the contracts only through $C$, except where the split into $D$'s and $J$'s is used; there, independence of the split is shown.
- **Inputs:** none; the package builds only on the base problem.

## Derivation

### Step 1. Outcome-averaged influence

Let $\delta\ge0$, $U:=e^{-\mathrm iC\delta}$ and $Q_t:=\Pi'_t(\delta)=U^\dagger\Pi'_tU$. By (1.3), outcome $s$ of $R_i$ occurs with probability $\lVert\Pi_s\Psi_1\rVert^2$ and leaves $\Pi_s\Psi_1/\lVert\Pi_s\Psi_1\rVert$. This state evolves by $U$, and $R_j$ then gives $t$ with probability $\lVert\Pi'_tU\Pi_s\Psi_1\rVert^2/\lVert\Pi_s\Psi_1\rVert^2$. Averaging over $s$ gives $P_j(t\mid R_i)=\sum_s\lVert\Pi'_tU\Pi_s\Psi_1\rVert^2=\langle\Psi_1\vert\mathcal M_i(Q_t)\vert\Psi_1\rangle$, while $P_j(t)=\langle\Psi_1\vert Q_t\vert\Psi_1\rangle$. Using $\sum_s\Pi_s=\mathbb 1$:

$$
\Delta_t:=P_j(t\mid R_i)-P_j(t)=\langle\Psi_1\vert\,\mathcal M_i(Q_t)-Q_t\,\vert\Psi_1\rangle,\qquad \mathcal M_i(X)-X=\sum_s\Pi_s\,[X,\Pi_s].
\tag{7.1}
$$

Hence $\mathcal M_i(X)=X$ if and only if $[X,\Pi_s]=0$ for all $s$: if $\mathcal M_i(X)=X$, then $X$ is block diagonal. Moreover $\delta P=\sum_t\lvert\Delta_t\rvert$ and $\sum_t\Delta_t=0$.

### Step 2. No influence without delay (item 1)

- **$\lambda_2=\lambda_1$.** Here $Q_t=\Pi'_t$. Both $\Pi'_t$ and $\Pi_s$ are diagonal in the joint place basis, so they commute, and $\Delta_t=0$ by (7.1). This holds for all $i,j$, including $i=j$. The order in which the two readings are applied does not matter.
- **$\lambda_2<\lambda_1$.** $R_j$ acts first, on the state $\Psi_2$ at $\lambda_2$. $R_i$ acts after the evolution $U':=e^{-\mathrm iC(\lambda_1-\lambda_2)}$. The joint probability of $t$ and then $s$ is $\lVert\Pi_sU'\Pi'_t\Psi_2\rVert^2$. Summed over $s$, it gives $\lVert U'\Pi'_t\Psi_2\rVert^2=\lVert\Pi'_t\Psi_2\rVert^2=P_j(t)$.

$$
P_j(\cdot\mid R_i)=P_j(\cdot)\qquad\text{whenever }\lambda_2\le\lambda_1,\ \text{for all cells } i,j,\ \text{all starts and all readings.}
\tag{7.2}
$$

### Step 3. Locality of nested commutators

Let $i>j$. Expand $\mathrm{ad}_C^k\Pi'_t=\sum\mathrm{ad}_{X_k}\cdots\mathrm{ad}_{X_1}\Pi'_t$ with every $X_l\in\{D_1,\dots,D_n,J_1,\dots,J_{n-1}\}$. After $l$ commutators, the support of a term lies in cells $\le b_l$, with $b_0=j$. A commutator with $X_l$ is nonzero only if $\operatorname{supp}X_l$ meets the current support. Among such terms, only $X_l=J_{b_{l-1}}$ reaches cell $b_{l-1}+1$. Hence $b_l\le b_{l-1}+1$, with equality only for $X_l=J_{b_{l-1}}$. A term that fails to commute with $\Pi_s$ needs $b_k\ge i=j+r$. Therefore

$$
[\mathrm{ad}_C^k\Pi'_t,\Pi_s]=0\ \ (k<r),\qquad [\mathrm{ad}_C^r\Pi'_t,\Pi_s]=[K_t,\Pi_s],\qquad K_t:=\mathrm{ad}_{J_{i-1}}\cdots\mathrm{ad}_{J_{j+1}}\mathrm{ad}_{J_j}\Pi'_t .
\tag{7.3}
$$

For $i<j$, the mirror argument gives $K_t:=\mathrm{ad}_{J_i}\cdots\mathrm{ad}_{J_{j-2}}\mathrm{ad}_{J_{j-1}}\Pi'_t$. Another booking (A5) replaces $J_m$ by $J_m+D_m$ or by $J_m+D_{m+1}$. This adds only terms with $b_r<i$, so $[K_t,\Pi_s]$ depends neither on the booking (M6) nor on the $D$'s.

### Step 4. Order of the influence (item 2)

In finite dimension, $Q_t=\sum_{k\ge0}\frac{(\mathrm i\delta)^k}{k!}\mathrm{ad}_C^k\Pi'_t$ converges for every $\delta$. Inserting this series into (7.1) and using (7.3):

$$
\Delta_t=\Gamma_t\,\delta^r+O(\delta^{r+1}),\qquad
\Gamma_t=\frac{1}{r!}\langle\Psi_1\vert\,\mathcal M_i(\mathrm i^rK_t)-\mathrm i^rK_t\,\vert\Psi_1\rangle=\frac{1}{r!}\operatorname{Tr}\bigl[(\mathcal M_i(V)-V)\,\mathrm i^rK_t\bigr],
\tag{7.4}
$$

$$
\delta P=\delta^r\sum_t\lvert\Gamma_t\rvert+O(\delta^{r+1}).
\tag{7.5}
$$

- **Second form of (7.4).** $V$ is the view (1.5), at $\lambda_1$, of the part formed by the cells from $j$ to $i$. The second form holds because $K_t$ and $\Pi_s$ act inside this part and the partial trace commutes with $\mathcal M_i$.
- **Properties of $\Gamma_t$.** $\Gamma_t$ is real, because $\mathrm i\,\mathrm{ad}_{J}$ and $\mathcal M_i$ map self-adjoint operators to self-adjoint ones. Also $\sum_t\Gamma_t=0$, because $\sum_t\Pi'_t=\mathbb 1$ gives $\sum_tK_t=0$.
- **Role of the reading.** $V-\mathcal M_i(V)=\sum_{s\neq s'}\Pi_sV\Pi_{s'}$ is the part of the view between places of cell $i$ in different classes. At leading order, $R_i$ therefore acts only through the cross terms that it erases.
- **Answer.** The lowest possible power is $\delta^r$, with coefficient $\sum_t\lvert\Gamma_t\rvert$. It depends on $J_j,\dots,J_{i-1}$, on the projectors and on $V$, but not on the $D_m$. Step 5 shows that it can be nonzero.

### Step 5. Examples with qubit cells (item 2)

**Conventions.** The cells are qubits with places $0,1$. The Pauli matrices are written in the place basis: $\sigma^z=\lvert0\rangle\langle0\rvert-\lvert1\rangle\langle1\rvert$, $\sigma^x=\lvert0\rangle\langle1\rvert+\lvert1\rangle\langle0\rvert$, $\sigma^y=-\mathrm i\lvert0\rangle\langle1\rvert+\mathrm i\lvert1\rangle\langle0\rvert$, so that $[\sigma^z,\sigma^x]=2\mathrm i\sigma^y$. The states $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ and $\lvert{+\mathrm i}\rangle=(\lvert0\rangle+\mathrm i\lvert1\rangle)/\sqrt2$ satisfy $\langle\sigma^x\rangle_+=1$ and $\langle\sigma^y\rangle_{+\mathrm i}=1$.

**Family.** Take $n=r+1$, $i=1$, $j=r+1$, and place readings on both cells, so that $\Pi'_{0,1}=(\mathbb 1\pm\sigma^z_{r+1})/2$. The $D_m$ are arbitrary, $g>0$, and

$$
J_1=g\,\sigma^x_1\sigma^x_2,\qquad J_m=g\,\sigma^z_m\sigma^x_{m+1}\ \ (2\le m\le r),\qquad \lVert J\rVert=g.
\tag{7.6}
$$

**Coefficient.** For $m\ge2$, $\mathrm i\,\mathrm{ad}_{J_m}$ maps $\sigma^z_{m+1}\sigma^y_{m+2}\cdots\sigma^y_{r+1}$ to $2g\,\sigma^z_m\sigma^y_{m+1}\cdots\sigma^y_{r+1}$. Then $\mathrm i\,\mathrm{ad}_{J_1}$ maps $\sigma^z_2\cdots$ to $2g\,\sigma^x_1\sigma^y_2\cdots$. Hence $\mathrm i^rK_0=-\mathrm i^rK_1=2^{r-1}g^r\,\sigma^x_1\sigma^y_2\cdots\sigma^y_{r+1}$. $\mathcal M_1$ annihilates this operator, because $\sum_s\Pi_s\sigma^x\Pi_s=0$. With $\Psi_1=\lvert+\rangle\otimes\lvert{+\mathrm i}\rangle^{\otimes r}$, (7.4) and (7.5) give

$$
\Gamma_0=-\Gamma_1=-\frac{2^{r-1}g^r}{r!},\qquad \delta P=\frac{(2g\delta)^r}{r!}+O(\delta^{r+1})\qquad(r\ge1).
\tag{7.7}
$$

- **$r=1$.** Take $n=2$, $J_1=g\sigma^x_1\sigma^x_2$ and $\Psi_1=\lvert+\rangle\lvert{+\mathrm i}\rangle$. Then $\delta P=2g\delta+O(\delta^2)$. For $D_1=D_2=0$ the result is exact: $\sigma^z_2$ anticommutes with $J_1$, so $Q_0=\tfrac12\bigl(\mathbb 1+\cos(2g\delta)\,\sigma^z_2+\sin(2g\delta)\,\sigma^x_1\sigma^y_2\bigr)$, and

$$
\delta P=\lvert\sin(2g\delta)\rvert\qquad(n=2,\ D_1=D_2=0,\ J_1=g\sigma^x_1\sigma^x_2,\ \Psi_1=\lvert+\rangle\lvert{+\mathrm i}\rangle).
\tag{7.8}
$$

- **$r=2$.** Take $n=3$, $J_1=g\sigma^x_1\sigma^x_2$, $J_2=g\sigma^z_2\sigma^x_3$ and $\Psi_1=\lvert+\rangle\lvert{+\mathrm i}\rangle\lvert{+\mathrm i}\rangle$. Then $\delta P=2g^2\delta^2+O(\delta^3)$.

The power $\delta^r$ in (7.5) is thus attained for every $r\ge1$.

### Step 6. Bound (item 3)

**(a) Reduction.** Let $\epsilon_t:=\operatorname{sgn}\Delta_t$ and $Z:=\sum_t\epsilon_t\Pi'_t$, which acts on cell $j$ and has $\lVert Z\rVert\le1$. By linearity of (7.1), $\delta P=\langle\Psi_1\vert\mathcal M_i(Z(\delta))-Z(\delta)\vert\Psi_1\rangle\le\lVert\mathcal M_i(Z(\delta))-Z(\delta)\rVert$.

Label the classes of $R_i$ by $s=0,\dots,m-1$, let $\omega:=e^{2\pi\mathrm i/m}$, and define $W_k:=\sum_s\omega^{ks}\Pi_s$. These are unitaries on cell $i$, with $W_0=\mathbb 1$. Since $\frac1m\sum_k\omega^{k(s-s')}$ is $1$ for $s=s'$ and $0$ otherwise, $\mathcal M_i(X)=\frac1m\sum_kW_kXW_k^\dagger$. Hence $\mathcal M_i(X)-X=\frac1m\sum_{k=1}^{m-1}[W_k,X]W_k^\dagger$, and

$$
\delta P\le\Bigl(1-\frac1m\Bigr)\max_{1\le k<m}\bigl\lVert[Z(\delta),W_k]\bigr\rVert .
\tag{7.9}
$$

**(b) Commutator bound** (Lieb–Robinson type). Let $A$ act on cell $j$ and $B$ on cell $i=j+r>j$.

- **Truncated evolutions.** For $m\ge j$, let $C_{\le m}:=\sum_{l\le m}D_l+\sum_{l<m}J_l$ be the terms inside cells $1,\dots,m$, so that $C_{\le n}=C$. Let $A_m(\mu):=e^{\mathrm iC_{\le m}\mu}Ae^{-\mathrm iC_{\le m}\mu}$; it is supported in cells $\le m$.
- **Duhamel step.** $C_{\le m}-C_{\le m-1}=D_m+J_{m-1}$, and $D_m$ commutes with $A_{m-1}(\mu)$. Hence $\frac{\mathrm d}{\mathrm d\mu}\bigl[e^{\mathrm iC_{\le m}(\delta-\mu)}A_{m-1}(\mu)e^{-\mathrm iC_{\le m}(\delta-\mu)}\bigr]=-\mathrm i\,e^{\mathrm iC_{\le m}(\delta-\mu)}[J_{m-1},A_{m-1}(\mu)]e^{-\mathrm iC_{\le m}(\delta-\mu)}$. Integrating over $\mu\in[0,\delta]$ gives

$$
\lVert A_m(\delta)-A_{m-1}(\delta)\rVert\le\int_0^\delta\beta_{m-1}(\mu)\,\mathrm d\mu,\qquad \beta_m(\mu):=\bigl\lVert[J_m,A_m(\mu)]\bigr\rVert .
\tag{7.10}
$$

- **Recursion.** $J_m$ commutes with $A_{m-1}(\mu)$, which lives on cells $\le m-1$. So for $m>j$, $\beta_m(\mu)=\lVert[J_m,A_m(\mu)-A_{m-1}(\mu)]\rVert\le2\lVert J\rVert\int_0^\mu\beta_{m-1}$, and $\beta_j\le2\lVert J\rVert\lVert A\rVert$. By induction, $\beta_m(\mu)\le2\lVert J\rVert\lVert A\rVert(2\lVert J\rVert\mu)^{m-j}/(m-j)!$. Then (7.10) gives $\lVert A_m(\delta)-A_{m-1}(\delta)\rVert\le\lVert A\rVert(2\lVert J\rVert\delta)^{m-j}/(m-j)!$.
- **Result.** $[A_{i-1}(\delta),B]=0$, and $A(\delta)=A_n(\delta)=A_{i-1}(\delta)+\sum_{m=i}^n\bigl(A_m(\delta)-A_{m-1}(\delta)\bigr)$. Therefore

$$
\bigl\lVert[A(\delta),B]\bigr\rVert\le2\lVert A\rVert\lVert B\rVert\sum_{k=r}^{\infty}\frac{(2\lVert J\rVert\delta)^k}{k!}\qquad(\delta\ge0,\ r\ge1).
\tag{7.11}
$$

For $i<j$, the same bound follows with $C_{\ge m}$, the terms inside cells $m,\dots,n$. The bound (7.11) is uniform in $n$: the finite sum over $m$ is bounded by the series. No $D_l$ enters it, since the $D$'s appear only inside unitary conjugations, which preserve norms.

**(c) Bound on $\delta P$.** Inserting (7.11) with $A=Z$ and $B=W_k$ into (7.9), and using $(r+l)!\ge r!\,l!$ for the last step:

$$
\delta P\le2\Bigl(1-\frac1m\Bigr)\sum_{k\ge r}\frac{(2\lVert J\rVert\delta)^k}{k!}\le2\sum_{k\ge r}\frac{(2\lVert J\rVert\delta)^k}{k!}\le\frac{2\,(2\lVert J\rVert\delta)^r}{r!}\,e^{2\lVert J\rVert\delta}.
\tag{7.12}
$$

- **Validity.** (7.12) holds for every start, every pair of readings and every $\delta\ge0$; trivially, also $\delta P\le2$.
- **Dependence on the $D_i$.** The bound does not depend on the $D_i$.
- **Booking (M6).** (7.12) holds for every split of $C$ into single-cell and pair terms, so it holds with the smallest $\lVert J\rVert$ over all splits. That value does not depend on the booking.
- **Leading order.** The bound is attained at leading order. For $m=2$, its first term is $(2\lVert J\rVert\delta)^r/r!$, which equals (7.7).

### Step 7. Sharpness (item 4)

**Analyticity.** Write the spectral decomposition $C=\sum_ac_aP_a$. Then $\Delta_t(\delta)=\sum_{a,b}e^{\mathrm i(c_a-c_b)\delta}\langle\Psi_1\vert(\mathcal M_i-\mathrm{id})(P_a\Pi'_tP_b)\vert\Psi_1\rangle$ is an entire function of $\delta$ (A1). If $\delta P=0$ on $(0,\varepsilon)$, every $\Delta_t$ vanishes on an interval, hence identically by the identity theorem:

$$
\delta P=0\ \text{on}\ (0,\varepsilon)\ \Longrightarrow\ \delta P=0\ \text{for all}\ \delta\ge0 .
\tag{7.13}
$$

So $\delta P$ cannot vanish on $(0,\varepsilon)$ and be nonzero at a larger $\delta$. For a fixed start and fixed readings, either $\delta P\equiv0$, or the zeros of $\delta P$ in $(0,\infty)$ are isolated and $\delta P\simeq c\,\delta^q$ near $0$, with $c>0$ and $q\ge r$.

**Delay sets.** Let $\mathcal I_{ij}$ be the set of $\delta>0$ for which some start and some readings give $\delta P>0$. Its complement in $(0,\infty)$ is contained in the zero set of one $\delta P\not\equiv0$. Hence, for $r\ge0$:

$$
\mathcal I_{ij}=\emptyset\quad\text{or}\quad \mathcal I_{ij}=(0,\infty)\setminus E_{ij},\ \ E_{ij}\ \text{discrete in }[0,\infty),\ \ E_{ij}\cap(0,\delta_0)=\emptyset\ \text{for some}\ \delta_0>0 .
\tag{7.14}
$$

**Conclusion: no sharp boundary.** By (7.7), for every $r\ge1$ there are chains with $\mathcal I_{ij}\supseteq(0,\delta_0)$. So no cone $\{\delta\ge r/v\}$ with finite $v$ contains all influence. The only sharp boundary is $\delta=0$, from (7.2). In the distance $r$ the order is approximate: influence outside the cone of Step 8 is small, but it is not zero.

### Step 8. Region of small influence and its speed (item 4)

**The set.** By (7.12), $\delta P<\epsilon$ for every start and every pair of readings on

$$
\mathcal S_\epsilon:=\Bigl\{(r,\delta):\ r\ge1,\ \delta\ge0,\ 2\sum_{k\ge r}\frac{(2\lVert J\rVert\delta)^k}{k!}<\epsilon\Bigr\}=\bigl\{(r,\delta):\ 0\le\delta<\delta_\epsilon(r)\bigr\}.
\tag{7.15}
$$

Here $\delta_\epsilon(r)$ is the unique root of $2\sum_{k\ge r}(2\lVert J\rVert\delta)^k/k!=\epsilon$. The left side increases strictly in $\delta$, from $0$ to $\infty$, and decreases in $r$; hence $\delta_\epsilon(r)$ increases with $r$.

**Explicit inner cone.** Let $v>2e\lVert J\rVert$ and $u:=2\lVert J\rVert/v<1/e$. For $\delta\le r/v$, $x:=2\lVert J\rVert\delta\le ur$, so the terms $x^k/k!$ with $k\ge r$ decrease with ratio $\le u$. Using $r!\ge(r/e)^r$, $\sum_{k\ge r}x^k/k!\le x^r/(r!(1-u))\le(eu)^r/(1-u)$. Hence

$$
\delta P\le\frac{2}{1-2\lVert J\rVert/v}\Bigl(\frac{2e\lVert J\rVert}{v}\Bigr)^{r}\qquad\text{for all starts, readings and } 0\le\delta\le r/v ,
\tag{7.16}
$$

which is exponentially small in $r$.

**Outside.** Conversely, let $v<2e\lVert J\rVert$ and $\delta=r/v$. Using $r!\le e\sqrt r\,(r/e)^r$, the bound is at least $2(2\lVert J\rVert\delta)^r/r!\ge\frac{2}{e\sqrt r}\bigl(\frac{2e\lVert J\rVert}{v}\bigr)^r\to\infty$. So $(r,r/v)\notin\mathcal S_\epsilon$ for large $r$.

**Speed.** Together, for every $\epsilon>0$ the boundary moves at

$$
\lim_{r\to\infty}\frac{r}{\delta_\epsilon(r)}=v_{\rm LR}:=2e\lVert J\rVert\quad\text{cells per unit }\lambda .
\tag{7.17}
$$

This speed is independent of $\epsilon$, of $n$ (uniformity of (7.11)) and of the $D_i$.

### Step 9. The order of events (item 4)

An event is a pair $e=(i,\lambda)$ of a cell and a value of $\lambda$, at which a reading is applied. Following the question,

$$
(i,\lambda_1)\prec(j,\lambda_2)\ :\Longleftrightarrow\ \text{some reading at }(i,\lambda_1)\text{ influences some reading at }(j,\lambda_2)\text{ for some start.}
\tag{7.18}
$$

**(i) Order in $\lambda$.** By (7.2):

$$
e_1\prec e_2\ \Longrightarrow\ \lambda_1<\lambda_2 .
\tag{7.19}
$$

**(ii) Pairs with $\lambda_1<\lambda_2$.**

- **Criterion.** By (7.1) and Step 1, influence occurs for some start if and only if some $\mathcal M_i(Q_t)-Q_t\neq0$; for $\Psi_1$ one takes an eigenvector of this self-adjoint operator for a nonzero eigenvalue. Equivalently, some $Q_t$ fails to commute with some $\Pi_s$. Coarse projectors are sums of place projectors $p_h$, so it suffices to test place readings. Hence $e_1\prec e_2$ if and only if $F_\delta(C):=\sum_{h,h'}\lVert[\,p^{(j)}_{h'}(\delta),\,p^{(i)}_h\,]\rVert_{2}^{2}>0$, with $\lVert\cdot\rVert_2$ the Hilbert–Schmidt norm.
- **Analyticity in the contract terms.** $F_\delta$ is real-analytic in the real parameters of $(D_1,\dots,D_n,J_1,\dots,J_{n-1})$. By A5 the $D$'s are booked in pair terms, so these are the parameters of the pair terms. Also $F_\delta(C)=F_1(\delta C)$.
- **$F_\delta\not\equiv0$, case $r\ge1$.** Place the chain (7.6) on the cells between $i$ and $j$, mirrored if $i>j$. For $d>2$ it acts on places $0,1$; all other terms are $0$. Then $F_{\delta'}>0$ for small $\delta'$ by (7.7), and rescaling $C$ by $\delta'/\delta$ gives $F_\delta>0$.
- **$F_\delta\not\equiv0$, case $r=0$.** Take $D_i=g\sigma^x_i$ and all other terms zero. Then $p^{(i)}_0(\delta)$ fails to commute with $p^{(i)}_0$ unless $\sin(2g\delta)=0$.
- **Generic statement.** The zero set of a nonzero real-analytic function has measure zero. Hence:

$$
\text{for fixed } i,j \text{ and } \lambda_1<\lambda_2:\ \ e_1\prec e_2\ \text{for all contract terms outside a closed set of measure zero.}
\tag{7.20}
$$

- **Fixed $C$.** For a fixed $C$, the pairs with $\lambda_1<\lambda_2$ but $e_1\nprec e_2$ are those with $\lambda_2-\lambda_1\in E_{ij}$ of (7.14). There are none for $0<\lambda_2-\lambda_1<\delta_0$, and they form a discrete set otherwise. For generic $C$, outside a countable union of null sets, there are none at rational delays.
- **Exceptional delays exist.** They do not vanish for every $C$. For $n=2$, $D=0$ and $J_1=g\sigma^x_1\sigma^x_2$, the evolution $e^{-\mathrm iC\pi/(2g)}=-\mathrm i\sigma^x_1\sigma^x_2$ is a product of single-cell unitaries. Every $p_h(\pi/(2g))$ is then a place projector of its own cell, so no reading influences any other at delay $\pi/(2g)$.

**(iii) Transitivity.** In the same chain, by (7.8) at $2g\delta=\pi/2$ and by the $1\leftrightarrow2$ symmetry of $C$ (start $\lvert{+\mathrm i}\rangle\lvert+\rangle$ for the second relation), together with (ii):

$$
e_1=(1,0)\prec e_2=\bigl(2,\tfrac{\pi}{4g}\bigr)\prec e_3=\bigl(1,\tfrac{\pi}{2g}\bigr),\qquad e_1\nprec e_3\qquad(n=2,\ D=0,\ J_1=g\sigma^x_1\sigma^x_2).
\tag{7.21}
$$

So $\prec$ is not transitive in general. If all $\mathcal I_{ij}\neq\emptyset$ and all $E_{ij}=\emptyset$, then (7.14) and (7.19) give $e_1\prec e_2\Leftrightarrow\lambda_1<\lambda_2$, which is transitive. For generic $C$, a triple can violate transitivity only if $\lambda_3-\lambda_1$ lies in the discrete set $E_{ik}$.

## Result

1. **Item 1.** $R_i$ cannot influence $R_j$ when $\lambda_2=\lambda_1$, $i\neq j$ (no), nor when $\lambda_2<\lambda_1$ (no): (7.2). The equal-$\lambda$ statement holds even for $i=j$.
2. **Item 2.**
   - The influence is $\Delta_t=\langle\Psi_1\vert\mathcal M_i(Q_t)-Q_t\vert\Psi_1\rangle$ (7.1).
   - The lowest power is $\delta^r$. The exact coefficient $\Gamma_t$ of (7.4) is built from the nested commutator $K_t$ of the links between the cells (7.3), the projectors and the view of the cells between; $\delta P$ is given by (7.5). The $D_i$ do not enter.
   - Nonzero examples exist for every $r\ge1$, (7.6)–(7.7), including $r=1$ (exactly $\delta P=\lvert\sin 2g\delta\rvert$, (7.8)) and $r=2$ ($\delta P=2g^2\delta^2+O(\delta^3)$).
3. **Item 3.** The bound is (7.12), $\delta P\le2\sum_{k\ge r}(2\lVert J\rVert\delta)^k/k!$, obtained from the chain Lieb–Robinson bound (7.11). It is uniform in $n$, attained at leading order, and does not depend on the $D_i$.
4. **Item 4.**
   - No: $\delta P$ cannot vanish on $(0,\varepsilon)$ and be nonzero later (7.13).
   - Hence there is no sharp cone: influence at some delay implies influence at all delays except a discrete set, in particular at arbitrarily small ones (7.14). The order is sharp only at $\delta=0$.
   - Below $\epsilon$ for every start: the set $\mathcal S_\epsilon$ (7.15), with explicit inner cone (7.16). Its boundary moves at $v_{\rm LR}=2e\lVert J\rVert$ cells per unit $\lambda$ (7.17).
   - For $\prec$ (7.18):
     - $\prec$ implies $\lambda_1<\lambda_2$: yes (7.19).
     - $\prec$ holds for every pair with $\lambda_1<\lambda_2$ only with conditions. For generic contract terms it holds at every prescribed delay (7.20) and at all small delays. It is not true for every $C$ at every delay, since exceptional delays exist (Step 9 (ii)).
     - $\prec$ is not transitive in general (7.21); it is transitive when no exceptional delays exist.

## Consistency checks

1. **Cut link** ($J_m=0$ for some $j\le m<i$). Then $C=C_L+C_R$ with commuting parts on cells $\le m$ and $\ge m+1$. So $Q_t$ stays in cells $\le m<i$, and $\delta P\equiv0$ exactly. This agrees with $K_t=0$ in (7.4), with $\beta_m\equiv0$ in the recursion behind (7.11), which forces every later term to vanish, and with $\mathcal I_{ij}=\emptyset$ in (7.14). Passed.
2. **$n=2$, exact solution (7.8).** The expansion of $\lvert\sin2g\delta\rvert$ begins with $2g\delta$, which agrees with (7.7) for $r=1$. The bound (7.12) with $m=2$ gives $e^{2g\delta}-1\ge2g\delta\ge\lvert\sin2g\delta\rvert$. The zeros $\delta=k\pi/(2g)$ are isolated, as (7.13)–(7.14) require. Passed.
3. **$\delta\to0$ and dimensions.** (7.5) and (7.12) give $\delta P\to0$ as $\delta\to0$, continuous with (7.2), and $\sum_t\Gamma_t=0$ preserves normalization. In (1.9), $C\lambda$ is dimensionless, so $\lVert J\rVert\delta$ in (7.11)–(7.16) is dimensionless and $v_{\rm LR}$ has the dimension of $\lVert J\rVert$, i.e. cells per unit $\lambda$. Passed.

## Open issues

- It is not decided whether $E_{ij}=\emptyset$ for generic contract terms; a dimension count suggests yes. If so, $\prec$ would coincide with the order of $\lambda$ and would be transitive.
- $v_{\rm LR}=2e\lVert J\rVert$ is the speed of the bound's $\epsilon$-boundary. The actual front speed of a given chain, for example (7.6), may be lower and is not determined; the bound is shown to be tight only at leading order.
- Out of scope: events defined by records, sequences of more than two readings, other contract graphs, the relation to the maximal speed of 03-ilang-space, and whether the order of $\lambda$ is detectable (criterion 8).

## Methods used

- Heisenberg picture; outcome-averaged reading as the dephasing map $\mathcal M_i$
- Nested-commutator (Taylor) expansion; support counting on a chain
- Duhamel formula; truncated evolutions; Grönwall-type iteration (Lieb–Robinson bound)
- Twirling over phase unitaries
- Real-analyticity and the identity theorem; measure-zero zero sets of analytic functions
- Stirling bounds; Pauli algebra

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