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

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

- **Subproject:** 04-ilang-time
- **Package:** 07-causal-order
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-10

## Changes from previous version

- **New Step 10** (tags (7.22)–(7.25)): the question "does one fixed generic chain have influence at every positive delay" is now treated on its own, separately from the fixed-delay statement (7.20). It is decided for $n=2$ (yes), reduced to blind delays for different cells when $n\ge3$, and the cases that remain undecided are named in Step 10 (d) and in the Result.
- **Step 9 (ii)**: (7.20) is labelled as a fixed-delay statement; the paragraph on a fixed $C$ now separates $\mathcal I_{ij}=\emptyset$ from $\mathcal I_{ij}\neq\emptyset$. **Step 9 (iii)** uses (7.25).
- **Step 6 (a)**: the one-class reading $m=1$ is treated separately. **Step 6 (c)** and Result item 3: leading-order attainment is attributed to the reading-dependent bound only.
- **Step 8**: the root $\delta_\epsilon(r)$ and the speed are stated for $\lVert J\rVert>0$; the case $\lVert J\rVert=0$ is added.
- Steps 1–5 and 7 are unchanged. Tags (7.1)–(7.21) keep their meaning; (7.22)–(7.25) are new. Checks 1 and 2 are extended to the new results.

## Response to verification

- **I1: Accepted.** Step 10 fixes the chain first and takes all delays at once. Result: yes for $n=2$, (7.25). For $n\ge3$ and different cells, (7.24) proves that a generic chain can fail only at blind delays (7.23); whether generic chains have blind delays, and the same-cell pairs for $n\ge3$, are not decided. Result item 4 marks the sub-item as incomplete for exactly these cases. The fixed-delay result (7.20) and the countable-delay remark are kept distinct in Step 9 (ii).
- **I2: Accepted and fixed in Step 9 (ii)**, paragraph "Fixed $C$": both alternatives of (7.14) are stated.
- **I3: Accepted and fixed in Step 6 (a)** ($m=1$ gives $\delta P=0$; (7.9) is stated for $m\ge2$) **and in Step 8** ($\lVert J\rVert>0$ for the root and the speed; $\lVert J\rVert=0$ treated separately).
- **I4: Accepted and fixed in Step 6 (c)** and Result item 3: only the bound with the factor $1-1/m$ is attained at leading order, for $m=2$.

## 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.
- **Genericity** (Steps 9 and 10). "Generic" means: outside a set of Lebesgue measure zero (a null set) in the real parameter space of the contract terms. Step 10 makes this precise.
- **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), $\delta P=\langle\Psi_1\vert\mathcal M_i(Z(\delta))-Z(\delta)\vert\Psi_1\rangle\le\lVert\mathcal M_i(Z(\delta))-Z(\delta)\rVert$.

- **One class, $m=1$.** Then $\Pi_1=\mathbb 1$, $\mathcal M_i$ is the identity map and $\delta P=0$ for all $\delta$: a reading with one outcome influences nothing.
- **$m\ge2$.** 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

$$
\delta P\le\Bigl(1-\frac1m\Bigr)\max_{1\le k<m}\bigl\lVert[Z(\delta),W_k]\bigr\rVert\qquad(m\ge2).
\tag{7.9}
$$

**(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 $\delta\ge0$, with $r\ge1$. For $m=1$ its first member is $0$, in agreement with (a). 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.** Only the first, reading-dependent member of (7.12) is attained at leading order, and this is shown for $m=2$: its first term is then $(2\lVert J\rVert\delta)^r/r!$, which equals (7.7). The second and third members, which do not depend on the reading, are not attained: their leading coefficient $2(2\lVert J\rVert)^r/r!$ exceeds that of the first member for every finite $m$.

### 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

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

- **$\lVert J\rVert>0$.** $\delta_\epsilon(r)$ 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 $r$.
- **$\lVert J\rVert=0$** (in some booking: $C$ is a sum of single-cell terms). Then (7.12) gives $\delta P\equiv0$ for all $r\ge1$. $\mathcal S_\epsilon$ is the whole domain, $\delta_\epsilon(r)=\infty$, and there is no boundary and no speed.

**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** ($\lVert J\rVert>0$). 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** ($\lVert J\rVert>0$). 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 $\lambda_1<\lambda_2$: one prescribed delay.**

- **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 $\sin(2g\delta)=0$.
- **Fixed-delay statement.** The zero set of a nonzero real-analytic function has measure zero. Hence:

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

- **What (7.20) does not say.** Its null set depends on $i$, $j$ and $\delta$. It does not state that one generic chain has influence at all delays. A countable union of null sets is null, so a generic chain has no exceptional delay in any prescribed countable set, for example at rational delays; in particular $\mathcal I_{ij}\neq\emptyset$ for all $i,j$ for a generic chain. All real delays at once are the subject of Step 10.
- **Fixed $C$.** By (7.14) there are two cases for each pair of cells.
  - $\mathcal I_{ij}=\emptyset$: then $e_1\nprec e_2$ at every delay. Example: a cut link between $i\neq j$ (check 1).
  - $\mathcal I_{ij}\neq\emptyset$: the pairs with $\lambda_1<\lambda_2$ but $e_1\nprec e_2$ are those with $\lambda_2-\lambda_1\in E_{ij}$. There are none for $0<\lambda_2-\lambda_1<\delta_0$, and they form a discrete set otherwise.
- **Exceptional delays exist.** 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; by (7.25) this is the generic case for $n=2$. If all $\mathcal I_{ij}\neq\emptyset$, a triple can violate transitivity only if $\lambda_3-\lambda_1$ lies in the discrete set $E_{ik}$.

### Step 10. One fixed chain and all delays (item 4)

Here the chain is fixed first and all delays $\delta>0$ are taken at once. Uncountably many delays cannot be handled by a union of the null sets of (7.20).

**Setting.**
- $\mathcal P:=\sum_{m=1}^{n-1}\mathrm{Herm}(\mathcal H_m\otimes\mathcal H_{m+1})\otimes\mathbb 1$ is the real vector space of all $C$ of the chain form, with the Lebesgue measure $\operatorname{vol}$ of the Hilbert–Schmidt inner product. **Generic** means: outside a null set of $\mathcal P$. The map $(J_1,\dots,J_{n-1})\mapsto C$, with the $D$'s booked by A5, is linear and onto, so a null set of $\mathcal P$ is a null set of pair terms. The sets below depend on $C$ only (M6).
- $\mathcal O_k$ is the algebra of all operators on cell $k$, extended by the identity. $\mathcal D_k\subset\mathcal O_k$ is the span of the place projectors $p^{(k)}_h$. $U_\delta:=e^{-\mathrm iC\delta}$.
- A set has **codimension $\ge2$** in a manifold if it lies in a countable union of submanifolds of codimension $\ge2$.

By the criterion of Step 9 (ii), $(i,\lambda_1)\nprec(j,\lambda_1+\delta)$ if and only if $[U_\delta^\dagger\mathcal D_jU_\delta,\mathcal D_i]=0$. The chains with an exceptional delay are

$$
\mathcal X_{ij}:=\bigl\{C\in\mathcal P:\ [U_\delta^\dagger\mathcal D_jU_\delta,\mathcal D_i]=0\ \text{for some }\delta>0\bigr\},\qquad \mathcal X:=\bigcup_{i,j=1}^{n}\mathcal X_{ij}.
\tag{7.22}
$$

With (7.19): $C\notin\mathcal X$ if and only if $e_1\prec e_2\Leftrightarrow\lambda_1<\lambda_2$ for all events. The question is whether $\mathcal X$ is null. A delay is **blind** for $(i,j)$ if nothing on cell $i$ influences anything on cell $j$, whatever the place bases:

$$
\Lambda^{ij}_C:=\bigl\{\delta>0:\ [U_\delta^\dagger\mathcal O_jU_\delta,\mathcal O_i]=0\bigr\},\qquad \mathcal X^{\rm bl}_{ij}:=\{C\in\mathcal P:\ \Lambda^{ij}_C\neq\emptyset\}\subseteq\mathcal X_{ij}.
\tag{7.23}
$$

$\Lambda^{ij}_C$ is closed in $(0,\infty)$. All sets of chains used below are projections of countable unions of compact sets, hence measurable.

**(a) Three tools.**

- **T1 (rotation).** Let $\mathcal G$ be a compact group of unitaries with $W\mathcal PW^\dagger=\mathcal P$. Conjugation preserves the Hilbert–Schmidt norm, hence $\operatorname{vol}$. By Tonelli's theorem, $\operatorname{vol}(\mathcal Y)=\int_{\mathcal P}\mathrm dC\,\operatorname{Haar}\{W\in\mathcal G:\ WCW^\dagger\in\mathcal Y\}$ for every measurable $\mathcal Y\subseteq\mathcal P$. So $\mathcal Y$ is null if this Haar measure vanishes for almost every $C$.
- **T2 (one-parameter union).** Let $I\subseteq(0,\infty)$ be open, and let $\Phi\ge0$ be real-analytic on $I\times\mathcal G$ such that $\{W:\Phi(\delta,W)=0\}$ has codimension $\ge2$ in $\mathcal G$ for every $\delta\in I$. Then $\{W:\ \Phi(\delta,W)=0\text{ for some }\delta\in I\}$ is Haar-null. Reason: the zero set of $\Phi$ is a countable union of submanifolds $S$ (stratification of analytic sets). By the rank theorem, $S$ has fibres over $\delta$ of dimension $\ge\dim S-1$, and they lie in a set of dimension $\le\dim\mathcal G-2$. So $\dim S\le\dim\mathcal G-1$, and the projection of $S$ to $\mathcal G$ is null (Sard).
- **T3 (bases inside a subalgebra).** Let $\mathcal S\subsetneq M_d$ be a unital $*$-subalgebra. The $W\in U(d)$ with $W^\dagger\lvert h\rangle\langle h\rvert W\in\mathcal S$ for all $h$ form a set of codimension $\ge2$. Reason: $\mathcal S\cong\bigoplus_kM_{m_k}\otimes\mathbb 1_{r_k}$ (structure theorem). It contains $d$ orthogonal rank-one projectors only if all $r_k=1$, that is, $\mathcal S=\bigoplus_kM(V_k)$ for an orthogonal decomposition $\mathbb C^d=\bigoplus_kV_k$ with at least two blocks. The bases adapted to it form a set of dimension $\sum_km_k^2\le(d-1)^2+1\le d^2-2$.

**(b) Different cells, every $n$.** Let $i\neq j$ and $\mathcal G=\{W_i\otimes W_j\}\cong U(d_i)\times U(d_j)$, acting on cells $i,j$. It preserves $\mathcal P$ and the algebras $\mathcal O_k$, so $\Lambda^{ij}_{WCW^\dagger}=\Lambda^{ij}_C$. Since $e^{-\mathrm iWCW^\dagger\delta}=WU_\delta W^\dagger$, the chain $WCW^\dagger$ has no influence from $i$ to $j$ at delay $\delta$ if and only if $[U_\delta^\dagger\mathcal A_jU_\delta,\mathcal A_i]=0$ with $\mathcal A_k:=W_k^\dagger\mathcal D_kW_k$. Rotating the contracts is the same as rotating the place bases of cells $i$ and $j$. Fix $\delta\notin\Lambda^{ij}_C$ and write $U=U_\delta$.

- For fixed $W_i$, $\mathcal S:=\{a\in\mathcal O_j:\ [U^\dagger aU,\mathcal A_i]=0\}$ is a unital $*$-subalgebra of $\mathcal O_j\cong M_{d_j}$, and $W$ is bad if and only if $W_j^\dagger p^{(j)}_hW_j\in\mathcal S$ for all $h$. By T3 the bad $W_j$ have codimension $\ge2$ in $U(d_j)$, unless $\mathcal S=\mathcal O_j$.
- $\mathcal S=\mathcal O_j$ if and only if $\mathcal A_i\subseteq\mathcal S':=\{b\in\mathcal O_i:\ [U^\dagger\mathcal O_jU,b]=0\}$. This is a unital $*$-subalgebra of $\mathcal O_i$, and it is proper because $\delta$ is not blind. By T3 such $W_i$ have codimension $\ge2$ in $U(d_i)$.

So at every delay that is not blind, the bad $W$ have codimension $\ge2$ in $\mathcal G$. T2 with $I=(0,\infty)\setminus\Lambda^{ij}_C$ and $\Phi(\delta,W)=F_\delta(WCW^\dagger)$, followed by T1, gives

$$
\operatorname{vol}\bigl\{C\in\mathcal P:\ [U_\delta^\dagger\mathcal D_jU_\delta,\mathcal D_i]=0\ \text{for some }\delta>0\text{ with }\delta\notin\Lambda^{ij}_C\bigr\}=0\qquad(i\neq j,\ n\ge2).
\tag{7.24}
$$

For a generic chain, a positive delay without influence between two different cells is therefore a blind delay. In particular $\mathcal X_{ij}\setminus\mathcal X^{\rm bl}_{ij}$ is null.

**(c) Two cells, $n=2$.** Now $\mathcal P=\mathrm{Herm}(\mathcal H)$, $N:=d_1d_2\ge4$, and T1 applies with $\mathcal G=U(N)$. Let (G) be the property: the eigenvalues $c_a$ of $C$ are simple, and no two gaps $c_a-c_b$, $c_{a'}-c_{b'}$ with $\{a,b\}\neq\{a',b'\}$ have a rational ratio. (G) holds for almost every $C$ (it excludes countably many hyperplanes of spectra) and is invariant under conjugation. Under (G), at most one pair of the numbers $e^{-\mathrm ic_a\delta}$ coincides at any $\delta>0$, so $U_\delta$ has at least $N-1\ge3$ distinct eigenvalues.

- **No blind delays.** For $n=2$ the commutant of $\mathcal O_1$ is $\mathcal O_2$. So $[U^\dagger\mathcal O_2U,\mathcal O_1]=0$ gives $U^\dagger\mathcal O_2U=\mathcal O_2$ (equal dimensions), then $U^\dagger\mathcal O_1U=\mathcal O_1$ (commutants), and $U=u_1\otimes u_2$ because automorphisms of full matrix algebras are inner. Hence $\delta$ is blind for $VCV^\dagger$ if and only if $VZV^\dagger$ is a product unitary, with $Z=U_\delta$. Let $Z$ be any non-scalar unitary with eigenvalues $z_a$.
  - $V\mapsto VZV^\dagger$ is a fibration of $U(N)$ over the conjugacy class of $Z$, of dimension $\#\{(a,b):z_a\neq z_b\}$ (ordered pairs).
  - A product unitary in this class has factors with eigenvalues $\alpha_k$ ($k\le d_1$) and $\beta_l$ ($l\le d_2$) such that $\{\alpha_k\beta_l\}=\{z_a\}$ as multisets. Up to $(\alpha,\beta)\to(\tau\alpha,\beta/\tau)$ there are finitely many such $(\alpha,\beta)$. For each, the product unitaries form a set of dimension $\le n_\alpha+n_\beta$, with $n_\alpha:=\#\{(k,k'):\alpha_k\neq\alpha_{k'}\}$ and $n_\beta$ likewise.
  - The class has dimension $\#\{(kl,k'l'):\alpha_k\beta_l\neq\alpha_{k'}\beta_{l'}\}\ge d_2n_\alpha+d_1n_\beta$ (pairs with $l=l'$, and pairs with $k=k'$). So the codimension is $\ge(d_2-1)n_\alpha+(d_1-1)n_\beta\ge n_\alpha+n_\beta\ge2$, since $n_\alpha+n_\beta$ is even and nonzero for non-scalar $Z$.

  So under (G) the bad $V$ have codimension $\ge2$ at every $\delta>0$. By T2 and T1, $\mathcal X^{\rm bl}_{12}=\mathcal X^{\rm bl}_{21}$ is null, and by (7.24) $\mathcal X_{12}$ and $\mathcal X_{21}$ are null.
- **Same cell.** Let $i=j=1$; cell $2$ is alike with $d_1\leftrightarrow d_2$. The chain $VCV^\dagger$ has no influence at delay $\delta$ if and only if $[Z^\dagger q_hZ,q_{h'}]=0$ for all $h,h'$, where $Z=U_\delta$ and $q_h:=V^\dagger p^{(1)}_hV$ projects on $E_h:=V^\dagger(\lvert h\rangle\otimes\mathcal H_2)$, of dimension $d_2$. Fix one $h$. Commuting projectors and $\sum_{h'}Z^\dagger q_{h'}Z=\mathbb 1$ give $E_h=\bigoplus_{h'}\bigl(E_h\cap Z^\dagger E_{h'}\bigr)$. So $E:=E_h$ satisfies (α) or (β):
  - (α) some unit vector $f\in E$ has $Zf\in E_{h'}$ with $h'\neq h$, hence $Zf\perp E$;
  - (β) $ZE=E$.

  Let $Z$ have at least three distinct eigenvalues $\zeta$, with eigenprojectors $\Pi_\zeta$ and multiplicities $\mu_\zeta$. Both sets have codimension $\ge2$ in the Grassmannian $\mathrm{Gr}(d_2,N)$, of real dimension $2d_2(N-d_2)$:
  - (α): $\langle f\vert Z\vert f\rangle=\sum_\zeta\zeta\,\lVert\Pi_\zeta f\rVert^2=0$. Three distinct points of the unit circle are not collinear, so these are two independent real conditions on the weights $\lVert\Pi_\zeta f\rVert^2$ when at least three weights are nonzero; otherwise $f$ lies in the sum of two eigenspaces, a proper subspace. Such $[f]$ thus have real dimension $\le2N-4$. Given $f$, $E=\mathbb Cf\oplus E'$ with $E'\subseteq\{f,Zf\}^\perp$, a Grassmannian of real dimension $2(d_2-1)(N-d_2-1)$. The total is $\le2N-4+2(d_2-1)(N-d_2-1)=2d_2(N-d_2)-2$.
  - (β): $E=\bigoplus_\zeta(E\cap\operatorname{ran}\Pi_\zeta)$ with dimensions $k_\zeta$, $\sum_\zeta k_\zeta=d_2$. These $E$ have dimension $2\sum_\zeta k_\zeta(\mu_\zeta-k_\zeta)$. It is smaller than $2d_2(N-d_2)=2\sum_{\zeta,\zeta'}k_\zeta(\mu_{\zeta'}-k_{\zeta'})$ by the even number $2\sum_{\zeta\neq\zeta'}k_\zeta(\mu_{\zeta'}-k_{\zeta'})$, which is positive because $0<d_2<N$ and $Z$ has more than one eigenvalue.

  $V\mapsto E_h$ is a fibration of $U(N)$ over $\mathrm{Gr}(d_2,N)$. So under (G) the bad $V$ have codimension $\ge2$ at every $\delta>0$. By T2 and T1, $\mathcal X_{11}$ and $\mathcal X_{22}$ are null.

Together:

$$
n=2:\qquad \operatorname{vol}(\mathcal X)=0,\quad\text{that is, for almost every }C:\quad e_1\prec e_2\iff\lambda_1<\lambda_2\ \ \text{for all events.}
\tag{7.25}
$$

**(d) Longer chains, $n\ge3$: what is decided and what is not.**

- **Decided.** (7.24): between different cells, a generic chain has no exceptional delay other than blind delays. Step 9 (ii): for a generic chain every $\mathcal I_{ij}\neq\emptyset$, every $E_{ij}$ is discrete, and no exceptional delay lies in a prescribed countable set.
- **Undecided (U1).** Whether $\operatorname{vol}(\mathcal X^{\rm bl}_{ij})=0$ for $i\neq j$, that is, whether a generic chain with $n\ge3$ has no blind delay. Blind delays are invariant under all products of single-cell unitaries, so (b) cannot remove them. The proof of (c) conjugates by all of $U(N)$, which varies the eigenvectors of $C$ at fixed spectrum; for $n\ge3$ this does not preserve $\mathcal P$.
- **Undecided (U2).** Whether $\operatorname{vol}(\mathcal X_{ii})=0$ for $n\ge3$ (two readings on the same cell). Here only the one place basis of cell $i$ can be rotated, and the bad bases can have codimension $1$: for a qubit cell on which $U_\delta$ acts as a rotation by $\pi$, they form a curve. T2 then does not apply unless such delays are excluded.
- A count of conditions against parameters suggests that both sets are null. A count is not a proof; no chain with $n\ge3$ is claimed to be free of exceptional delays.

## 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): $\delta P\le2(1-\tfrac1m)\sum_{k\ge r}(2\lVert J\rVert\delta)^k/k!\le2\sum_{k\ge r}(2\lVert J\rVert\delta)^k/k!$, obtained from the chain Lieb–Robinson bound (7.11). It is uniform in $n$ and does not depend on the $D_i$. Only the first form, with the number $m$ of classes of $R_i$, is attained at leading order (for $m=2$, by (7.7)); the reading-independent second form is not.
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). For $\lVert J\rVert>0$ its boundary moves at $v_{\rm LR}=2e\lVert J\rVert$ cells per unit $\lambda$ (7.17); for $\lVert J\rVert=0$ there is no influence between different cells and no boundary.
   - For $\prec$ (7.18):
     - $\prec$ implies $\lambda_1<\lambda_2$: yes (7.19).
     - Does $\prec$ hold for every pair with $\lambda_1<\lambda_2$ when the pair terms are generic? Two statements must be kept apart.
       - **One prescribed delay:** yes, outside a null set of contract terms that depends on the delay (7.20).
       - **One fixed generic chain, all delays** (exceptional set $\mathcal X$, (7.22)):
         - $n=2$: **yes**. For almost every $C$, $e_1\prec e_2\Leftrightarrow\lambda_1<\lambda_2$ for all events (7.25).
         - $n\ge3$, different cells: **partly decided**. A generic chain can fail only at blind delays (7.23), (7.24). Whether a generic chain has blind delays is **not decided** (U1).
         - $n\ge3$, same cell: **not decided** (U2).
       - It does not hold for every chain: (7.21), and chains with a cut link.

       For $n\ge3$ this sub-item is therefore incomplete.
     - $\prec$ is not transitive in general (7.21). It is transitive whenever $C\notin\mathcal X$, which is the generic case for $n=2$ (7.25).

## 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). Every delay is then blind, $\Lambda^{ij}_C=(0,\infty)$, as (7.24) requires of exceptional delays between different cells. 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 first member of (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. This chain has the double eigenvalues $\pm g$, so it violates (G), and at $\delta=\pi/(2g)$ its evolution is a product unitary with only two eigenvalues: the delay is blind, and the chain lies in the null set $\mathcal X$ that (7.25) allows. 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

- **U1** ($n\ge3$, different cells): it is not decided whether a generic chain has no blind delay, i.e. whether $\mathcal X^{\rm bl}_{ij}$ of (7.23) is null. If it is, (7.24) gives $\operatorname{vol}(\mathcal X_{ij})=0$ for $i\neq j$.
- **U2** ($n\ge3$, same cell): it is not decided whether $\mathcal X_{ii}$ is null. If U1 and U2 both hold, (7.25) extends to every $n$, and $\prec$ is then generically the order of $\lambda$ and 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; attainment of the bound is shown only at leading order, for $m=2$.
- 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
- Invariance of Lebesgue measure under conjugation, Tonelli's theorem, Haar measure
- Structure theorem of finite-dimensional $*$-algebras, commutants; dimension counts on conjugacy classes and Grassmannians; stratification of analytic sets and Sard's theorem
- Stirling bounds; Pauli algebra

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