Entanglement is bounded by the configuration's deviation (Lem. 2 / 8)

Fix a finite type $S$ with decidable equality (its cardinality is $|S| = \text{card}\,S$), and write $[N]$ for the $N$-element index type $\mathrm{Fin}\,N$. The statement asserts: for every choice of the following data, satisfying all of the listed hypotheses, the displayed inequality holds.

Data and hypotheses.

• Two real matrices $P^0, P^1$ indexed by $S \times S$, each a transition matrix: $P^0_{x y} \ge 0$ and $\sum_{y \in S} P^0_{x y} = 1$ for every $x$, and likewise for $P^1$.
• A function $\nu : S \to \mathbb{R}$ that is injective.
• A real number $\alpha$ with $0 < \alpha < 1$.
• A function $m^\star : S \to \mathbb{R}$ that is a configuration: $m^\star(x) \ge 0$ for all $x$ and $\sum_{x \in S} m^\star(x) = 1$. (No relation whatsoever is assumed between $m^\star$ and $P^0, P^1, \nu, \alpha$, or the dynamics below; it is an arbitrary probability vector on $S$.)
• A natural number $N$ with $N > 0$. Set $M := \lfloor \alpha N \rfloor$, the natural-number floor (so $M = 0$ whenever $\alpha N < 1$, e.g. when $N = 1$).
• A function $\pi : (S^{[N]}) \to (\{\mathrm{false},\mathrm{true}\}^{[N]}) \to \mathbb{R}$ assigning to each joint state $s = (s_j)_{j \in [N]}$ and each joint action $a = (a_j)_{j \in [N]}$ (each $a_j$ a Boolean) a real number, subject to the three conditions of being an index policy for $\nu$ with budget $M$:
  1. $\pi(s,a) \ge 0$ for all $s, a$, and $\sum_{a \in \{\mathrm{false},\mathrm{true\}}^{[N]}} \pi(s,a) = 1$ for every $s$;
  2. for all $s,a$: if $\pi(s,a) \neq 0$ then $a$ is exactly $M$-budgeted, i.e. $\#\{j \in [N] : a_j = \mathrm{true}\} = M$ (equality, not $\le$);
  3. for every joint state $s$ and every agent $j \in [N]$, the marginal probability that agent $j$ is activated,

$$\sum_{a\,:\,a_j = \mathrm{true}} \pi(s,a),$$

 equals $\mathrm{iap}_\nu^M(s, s_j)$, where for $x \in S$ one writes $c_s(x) := \#\{k \in [N] : s_k = x\}$ (the count of agents in state $x$), $h_s(x) := \sum_{y : \nu(x) < \nu(y)} c_s(y)$ (the number of agents in strictly higher-$\nu$ states), and
 

$$\mathrm{iap}_\nu^M(s,x) := \begin{cases} 0, & c_s(x) = 0,\\[2pt] \dfrac{\min\bigl(c_s(x),\; M \dot- h_s(x)\bigr)}{c_s(x)}, & \text{otherwise,}\end{cases}$$

 with $\dot-$ denoting **truncated natural-number subtraction** (so $M \dot- h_s(x) = 0$ when $h_s(x) \ge M$).

• A function $\mu$ on the joint state–action space $\Omega := \prod_{j \in [N]} (S \times \{\mathrm{false},\mathrm{true}\})$ — i.e. $\mu$ assigns a real number to each tuple $p = (p_j)_{j\in[N]}$ where $p_j = (\text{state}_j, \text{action}_j)$ — such that:
  • $\mu(p) \ge 0$ for all $p$ and $\sum_{p \in \Omega} \mu(p) = 1$;
  • $\mu$ is exchangeable: $\mu(p \circ \sigma) = \mu(p)$ for every permutation $\sigma$ of $[N]$ and every $p$;
  • $\mu$ is stationary for the matrix $T$ on $\Omega \times \Omega$ defined by

$$T(p, q) \;=\; \Bigl(\prod_{j \in [N]} K\bigl(\text{state}_j(p),\, \text{action}_j(p),\, \text{state}_j(q)\bigr)\Bigr)\cdot \pi\bigl(\text{states}(q),\, \text{actions}(q)\bigr),$$

where $K(x,a,y) = P^1_{x y}$ if $a = \mathrm{true}$ and $K(x,a,y) = P^0_{x y}$ if $a = \mathrm{false}$; stationarity meaning $\sum_{p \in \Omega} \mu(p)\, T(p,q) = \mu(q)$ for every $q \in \Omega$. (Note the next action in $q$ is drawn by $\pi$ from the *next* state in $q$, and $T$ is not itself asserted to be a transition matrix.)

• A distinguished agent index $i \in [N]$.

Conclusion. With $T$ as above,

$$\mathcal{E}_i(\mu, T) \;\le\; |S|^2 \cdot \sum_{p \in \Omega} \mu(p)\, \Bigl(\sup_{x \in S} \bigl| c_{\mathrm{st}(p)}(x)/N \;-\; m^\star(x) \bigr|\Bigr),$$

where $\mathrm{st}(p) = (\text{state}_j(p))_{j \in [N]}$ is the state part of $p$, the inner quantity $c_{\mathrm{st}(p)}(x)/N$ is the empirical configuration (fraction of the $N$ agents whose state is $x$, a real division by $N > 0$), the supremum is the supremum over $x \in S$ of $|\cdot|$ (a maximum, since $S$ is finite and, as noted below, nonempty), and $|S|^2$ is the square of the real cast of the cardinality of $S$. The left-hand side $\mathcal{E}_i(\mu, T)$ is the agent-$i$ entanglement, defined as the real infimum

$$\mathcal{E}_i(\mu, T) \;=\; \inf\Bigl\{\, r \in \mathbb{R} \;\Bigm|\; \exists\, Q \text{ a transition matrix on } S \times \{\mathrm{false},\mathrm{true}\} \text{ with } r = D(Q) \,\Bigr\},$$ $$D(Q) \;=\; \sum_{p \in \Omega} \mu(p)\cdot \frac{1}{2}\sum_{t \in S \times \{\mathrm{false},\mathrm{true}\}} \Bigl| \Bigl(\sum_{q \in \Omega\,:\, q_i = t} T(p,q)\Bigr) \;-\; Q(p_i,\, t) \Bigr| .$$

Here the competitor $Q$ ranges over all matrices indexed by the pair space $S \times \{\mathrm{false},\mathrm{true}\}$ (agent $i$'s local coordinate is a state–action pair, not a state) whose entries are nonnegative with each row summing to $1$; $Q$ is not required to arise from $P^0, P^1$ or $\pi$ in any way. The quantity $\sum_{q : q_i = t} T(p,q)$ is agent $i$'s marginal one-step weight from the joint point $p$ to the local pair $t$.

Degenerate and edge cases silently included. $N$ may be $1$, in which case $M = \lfloor \alpha \rfloor = 0$ (since $0 < \alpha < 1$), so every action profile carrying positive $\pi$-mass activates exactly zero agents and $\mathrm{iap}$ is identically $0$; more generally $M \dot- h_s(x)$ is truncated at $0$. $S$ cannot be empty: with $N > 0$ an empty $S$ makes $\Omega$ empty and $\sum_p \mu(p) = 0 \ne 1$, so the hypothesis that $\mu$ is a distribution is unsatisfiable there — the statement is vacuous for empty $S$. The set whose infimum defines $\mathcal{E}_i$ is nonempty and bounded below by $0$, so the infimum is the ordinary real infimum. The hypotheses on $\pi$ and on $\mu$ (index policy, distribution, exchangeability, stationarity for $T$) are assumptions only; the statement makes no claim that such $\pi$ or $\mu$ exist for given $P^0, P^1, \nu, \alpha, N$, and if for some data no such $\pi$ or $\mu$ exists the assertion is vacuously true for that data. The hypotheses of injectivity of $\nu$, of $0 < \alpha < 1$, and that $m^\star$ is a configuration are assumed but the right-hand side depends on $m^\star$ only through the displayed deviation; nothing forces $m^\star$ to be a fixed point of any mean-field map, and nothing forces $N$ to be large — the bound is claimed for every $N \ge 1$.