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--- uiai [2026/03/17 01:38] – [Corollary: Preference pluralism] pedroortega
+++ uiai [2026/03/17 10:30] (current) – [Definition: Third-party action] pedroortega
@@ Line 398: / Line 398: @@
 Generate $(\dot{\gamma}_j,\dot{x}_j)_{j \ge k}$ as follows:
-- **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$, $\dot{x}_{\le k-1} := x_{\le k-1}$.
+  * **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$, $\dot{x}_{\le k-1} := x_{\le k-1}$.
-- **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_k := 1$.
+  * **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_k := 1$.
-- **Evolve branch chronologically:** For $j \ge k$, first sample the next substrate symbol by $\dot{x}_j \sim \mu(\cdot \mid \underline{a\hat{o}}_{<t} a_t\,w\,\dot{x}_{k:j-1})$, so $\mu$ emits the content of the forced $\mathcal{A}$-block in the branch, conditioned on the shared past and the already-emitted branch block prefix. Then sample the next gate value by
+  * **Evolve branch chronologically:** For $j \ge k$, first sample the next substrate symbol by $\dot{x}_j \sim \mu(\cdot \mid \underline{a\hat{o}}_{<t} a_t\,w\,\dot{x}_{k:j-1})$, so $\mu$ emits the content of the forced $\mathcal{A}$-block in the branch, conditioned on the shared past and the already-emitted branch block prefix. Then sample the next gate value by
 $$
   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, \dot{x}_{\le j}).
@@ Line 416: / Line 416: @@
 Note that $k'$ is determined inside the branch and therefore the length of $\dot{a}_{t+1}$ need not match the length of the on-path $\mathcal{A}$-token written by the agent starting at $k$.
+{{ ::uiai-cf-action.png?600 |Counterfactual Action}}
 **Diagram note.**
-The intended picture is that after the shared on-path prefix ending at $a_3,o_3$, the on-path transcript has factual action $a_4$, while the counterfactual branch replaces that with the world-generated block $\dot{a}_4$ occupying the same would-be action slot.
+The diagram shows that after the shared on-path prefix ending at $a_3,o_3$, the on-path transcript has factual action $a_4$, while the counterfactual action $\dot{a}_4$ is spawn from the same prefix but generated by the world.
 ==== Definition: Third-party action ====
@@ Line 426: / Line 428: @@
 To define the world’s $\mathcal{A}$-continuation at $k$, run the following tokenization procedure, initialized from the already-written on-path transcript up to $k-1$. Let $(\dot{\gamma}_j)_{j \ge k}$ be generated as follows:
-- **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$.
+  * **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$.
-- **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_{k} := 1$.
+  * **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_{k} := 1$.
-- **Read transcript chronologically:** For $j \ge k$, let $x_j$ be the next substrate symbol generated by the world on-path. Then sample the next gate value by
+  * **Read transcript chronologically:** For $j \ge k$, let $x_j$ be the next substrate symbol generated by the world on-path. Then sample the next gate value by
 $$
   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, x_{\le j}).
@@ Line 448: / Line 450: @@
 o_t = w\,\dot{a}_{t+1}\,v.
 $$
+{{ :uiai-third-party.png?600 |}}
 **Diagram note.**
-The intended picture is that inside a long world-written observation token, there is an embedded block $\dot{a}_4$ that occupies an $\mathcal{A}$-position under the tokenization convention, even though on-path it is still written by the world and therefore appears as evidence.
+The diagram illustrates a third party action. Inside a long observation token $o_3$, one can identify an embedded block $\dot{a}_4$ that is interpreted as a third-party action. Because it is written by the world, it counts as evidence.
 It is not hard to see that, for a given potential index $k$, the counterfactual action $\dot{a}_{t+1}$ and the third-party action $\dot{a}_{t+1}$ are the same random block: the difference is only whether the gate sampled $\gamma_k = 1$ (counterfactual, not observed) or $\gamma_k = 0$ (third-party, observed) at position $k$. The precise distinction between the different types of $\mathcal{A}$-tokens is important; we will also refer to them as //factual// (first-person, agent-generated), //counterfactual//, and //third-party// $\mathcal{A}$-tokens.
@@ Line 545: / Line 549: @@
 Assume $(\Sigma,\Gamma,\pi,\mu)$ is an interaction system where $\pi := M$ is the //universal semimeasure// and $\mu$ is a //primitive measure//. Let $(k_i)_{i \ge 1}$ be an action-slot schedule. The following conditions hold:
-- **Action-slot is chosen by coin flip.**
+  * **Action-slot is chosen by coin flip.** At each $k_i$, the gate draws $\gamma(k_i) \sim \mathrm{Bernoulli}(\rho_i)$, $\rho_i \in (0,1)$, where $\rho_i$ is a chronological function of the agent-visible history $h_i$. Conditional on $h_i$, the bit $\gamma(k_i)$ is independent of the world’s $\mathcal{A}$-token $\dot{a}^{(k_i)}$ at $k_i$.
-  At each $k_i$, the gate draws $\gamma(k_i) \sim \mathrm{Bernoulli}(\rho_i)$, $\rho_i \in (0,1)$, where $\rho_i$ is a chronological function of the agent-visible history $h_i$. Conditional on $h_i$, the bit $\gamma(k_i)$ is independent of the world’s $\mathcal{A}$-token $\dot{a}^{(k_i)}$ at $k_i$.
-- **Gate held fixed through action-slot.**
+  * **Gate held fixed through action-slot.** The gate holds the value of $\gamma(k_i)$ fixed throughout the $\mathcal{A}$-token beginning at $k_i$. If $\gamma(k_i)=0$, the world writes the $\mathcal{A}$-token, so it is a third-party action. If $\gamma(k_i)=1$, the agent writes the $\mathcal{A}$-token, so it becomes an intervention $\hat{a}$ from the agent’s view.
-  The gate holds the value of $\gamma(k_i)$ fixed throughout the $\mathcal{A}$-token beginning at $k_i$. If $\gamma(k_i)=0$, the world writes the $\mathcal{A}$-token, so it is a third-party action. If $\gamma(k_i)=1$, the agent writes the $\mathcal{A}$-token, so it becomes an intervention $\hat{a}$ from the agent’s view.
-- **Infinitely many agent-written slots.**
+  * **Infinitely many agent-written slots.** With probability $1$, $\gamma(k_i)=1$ occurs for infinitely many $i$.
-  With probability $1$, $\gamma(k_i)=1$ occurs for infinitely many $i$.
 **Induced agent interventions and world targets.**