Differences

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--- uiai [2026/03/17 01:39] – [Assumption: Standard setup] pedroortega
+++ uiai [2026/03/17 01:40] (current) – [Definition: Counterfactual 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
 $$
   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, \dot{x}_{\le j}).
@@ Line 426: / Line 426: @@
 To define the world’s $\mathcal{A}$-continuation at $k$, run the following tokenization procedure, initialized from the already-written on-path transcript up to $k-1$. Let $(\dot{\gamma}_j)_{j \ge k}$ be generated as follows:
-- **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$.
+  * **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$.
-- **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_{k} := 1$.
+  * **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_{k} := 1$.
-- **Read transcript chronologically:** For $j \ge k$, let $x_j$ be the next substrate symbol generated by the world on-path. Then sample the next gate value by
+  * **Read transcript chronologically:** For $j \ge k$, let $x_j$ be the next substrate symbol generated by the world on-path. Then sample the next gate value by
 $$
   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, x_{\le j}).
@@ Line 545: / Line 545: @@
 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.**