The Observer-Relative Identifiability Boundary Between Learning and Inference

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1. #1

   If a role induces transcript laws , then the optimal equal-prior misclassification probability is

   $$\boxed{ R^*=\frac{1-\lVert P_L-P_I\rVert_{\mathrm{TV}}}{2} }$$
2. #2

   Learning and inference are not primitive symbols of a probability space. A system boundary must first declare which events it calls or . Role typing may include

   $$(\text{information source},\ \text{write target},\ \text{persistence}, \ \text{reset/clone response},\ \text{future causal role}).$$
3. #3

   The minimum risk over all classifiers is

   $$R^*_{\mathcal F} = \mathbb E[\min\{\eta,1-\eta\}].$$
4. #4

   Fix an observer contract and horizon, and let the transcript space be . The optimal binary risk for the two simple hypothesis laws is given exactly by the total-variation identity

   $$R^*=\frac{1-\lVert P_L-P_I\rVert_{\mathrm{TV}}}{2}, \qquad \lVert P_L-P_I\rVert_{\mathrm{TV}} =\sup_{A\in\mathscr H}|P_L(A)-P_I(A)|.$$
5. #5

   Consequently,

   $$R^*=\frac12\iff P_L=P_I, \qquad R^*=0\iff P_L\perp P_I.$$
6. #6

   Suppose the observer can use a nonanticipating, role-blind policy that selects actions from the past transcript. Define the separation coefficient at horizon by

   $$\Delta_T(O) = \sup_{\pi\in\Pi_O} \lVert P_{L,T}^{\pi}-P_{I,T}^{\pi}\rVert_{\mathrm{TV}}$$
7. #7

   The infimal risk over permitted policies and classifiers is

   $$R_T^*(O)=\frac{1-\Delta_T(O)}{2}$$
8. #8

   Let be a set of hidden events and a role map. Identify events that produce the same response profile under every permitted policy and horizon:

   $$e\sim_O e' \iff P_{e,T}^{\pi}=P_{e',T}^{\pi} \quad\text{for all }(\pi,T)$$
9. #9

   A universal observer classifier valid for every event exists if and only if factors through this quotient:

   $$\rho=\bar\rho\circ q_O.$$
10. #10

    If an internal transcript is obtained by garbling an external transcript through a Markov kernel , then

    $$\lVert KP_L-KP_I\rVert_{\mathrm{TV}} \le \lVert P_L-P_I\rVert_{\mathrm{TV}}.$$
11. #11

    For one fixed infinite policy, let be the finite-prefix laws. Prefix TV converges to full path-law TV. The optimal finite-prefix error tends to zero if and only if the full path laws are mutually singular:

    $$R_t^*\to0 \iff P_L^{\infty}\perp P_I^{\infty}.$$
12. #12

    Let denote the probe law after event , the law after a typed reset , and the baseline law in which the event did not occur. Define

    $$D_r(e) = \sup_{\pi\in\Pi_{\mathrm{probe}}} \lVert P_{e,r}^\pi-P_{0,r}^\pi\rVert_{\mathrm{TV}}.$$
13. #13

    In a linear toy model, two systems with the same passive law can separate with TV 1 under a policy. Merely adding an operation named “reset,” however, is not sufficient. If distinct persistence operators satisfy

    $$HR_1B=HR_2B$$