The Observer-Relative Identifiability Boundary Between Learning and Inference

Conclusion#

There is no single observer-independent YES or NO to the question, “Can learning and inference be distinguished inside a model?” The exact answer is the following necessary-and-sufficient condition:

An observer can distinguish the current role exactly if and only if the declared role variable is measurable with respect to that observer's information filtration.

If a role $C\in\{L,I\}$ induces transcript laws $P_L,P_I$, then the optimal equal-prior misclassification probability is

1. Declare the role and observer first#

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

The information available to an observer through time $t$ is represented by a sub-$\sigma$-algebra $\mathcal F_t^O$. This filtration may include activations, inputs, outputs, a clock, observer memory, and the results of permitted interventions.

Location is not the decisive issue.

• An observer located inside the model remains limited if it cannot inspect optimiser state or reset responses.
• An observer outside the model can have the same information filtration if it receives the same latent stream.
• Even access to raw hidden state does not recover the role automatically when the role is not a function of that state.

2. Theorem A — Measurability characterisation#

For a probability space $(\Omega,\mathcal G,\mathbb P)$, a role $C$, and observer information $\mathcal F\subseteq\mathcal G$, the following statements are equivalent:

1. an $\mathcal F$-measurable classifier exists that predicts $C$ almost surely;
2. $C$ is $\mathcal F$-measurable;
3. the posterior $\eta=\mathbb P(C=L\mid\mathcal F)$ is almost surely either 0 or 1.

The minimum risk over all classifiers is

A larger internal network or a longer computation is only a measurable function of the existing $\sigma$-algebra. If the input information does not separate the roles, additional computation cannot create new role information.

3. Theorem B — Exact risk for a fixed transcript#

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

Consequently,

The verdict is determined not by different means or weights, but by the entire transcript law that actually reaches the observer.

4. Theorem C — Adaptive experiments#

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

The infimal risk over permitted policies and classifiers is

Positive TV is equivalent to performance above chance; finite-time zero error requires singularity under one permitted policy. Even if $\Delta_T=1$, the supremum need not be attained, so the existence of a single zero-error policy must be checked separately.

5. Theorem D — Response-fibre factorisation#

Let $E$ be a set of hidden events and $\rho:E\to\{L,I\}$ a role map. Identify events that produce the same response profile under every permitted policy and horizon:

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

If $L$ and $I$ conflict within the same response fibre, no amount of observer-side computation can resolve the conflict. Conversely, factorisation establishes in-principle classification by profile; it does not automatically provide a single finite test.

6. Theorem E — Garbling contraction#

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

Thus, additional computation by a weaker observer cannot recover provenance that has been lost. Neither the spatial claim that “inside always knows more than outside” nor its converse is valid. The declared channel determines the information ordering.

7. Theorem F — Infinite horizon#

For one fixed infinite policy, let $P_{L,t},P_{I,t}$ 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:

This is a result for one coherent infinite policy. It must not be confused with a profile that selects a different oracle policy at every horizon.

8. Theorem G — Common quotient kernel#

If two processes have the same observable quotient transition and their output kernels do not read the hidden fibre directly, different hidden-fibre updates can remain indistinguishable in the declared adaptive transcript. This result is stated as a sufficient condition. Its assumptions fail if the output reads the fibre directly or if a policy is given the role label in advance.

9. Reset-relative operational specialisation#

Let $P_e^\pi$ denote the probe law after event $e$, $P_{e,r}^\pi$ the law after a typed reset $r$, and $P_{0,r}^\pi$ the baseline law in which the event did not occur. Define

• $D_r(e)>0$ indicates persistent adaptation relative to this reset contract.
• If $D_r(e)=0$ but the pre-reset response differs, the event is reset-local inference.

This supplies one operational definition of learning and inference roles. Changing the reset may change the role; it is not a universal definition covering all fast learning and continual adaptation.

10. Strictness of passive and richer ports#

In a linear toy model, two systems with the same passive law can separate with TV 1 under a write; reset; read policy. Merely adding an operation named “reset,” however, is not sufficient. If distinct persistence operators $R_1,R_2$ satisfy

they remain in the same partial response fibre. The identifying power of a richer port is itself interface-relative.

11. Scope of proof and verification#

The canonical audit establishes the following scope:

The finite exact verifier passed 8/8 checks, including 225 distribution pairs, 8 deterministic garblings, 27 stochastic kernels, and 64 adaptive quotient policies. This computation does not replace the general measure-theoretic proof; it audits signs, normalisation, and finite counterexample implementations.

What is not proved#

• An observer-independent ontological definition of learning and inference
• Identification of the observer filtration in a real Transformer
• Finite-sample estimation rates for a classifier
• That this theorem constitutes a neural-specific ULR object

The Canon 24 increment is therefore not an ontology, but a complete observer-relative characterisation of the identifiability boundary for a declared role.