ora-0080
5.1 The doctrinal audit
Recall that an ORACLE problem identifies a compositional world only up to the equivalence detected by a declared query doctrine. A doctrinal audit asks what had to be fixed before the familiar learner could even state its objective.
A learning paradigm is a query-relative UOCL specialization when it specifies:
a structural doctrine \(\mathfrak D\) and a category \(\mathbf H_{\mathfrak D}\) of admissible theories, models, and structure-preserving comparisons;
a presentation protocol \(\mathbf{Pres}\) revealing finite evidence;
a query doctrine \(\mathcal Q\) of questions posed to hypotheses;
an internal solution construction, such as conditioning, minimization, a fixed point, final semantics, or predictive completion;
observational equivalence \(\simeq _{\mathcal Q}\) induced by those questions;
a non-anticipatory update, selection, or posterior mechanism; and
a finite, limiting, probabilistic, or regret-based success criterion, together with a boundary separating repairs internal to \(\mathfrak D\) from evidence against the doctrine itself.
It is structurally faithful when the morphisms and compositions of \(\mathbf H_{\mathfrak D}\) affect at least one admitted query, solution, update, repair, or transport claim.
The last condition prevents a vacuous recoding. Any set can be regarded as a discrete category, but doing so explains nothing about learning. A useful specialization identifies compositions that constrain predictions, relate models, transport solutions, or localize revisions.
A specialization is fixed-doctrine when every learner state and update remains in \(\mathbf H_{\mathfrak D}\). It becomes doctrine-learning only when the learner state also records \(\mathfrak D_t\) and updates may follow declared doctrine maps \(\mathfrak D_t\to \mathfrak D_{t+1}\), while transporting the structure that remains warranted.
Fixed-doctrine learning is not a defect. It is the source of the strong existence, convergence, and sample-complexity results available in mature fields. The distinction concerns the scope of what has been learned. A proof that an algorithm converges inside \(\mathfrak D\) cannot by itself establish that interaction selected \(\mathfrak D\) from competing doctrines.
Suppose a learning method supplies the items of Definition 5.1, and its update maps respect reindexing of presentations and the declared observational equivalence. Then it determines a UOCL instance in the sense of Definition 6.1. If the update also conservatively transports settled query answers, it determines a persistent UOCL instance on that settled doctrine.
Take \(\mathbf H_{\mathfrak D}\) as the hypothesis fiber, pull it back along the presentation protocol, and use the method’s updates as lift selectors. Reindexing supplies non-anticipation, while compatibility with \(\simeq _{\mathcal Q}\) makes the query quotient well defined. The stated conservativity condition is precisely persistence of settled answers.
The proposition is an interface theorem, not a claim of algorithmic equivalence. It says exactly what must be exhibited before calling an existing method a UOCL specialization.
Let \(\mathfrak D\) and \(\mathfrak D'\) admit hypotheses \(M\) and \(M'\) that induce the same presentation law and agree on every query in \(\mathcal Q\). No learner whose input and success criterion factor only through that presentation and query doctrine can determine whether the underlying world was \((\mathfrak D,M)\) or \((\mathfrak D',M')\).
The learner receives identically distributed evidence and every scored answer is equal in the two cases. Its observable execution therefore has the same law under both hypotheses. Any claimed doctrinal distinction would require an additional probe, intervention, or prior not present in the declaration.
This elementary obstruction is the recurring lesson of the chapter. Bellman convergence can certify an optimum in an MDP without certifying Markovity; likelihood can select a sequence law without identifying a grammar; and observational fit can select a causal graph only up to the equivalence exposed by the admitted data and assumptions.
|
Paradigm |
Fixed structural doctrine |
Internal solution construction |
Query quotient and repair boundary |
|---|---|---|---|
|
Automata |
Typed transition coalgebras with fixed alphabet, outputs, and often finite carriers |
Final behavior, minimization, and counterexample refinement |
Behavioral quotient; revise the endofunctor, types, or finiteness assumption when it fails |
|
System identification and PSRs |
Controlled stochastic processes with composable kernels and admitted tests |
Conditioning and finite predictive state update |
Predictive equivalence; revise stationarity, observability, or test sufficiency |
|
Topos World Models |
Context-indexed local models with restriction, provenance, and descent |
Typed extraction followed by local update and sheaf gluing |
Contextual predictive quotient; revise the cover, gluing law, or causal warrant |
|
Reinforcement learning |
Markov stochastic dynamics enriched by reward, policies, order, and Bellman structure |
Planning, Bellman fixed points, or stochastic approximation |
Bisimulation or policy-sufficient quotient; revise Markovity, observability, stationarity, or value structure |
|
Causal discovery |
DAGs or structural mechanisms with declared intervention semantics and independence assumptions |
Constraint, score, or functional-model identification |
Markov or interventional equivalence; revise acyclicity, sufficiency, modularity, or variable ontology |
|
Sequence modeling |
Token and prefix categories with conditional continuation laws and positional structure |
Likelihood fitting, attention, and autoregressive generation |
Predictive sequence equivalence; revise tokenization, grammar, grounding, or memory assumptions |
|
JEPA and learned world models |
Latent quotients with predictive targets, readouts, and sometimes action dynamics |
Representation prediction and model-based planning |
Representation- or control-sufficient quotient; revise the quotient when later probes need discarded distinctions |