Contextual state, transport, composition, and obstruction
Mechanistic interpretability recovers local objects. Algebraic interpretability asks whether those objects possess coherent identity across contextual state.
Mechanistic interpretability has become highly effective at discovering local regularities: directions that decode concepts, sparse features that summarize activation structure, circuits that mediate narrow tasks, Jacobians that describe local sensitivity, and interventions that alter behavior. The recurring difficulty is not the absence of local success. It is the inability to determine when local success belongs to a coherent global account.
A probe may work at one layer and fail at another. A feature may split under retokenization. A steering direction may reverse after context extension. A circuit may mediate one prompt family while an alternative path carries the same computation elsewhere. These are not merely engineering defects. They indicate that interpretation itself has algebra, transport, and obstruction.
This is a diagnostic vocabulary and a set of tests. It is not a proof, and it should not be cited as one.
Three statements must be kept apart, because they are routinely fused, and because the fusion is the exact error this document exists to name.
| Statement | Status here |
|---|---|
| Local linear structure is recoverable in neighborhoods of transformer state | Conceded. Measured constantly. Not in dispute. |
| Those neighborhoods glue into one smooth global chart carrying stable semantic coordinates | Denied. This is the load-bearing assumption, and it is unearned. |
| The denial is formally established | Not here. Argued in the companion falsification document. This page assumes neither its truth nor its falsity. |
Nothing in the sections that follow licenses a global smooth chart. Where this document uses manifold language, it uses it the way a surveyor uses a level: as a working instrument applied to a patch of ground, valid where it is placed, and silent about the next valley.
Context should not be reduced to surrounding text. It is the structured model state that determines how an activation, direction, feature, or circuit functions. Position, layer, residual relations, normalization, attention state, token history, cache state, prior interventions, and downstream transformation jointly determine computational role.
An internal object therefore has no complete meaning in isolation. Its identity is relational. What matters is not only the value of an activation, but the transformations it participates in, the states to which it can move, the paths that preserve it, and the operations under which it remains equivalent.
Interpretability methods construct local charts, coordinates, tangent directions, sparse decompositions, or causal summaries on a context-conditioned state space. The construction is legitimate. The question is what the constructions are permitted to be joined into.
Real model state may be stratified, piecewise smooth, dimension-changing, singular at boundaries, disconnected across tokenization regimes, or fractured by quantization and saturation. Smoothness is therefore a local finding to be reported, not a global premise to be inherited.
Linear probes, Jacobians, and steering vectors need not imply that the model is globally linear. They may succeed because nonlinear computation is locally linearizable on limited patches. A probe can recover a local coordinate. A Jacobian can describe tangent behavior. A steering direction can follow a locally meaningful path. A sparse dictionary can approximate one chart.
The error begins when a tangent-space object is promoted into a global semantic axis. Local linear success is evidence about one neighborhood. A global claim requires compatible continuation across the relevant state space, and the continuation is a separate experiment that is almost never run.
At each contextual state there may be a local family of admissible interpretations: candidate features, directions, causal descriptions, or semantic correspondences.
Without a transport rule, the phrase “the same feature across contexts” is undefined. A feature may rotate, split, merge, vanish, or exchange identity with another feature as the model moves. The bundle language is used here as a bookkeeping discipline for that problem. It is not an assertion that the state space is a fiber bundle.
A connection specifies how local interpretive objects correspond across neighboring states. In empirical terms, it asks how a direction at one layer relates to a direction at the next, how a sparse feature survives a prompt rewrite, how a causal role changes under retokenization, or how an internal variable persists through fine-tuning.
Transport should be tested for path dependence, reversibility, composition, and stability. If an object transported through two different contextual paths arrives differently, then its identity is not globally fixed. This is measurable today, with existing tooling, on existing models.
Carry an object around a closed contextual loop and return it to the starting state. Paraphrase, translate, extend, restore. If it comes back changed, the loop has recorded something the endpoints cannot.
In model terms this appears as prompt-history dependence, order effects, noncommuting interventions, or semantic drift after returning to an apparently equivalent context. A feature that returns rotated after a contextual loop is not a single global coordinate, even if it remains locally meaningful everywhere along the path. Loop error is the cheapest available test of a global claim, and it is the one most consistently omitted.
Interpretive descriptions may differ only by coordinate choice. Two probes, dictionaries, or circuit decompositions can appear different while preserving the same causal transitions and invariant relations.
Gauge structure separates real mechanistic disagreement from harmless reparameterization. The aim is not always to identify one privileged basis. It is to identify what remains invariant across admissible gauges. A feature-count dispute in which both parties preserve the same transitions is not an ontology war. It is a change of coordinates conducted at conference volume.
Some distinctions should be collapsed. If multiple states differ only in irrelevant detail while preserving the same causal role, the scientifically useful object may live in a quotient space. Quotienting removes nuisance variation and exposes equivalence classes of states, mechanisms, or interpretations.
The quotient is justified only when the collapsed distinctions are causally irrelevant over the declared domain. An unjustified quotient is not an abstraction. It is a deletion, performed silently, and recoverable only by the person who performed it.
Local interpretations may work in separate regions yet fail to combine into one global account. The gluing question asks whether compatible local sections agree on their overlaps.
Suppose one feature dictionary explains short factual prompts and another explains long compositional prompts. If the overlap between them admits no coherent correspondence, then the two local explanations do not define one global mechanism. The failure may be empirical, representational, or structural, and the three must be distinguished before any of them is excused.
Global understanding is therefore not the accumulation of local charts. It is the successful gluing of compatible local accounts, and accumulation has repeatedly been reported as if it were gluing.
A failed global interpretation may not indicate weak tooling. There may be a genuine obstruction. Local ontologies may be incompatible. Loops may carry nontrivial path dependence. Dimensions may change. Singular boundaries may prevent continuation. Disconnected regions may not support one coordinate system.
Obstruction analysis distinguishes “we have not yet found the global mechanism” from “no global mechanism of the proposed type exists.” This is a decisive difference. It prevents repeated local success from being treated as evidence that a single global ontology is merely waiting to be discovered, and it prevents an impossibility from being rescheduled as future work.
| Tier | Question | Failure looks like |
|---|---|---|
| Local | Does an object survive transport around a loop? | Path dependence, order effects, holonomy |
| Gluing | Do neighboring descriptions agree on their overlap? | Incompatible charts, contradictory dictionaries |
| Global | Does one coordinate system cover the space? | Disconnection, dimension change, no continuation |
The tiers are stated here as questions with observable failure signatures. Whether they are populated, and with what, is an empirical matter argued at length in the companion falsification document. It is not settled by the existence of this vocabulary, and a reader who leaves this page believing a theorem has been proved here has performed the promotion this page describes.
The model does not merely occupy states. It transforms them. A complete account therefore requires vector fields, flows, trajectories, attractors, bifurcations, transition operators, and path-dependent computation.
A mechanism may be better defined by a lawful flow than by a static feature. The stable object may be a transition rule, recurring trajectory, invariant set, or family of admissible paths.
Interpretability methods often study interventions independently. Internal operations may not commute. Applying intervention A then B may differ from B then A. The commutator exposes hidden ordering and interaction structure.
Noncommutativity can reveal causal hierarchy, feature dependence, phase ordering, and invalid assumptions of independent semantic axes. The algebra of composition may be more informative than isolated feature effects. Two safety levers that do not commute are not two safety levers.
Ordinary geometry captures proximity, but mechanism may depend on causal reachability rather than distance. Two states can be close in Euclidean activation space yet causally unrelated. Two distant states may be connected by one valid transition.
The relevant structure may therefore require a causal metric, directed adjacency, or reachability relation. Interpretability should distinguish geometric neighborhood from computational neighborhood, and the distance function in current use is usually inherited rather than chosen.
The state space requires a distribution over states. Otherwise rare, unreachable, or intervention-induced regions are treated as equivalent to naturally occupied regions.
Four notions should remain distinct: geometric possibility, natural occupancy, causal reachability, and deployment relevance. A state may exist mathematically while never arising under valid inputs. Another may be rare but central to safety.
Readable structure is not necessarily controllable. Controllable structure is not necessarily semantically identified. The distinction refines both interpretability and safety claims, and collapsing it is how a working monitor becomes a claimed understanding.
Context Integrity measures whether relevant relational and causal structure remains recoverable across transformation. It asks whether neighborhood relations, paths, transitions, feature correspondences, composition laws, dimensional structure, and causal dependencies survive.
A decoder can remain accurate while Context Integrity fails. The same label may be assigned to states whose internal roles differ. Context Integrity therefore concerns preservation of structured identity, not only classification accuracy.
Algebraic Interpretability studies transformations, symmetries, invariants, composition laws, quotient structure, and obstructions across contextual state. Its preferred object is not the isolated feature but the relation that persists under admissible transformation.
It resolves ambiguity about basis, coordinate choice, local continuation, feature equivalence, composition, and global extension. It does not by itself resolve human semantics, causal use, safety relevance, or completeness. Those still require empirical evidence, and this document confers none.
Two lists circulate in this corpus and they are not the same list. The distinction matters, because conflating them invites the objection that the fracture count is unstable.
A pressure is a transformation applied in the laboratory: retokenize, reformat, rewrite, extend, rescale, quantize, attack, recompose. There are twelve standard pressures, enumerated in the audit and argued individually in the falsification document. A class is the structural type of the failure that results. Several pressures may exhibit the same class; one pressure may exhibit several.
| Class | What fails | Pressures that exhibit it | Tier |
|---|---|---|---|
| Continuity | Nearby states produce abrupt interpretive change | Tokenization, formatting, syntax, adversarial, finite precision | Local |
| Transport | An object cannot be coherently carried across context | Layer and position, context length, model version | Local |
| Composition | Locally valid operations fail when combined | Composition, intervention ordering | Gluing |
| Dimension | Effective structure changes rank or dimension | Normalization, saturation, finite precision | Gluing / Global |
| Causal | Correspondence survives while causal role changes | Layer and position, context length | Local |
| Semantic | One label ceases to identify one stable internal role | Syntax, translation, composition | Gluing |
| Observer | The recovered object exists only under one instrument | Projection, dictionary, seed, width | Pre-tier |
| Restoration | Output returns while native relational structure does not | Ablate-and-repair, substitution, patching | Pre-tier |
The final two classes are marked pre-tier deliberately. They are not properties of the state space at all. Observer fracture concerns whether there was an object to obstruct; restoration fracture concerns whether the repair restored anything. Both must be cleared before a fracture can be attributed to the model rather than to the procedure, and both are routinely skipped in favor of the tiers, which are more interesting.
Restoration should recover more than an answer or a hidden vector. It should restore the correct region of state space, native trajectory, local tangent behavior, transition relations, downstream sensitivity, chart compatibility, and fracture boundaries.
A state can reproduce an output while lying on the wrong branch and following an artificial shortcut. Process restoration is therefore restoration of relational state, and a returned answer is evidence only that an answer returned.
A mechanism is incomplete when it recovers only one chart, one tangent direction, or one region while ignoring alternate components, paths, singularities, dimension changes, and residual causal structure.
Completeness should be reported as coverage of the relevant state space, transition structure, causal flow, and obstruction set. Explained variance alone cannot establish mechanistic coverage.
The framework becomes scientific only through measurable proxies. None of the following requires the mathematics to be settled first, and all of them can be run this quarter.
These provide operational pressure on claims of identity, coherence, and global mechanism. They are also, individually, cheap.
Three documents, three jobs, and they should not be quoted against each other.
| Document | Asks | Commitments |
|---|---|---|
| The audit | What has this evidence earned? | None. Framework-neutral by construction. |
| This document | What structure would be required for local results to become a global account? | Vocabulary and tests. No structural verdict. |
| The falsification | Does that structure exist in transformers? | The verdict, and the burden of defending it. |
The audit can remain neutral because its standards are empirical. This document explains why the same failures recur: local objects lack transport, incompatible charts fail to glue, coordinate choices are mistaken for ontology, and genuine obstructions are treated as temporary inconvenience. The falsification argues that the obstructions are there. A reader may accept the first two and reject the third, and the first two will still work.
Mechanistic interpretability recovers local objects and bounded causal roles. Algebraic interpretability determines whether those objects maintain coherent identity across contextual state.
The model’s internal state is not merely a vector containing features. It is a dynamically traversed, possibly stratified and fractured space with local interpretive fibers, transport laws, symmetries, singularities, and obstructions. Which of those possibilities hold is a question of measurement, and the measurements are available.