to2d

A space for ideas, notes, and ongoing work.

Domain Mismatch & Hallucination

Correct operators on incorrect representations

Hallucination is not a failure of intelligence. It is the natural outcome of applying a correct operator to an incorrect domain representation.

This section formalizes why LLMs produce coherent but wrong outputs, why these errors are structurally predictable, and how domain mismatch creates unstable latent trajectories.

1. Hallucination as a domain error

LLMs operate over vector representations, not meanings. When the input representation does not encode the intended domain:

  • the latent state lands in the wrong manifold region,
  • the operator applies the correct transformation to the wrong state,
  • the output appears coherent but is structurally invalid.

The model isn't inventing; it is following the physics of its latent space.

2. Latent-space manifold boundaries

Every domain corresponds to a region of the model's manifold. For example:

  • legal structures → one basin,
  • step-by-step reasoning → another basin,
  • creative writing → another basin.

When a prompt mixes structures from different domains:

  • embeddings overlap,
  • boundaries blur,
  • latent trajectories cross unintended regions.

The model resolves ambiguity by falling into the nearest stable attractor — which may not match the intended task.

3. Structural ambiguity as the root cause

Hallucination is not randomness. It is resolution of ambiguity.

Ambiguity arises when:

  • instructions conflict with examples,
  • schemas are incomplete,
  • traces from previous steps remain in context,
  • the prompt encodes multiple domains at once.

The model compresses all of this into a single latent state. This forces structural contradictions into one representation, guaranteeing drift.

4. Attractor misalignment

Human-style domains do not matter.
Latent attractors do.

If the encoded input lies closer to a pattern the model has reinforced heavily in training, the trajectory is pulled toward that basin.

Examples:

  • "Fix this JSON" drifting into explanation text,
  • "Extract fields" drifting into creative synthesis,
  • "Analyze" drifting into chain-of-thought templates,
  • "Summarize a document" drifting into hallucinated summaries when structure is missing.

The attractor wins, not the intent.

5. Coherence is not correctness

Coherence is a property of the attractor dynamics.
Correctness is a property of the domain.

When the domain is mismatched:

  • the model produces a trajectory that is internally consistent,
  • but the mapping between input and output is structurally invalid.

This is why hallucinations often read smooth and authoritative.

6. Domain collapse under mixed signals

When two domains coexist in the same prompt, the model must:

  • compress them,
  • unify them,
  • produce a single trajectory.

This collapse creates outputs that seem "invented" but are in fact statistical interpolations across incompatible regions of the manifold.

7. Why retrying doesn't fix it

If the domain is wrong:

  • retries land in the same region,
  • the attractor remains dominant,
  • the structure of the error persists.

Hallucinations repeat because the representation error repeats.

8. The fix: enforce domain alignment before the operator

Hallucination drops dramatically when:

  • the domain is isolated,
  • ambiguity is removed,
  • schemas are enforced,
  • the prompt is rewritten into a form that matches a stable manifold region.

This is the core mechanic behind:

  • 0-context,
  • Domain Intelligence,
  • verifiable agent loops,
  • deterministic planning pipelines.

These systems reduce hallucination not by improving the model but by controlling its initial conditions.

9. Formalizing domain mismatch

Let D₁ be the intended domain.
Let E(x) be the encoding of input x.
Let M be the model operator.
Let Y be the output domain.

If E(x) ∉ manifold(D₁), then:
    M(E(x)) → Y′ where Y′ ∉ Y

This is hallucination: a mapping to a valid region of Y, but not the correct one.

10. Research directions

Areas for deeper exploration:

  • mapping basin transitions caused by mixed-domain prompts,
  • measuring drift distance between intended vs actual manifold regions,
  • designing canonicalization processes that force domain purity,
  • attractor shaping through minimal representations,
  • formal evaluation metrics for domain alignment.

These elements form the empirical backbone for predictable, verifiable agentic systems.

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