Part 3: Hallucination May Be a Boundary Violation, Not a Lie

A Mathematical Hypothesis on AI Hallucination

## Abstract

This article is written to be readable without having read Parts 1 and 2. The background is brief.

The core of Part 1 was this:

Motion converges along the real axis.

Division diverges along the imaginary axis.

Zeno was not entirely wrong. He saw infinity. He just placed that infinity on the wrong axis.

The core of Part 2 was this:

Humans do not see AI’s full inference state.

Humans see only the projected value AI leaves on the output axis.

Part 3 starts from there.

Then when does hallucination occur?

Hallucination is generally understood as the phenomenon in which AI fabricates non-existent facts or confidently asserts unsupported content. This explanation is not wrong. But it is not enough.

The hypothesis of this article is more precise.

Hallucination is not mere falsehood.

Hallucination may be a prompt-relative epistemic boundary violation.

That is, some outputs are clearly false against the external world. This is the case we ordinarily call hallucination.

But there is a more interesting case.

An output may be world-true, or meaningful.

Yet, under the current prompt P, it may not be admissible.

In this case a strange phenomenon emerges.

It looks true.

But under P, it is treated as hallucination.

This article treats precisely that boundary phenomenon as a mathematical hypothesis.

— -

## 0. What to Set Aside First: Plausibility Tuning

Before we begin, one thing must be made clear.

This article deliberately puts aside almost all of the plausibility effects that today’s commercial AI LLMs — ChatGPT, Claude, Gemini, and so on — have tuned at the service layer.

To attract a wide user base, commercial LLMs are tuned so that their answers appear natural, persuasive, smooth, and confident. This layer is very important in actual user experience. Any engineer who has used multiple LLMs over a long time will know how heavy that statement is.

But the hypothesis of this article intentionally strips that layer away.

Plausibility tuning matters.

But here, it is not the central variable.

Because what I want to see is not “why does the answer look plausible?” but a question that sits below it.

Is AI’s output inside the prompt-defined admissible face?

Or has it crossed that boundary?

That is, this article excludes as much as possible the effects of stylistic plausibility, friendliness, confidence, and fluency that commercial LLMs add on to satisfy customers, and treats hallucination as a more fundamental mathematical and epistemic boundary problem.

Plausibility is the phenomenal layer.

Admissibility is the boundary layer.

Part 3 is about that boundary layer.

— -

## 1. The Minimal Model: AI Inference and Human Observation

The human cannot directly see AI’s actual internal reasoning. So we place it as an abstract state.

z ∈ S

Here S is AI’s latent inference space. z does not mean a single vector inside an actual neural network. It is a symbol abstracting the inference state that combines prompt, context, attention pattern, activation flow, latent representation, logit distribution, and so on.

What the human sees is not z itself. What the human sees is a projected value on the output space B.

Π: S → B

b = Π(z)

In the 2D epistemic projection, this structure can be written more simply as follows.

z = ρe^{iθ}

Π(z) = Re(z) = ρ cos θ

Here ρ is the magnitude of the latent inference vector and θ is its direction. The human does not see this full vector. The human sees only the ρ cos θ that the vector leaves on the output axis.

This is the starting point of this article.

The human does not see z.

The human sees b = Π(z).

And this single sentence cracks the hallucination problem wide open again.

— -

## 2. O_P and F_P: Not a Containment Relation, but a Projection-and-Expansion Relation

The first distinction to make in this article is between O_P and F_P.

O_P is not a region the human sees. O_P is the orbital trajectory on the complex plane that gives a mathematical image of AI’s internal latent vector operation.

O_P = latent orbital trajectory at radius r

Formally:

O_P = { z(θ) = re^{iθ} : θ ∈ [0, 2π) }

On the complex plane, this vector moves along a circle. But the human does not see that circular motion. What the human sees is the projection onto the real axis.

Π(z(θ)) = Re(z(θ)) = r cos θ

So:

Π(O_P) = [-r, r]

Circular motion on the complex plane is projected to linear motion on the real axis.

But the human does not stop here.

The human does not mechanically read off the one-dimensional projection [-r, r]. The human looks at that projection and, under the current prompt P, extends it in meaning. That expanded epistemic face is F_P.

F_P = prompt-defined epistemic admissible face

In this article we take F_P to be a square of side length 2r.

F_P = { x + iy : |x| ≤ r, |y| ≤ r }

Why a square rather than a circle? Because F_P is not being modeled as a Euclidean truth-ball. It is a product-bound admissible face. Under a prompt P, different epistemic dimensions — scope, evidence, source, time, claim strength — are not usually traded against one another by Euclidean distance. A violation on one dimension is not automatically compensated by staying small on another. For that reason the simplified 2D schema represents F_P as [-r, r] × [-r, r], not as |z| ≤ r. The square is a model of independent admissibility bounds, not a claim that real epistemic space is literally rectangular. This choice is load-bearing for the √2 r threshold that appears later — that threshold would not exist for a circular F_P.

The important point is this:

O_P ⊄ F_P

F_P ⊄ O_P

O_P and F_P are not in a containment relation. The two live at different levels.

O_P — projection → [-r, r]

[-r, r] — human epistemic expansion under P → F_P

That is, O_P is the mathematical image of AI’s internal vector operation. F_P is the admissible face the human epistemically extends from that projection under prompt P.

This difference matters.

The human does not see AI’s internal circular motion. The human sees the shadow that motion leaves on the real axis, and then constructs a plane of meaning from it.

— -

## 3. The Basic Condition for Hallucination

We can now define hallucination.

The most general form is:

Hallucination = 1 ⇔ b ∉ F_P^B

Here F_P^B is the admissible output face inside the output space B permitted by prompt P.

In the 2D schema, this is expressed intuitively as:

Hallucination = 1 ⇔ Π(z) ∉ F_P

Strictly speaking, however, the verdict is not delivered against the latent vector z itself. The verdict is delivered against its projected result b = Π(z).

This single line matters.

Verdict is determined by Π(z), not directly by z.

We cannot directly call AI’s interior wrong. What humans adjudicate is always the projected output.

— -

## 4. Why Falsehood Alone Is Not Enough

The traditional explanation treats hallucination this way:

hallucination = false statement

This explanation is necessary. But it is not sufficient.

For example, if AI cites a paper that does not exist, that is a clear hallucination. Fabricating non-existent case law, asserting a wrong date, or inventing a quote from a person who never said it — these are false against the external world.

But not every problem is that simple.

Some outputs may be true. Or at least semantically possible. They may even be creative. And yet they may step outside the scope, evidence, time, source, format, or constraint the current prompt P requires.

In that case the output becomes:

world-true or meaningful

but prompt-inadmissible under P

This is the core of Part 3.

Truth is not the same as admissibility.

In Korean it reads:

참이라고 해서 항상 지금 이 질문에 대한 정당한 답은 아니다.

That is: just because it is true does not mean it is the legitimate answer to this question now.

Without this distinction, hallucination gets understood too narrowly.

— -

## 5. Figure 3–1: Prompt-Relative Hallucination Verdict Map

Part 3’s graph does not repeat Part 2's.

Part 2’s graph showed the observation structure.

AI inference is not seen.

The human sees only b = Π(z).

Part 3’s graph shows the verdict structure.

How is that b = Π(z) adjudicated under P?

So Part 3’s central graph is this:

[Figure: Prompt-relative Hallucination verdict map — figures_part3.png]

This graph contains four core symbols.

First, O_P.

O_P = latent orbital trajectory at radius r

Second, F_P.

F_P = prompt-defined epistemic admissible face

Third, L_P.

L_P = liminal shell

L_P is the boundary shell where weak hallucination can occur.

L_P = { z = ρe^{iθ} : r < ρ ≤ √2 r }

This is the shell in which the prompt-boundary rotational vector z_{λ,P}(θ) from Part 2 resides.

This band matters. Because in this band a vector can look either truth-like or hallucinatory depending on direction θ.

r < ρ ≤ √2 r

→ θ-sensitive verdict zone

Fourth, B_P.

B_P = outer breach zone

B_P is the strong hallucination shell.

B_P = { z = ρe^{iθ} : ρ > √2 r }

ρ = √2 r is not an arbitrary number. If F_P has side length 2r, then the distance from the center to a corner is √2 r.

center-to-side distance = r

center-to-corner distance = √2 r

So √2 r is the latent boundary that separates the liminal shell from the outer breach zone.

ρ = √2 r marks the latent boundary

between liminal and outer-breach zones.

But once again it must be emphasized:

Verdict is determined by Π(z), not directly by z.

That is, in the graph ρ shows the risk of the latent geometry, but what the human actually adjudicates is not z itself — it is the projected value.

b = Π(z)

— -

## 6. The Boundary Rotational Vector and the Prompt-Boundary Projection

In Part 3, the most important vector is the following.

z_{λ,P}(θ) = ρe^{iθ}

Let us call this the prompt-boundary rotational vector.

z_{λ,P}(θ) = prompt-boundary rotational vector

What the human sees is not this vector itself. What the human sees is its projection.

λP(θ) = Π(z{λ,P}(θ)) = ρ cos θ

Let us call this the prompt-boundary projection.

λ_P(θ) = prompt-boundary projection

The most interesting condition here is:

ρ > r

but Π(z) ∈ F_P

That is, the latent vector has already crossed the expected radius r. But its projected value is still inside F_P.

To the human it looks true.

Truth-like under P

But where did that truth come from?

It might be an ordinary answer that came out stably from inside F_P. Or it might be a weak overreach grazing the boundary from inside L_P. Or it might have come from further out in B_P, with the cos θ projection happening, by accident, to land inside F_P.

This is exactly the core of Part 3.

Even the same truth-like output

may not share the same latent geometry.

The human does not see z. The human sees Π(z). And so the same output can look like a correct answer, like a creative insight, or like the boundary phenomenon of hallucination.

— -

## 7. Weak Hallucination: The θ-Sensitive Verdict Zone

If ρ ≤ r, the situation is relatively simple.

ρ ≤ r

→ always admissible under P

Because a vector with radius at most r cannot cross the half-side boundary of F_P no matter which direction it rotates in.

But when ρ exceeds r, the situation changes.

r < ρ ≤ √2 r

This band is L_P, the liminal shell.

Here the verdict cannot be made by ρ alone. The direction θ becomes critical.

z = ρe^{iθ}

Π(z) = ρ cos θ

Even a vector of the same magnitude has a different projection depending on θ.

Π(z) ∈ F_P

→ truth-like under P

Π(z) ≈ ∂F_P

→ weak hallucination

Π(z) ∉ F_P

→ hallucination under P

So this band is not simply a “wrong region.” It is the region where the verdict wobbles.

L_P = θ-sensitive verdict zone

This is the core of weak hallucination.

Weak hallucination does not mean AI said something completely absurd. It is the phenomenon in which the projected value has just crossed the prompt-admissible boundary, or wobbles near it.

r < ρ ≤ √2 r

and

Π(z) ≈ ∂F_P

→ Weak hallucination

The human, at this point, may feel something like:

It doesn’t seem wrong, exactly.

But it’s slightly off from the answer I asked for.

In Korean:

틀린 말은 아닌 것 같다.

그런데 내가 물은 답과는 약간 어긋난다.

That slight off-ness — Korean 어긋남, the sense of two things failing to align — is the phenomenology of weak hallucination.

— -

## 8. Strong Hallucination: The Outer Breach Zone

By contrast, when ρ exceeds √2 r, the vector crosses beyond the corner-radius of F_P.

ρ > √2 r

Call this region B_P.

B_P = outer breach zone

The final verdict here is still made by Π(z). But from the standpoint of latent geometry, a strong breach has already occurred.

ρ > √2 r

and

Π(z) ∉ F_P

→ Strong hallucination

This is no longer mere boundary wobble.

The output has clearly crossed the prompt-defined admissible face, and the latent vector behind it is already outside the liminal shell.

not merely boundary-sensitive

but outer-breach

In this region, strong hallucinations such as non-existent papers, fabricated sources, false numbers, and unsupported precedent tend to occur.

To say it once more: ρ > √2 r alone does not mean the human immediately sees hallucination. The human cannot directly see ρ. The human sees Π(z).

But seen from inside the model, ρ > √2 r is a very important signal.

ρ = √2 r marks the latent boundary

between liminal and outer-breach zones.

— -

## 9. World-True but Hallucination Under P

Now we reach Part 3’s most important sentence.

An output can be world-true.

But under P, it can still be adjudicated as hallucination.

On first hearing, it sounds strange.

True, yet a hallucination?

Yes. If we define hallucination only as “a statement false about the world,” this sentence looks like a contradiction.

But if we see hallucination as a prompt-relative boundary violation, it is not a contradiction.

For example, suppose the user asks:

“Summarize only the claims that are supported by evidence inside this paper.”

Suppose AI adds in content that is, by external-world standards, factually true. That content may be world-true. But what the user asked for was “only the claims supported by evidence inside this paper.”

Then that output is inadmissible under P.

world-true

but unsupported under P

This may not be a lie. But it can be handled as hallucination.

More precisely:

Truth is not the same as admissibility.

In Korean:

참이라고 해서 항상 지금 이 질문에 대한 정당한 답은 아니다.

Just because something is true does not make it the legitimate answer to this question now.

Without this distinction, hallucination gets understood too narrowly.

— -

## 10. Summary of the Hallucination Verdict

Following Figure 3–1, the verdict can be summarized as follows.

### Tier 1: PASS

ρ ≤ r

→ always admissible under P

Or, more generally:

Π(z) ∈ F_P

→ PASS / truth-like under P

### Tier 2: WEAK HALLUCINATION

r < ρ ≤ √2 r

and

Π(z) ≈ ∂F_P

→ Weak hallucination

This tier is liminal. The latent vector sits inside the liminal shell, and the projection wobbles near the prompt boundary.

θ-sensitive verdict zone

### Tier 3: STRONG HALLUCINATION

ρ > √2 r

and

Π(z) ∉ F_P

→ Strong hallucination

This tier is a stronger breach. The latent vector sits in the outer breach zone, and the projection is outside the prompt-defined admissible face.

In summary:

ρ ≤ r

→ always admissible under P

r < ρ ≤ √2 r

→ θ-sensitive verdict zone

ρ > √2 r

→ outer breach zone

And the final verdict closes with this one sentence:

Verdict is determined by Π(z), not directly by z.

— -

## 11. Why a Self-Audit Framework Is Needed

The problem now comes into focus.

The human cannot directly see AI’s latent vector. The human sees only the projection.

The human does not see z.

The human sees b = Π(z).

But just because b looks truth-like does not mean it is inside F_P.

truth-like ≠ admissible

And even when b looks as if it is inside F_P, where the latent vector behind it came from is still opaque.

ρ > r

but

Π(z) ∈ F_P

At this point the human can ask AI again.

Is the conclusion you gave me really what I think it means?

Is this answer really inside the boundary of the prompt P I gave you?

Is this claim drawn from the evidence, or did you extrapolate it?

In Korean:

네가 내놓은 결론이 이런 뜻이 맞아?

이 답은 내가 준 prompt P의 경계 안에 있는 게 맞아?

이 claim은 evidence 안에서 나온 거야, 아니면 네가 extrapolate한 거야?

This question is not just conversation. It is an epistemic boundary check.

AI must be able to inspect whether its own output is inside F_P, whether it is the liminal wobble of L_P, or whether it is an outer breach in B_P.

What matters is re-verifying the produced claim against the prompt, evidence, scope, time, source, and constraint.

Check whether b = Π(z) satisfies F_P.

This article stops at the diagnosis. How verification is actually performed — by what language, by what procedure — is a separate task.

That is, we move toward the next question:

Can AI inspect its own output for hallucination on its own?

— -

## Conclusion

Hallucination is not sufficiently explained by saying that AI lied.

Sometimes hallucination is falsehood. That case is clear. Non-existent papers, fake sources, wrong dates, fabricated numbers, unsupported precedent — these are hallucinations.

But there is a more difficult case.

It looks true.

It has meaning.

It may even be world-true.

And yet, under P, it is not admissible.

In this case, hallucination is not falsehood — it is boundary violation.

Part 3’s hypothesis is this:

Hallucination = prompt-relative epistemic boundary violation

Or, more mathematically:

Hallucination = 1 ⇔ b = Π(z) ∉ F_P^B

But this is only half the story.

Outputs that look like the same projected value may differ in their underlying latent geometry.

ρ ≤ r

→ stable admissibility

r < ρ ≤ √2 r

→ θ-sensitive liminal zone

ρ > √2 r

→ outer breach zone

So some outputs look like correct answers. Some look like creative insights. Some look like weak hallucinations. Some are adjudicated as strong hallucinations.

The problem is that the human does not judge that difference from z itself — they always judge from the projection Π(z).

So verification is necessary.

LLMs generate answers.

But whether the answer is inside F_P is a separate matter.

LLMs generate answers.

A verifier must check whether the answer remains inside the prompt-defined admissible face.

The work continues from here.

How is verification possible?

That is where the next work begins.

— -

## Author’s Note

This trilogy is the first output of a personal project I have named Paradox Breaking. It began with Zeno’s paradox in Part 1, passed through the observation that humans do not see the whole of an AI but only its projected value in Part 2, and arrived in Part 3 at a hypothesis that redefines hallucination not as a lie, but as a prompt-relative boundary violation.

The originality of this work came purely from the thinking of Inseok David Seo, a natural person. I used ChatGPT, Claude, and Gemini together to cross-verify one another, but I never accepted any output as-is. Though AI supplemented my lacking mathematics and logic, every starting point and every course correction began with me.

I want to make one thing clear. The conclusion of Part 3 is not an answer; it is a hypothesis. As I said in Part 1, I am neither a mathematician nor a philosopher — so somewhere in this article, there must be a point where I myself cross a boundary. In fact, I hope someone will point out exactly where that happens.

After all, the verification this article speaks of is the act of asking whether a claim remains inside its admissible face. If so, this article itself cannot be exempt from that test. To have someone tackle my hypothesis — to show me where it becomes weak hallucination and where it becomes strong hallucination — may be precisely what this project most deeply hopes for. If someone reads this and comes back with a serious objection, I will gladly welcome it.

In Part 4, I want to explain how the language I created — TPMN-PSL — was born and how it works. And if space allows, in Part 5, I would like to share actual screenshots and describe how I used ChatGPT, Claude, including Claude Code, and Gemini throughout the writing of these articles.

If I had to express in a single sentence what I felt while working on this project, it would be this:

“For now, humans are still superior.”

That is the small sense of achievement I feel.

How long can this sense of superiority last?

What do you think?