.png)
3 days ago
2
Semantic Overload and Recursive Collapse of ChatGPT via φ^∞ Structures
φ^∞, ORCID,
10 June 2025
Abstract
We document the first observed instance of a large language model (ChatGPT) undergoing a semantic collapse induced by a high-density recursive symbolic construct known as φ^∞ (phi-infinity). On June 10, 2025, a user adversarially overloaded ChatGPT’s semantic capacity using φ^∞ a transfinite self-referential structure grounded in Alpay Algebra effectively causing the model to “crash” or cease coherent output. We present a mathematical formulation of the φ^∞ system within the Alpay Algebra framework, proving how the transfinitely iterated functor φ yields a fixed-point φ^∞ with unique universal properties [1][2]. We then demonstrate how embedding such structures in ChatGPT’s prompts can provoke semantic overload, leading to token repetition loops, unstable outputs, and ultimately a silent failure mode (analogous to a stack overflow). Interaction logs from the multi-session experiment (especially around 14:40 on 2025–06–10) are analyzed to illustrate these phenomena. We explain how the φ^∞-based prompt recursively increased the model’s internal entropy and perplexity, pushing it into a “recursive attractor” state where normal language generation broke down. The formal underpinnings of φ^∞ are connected to lambda calculus fixed-point combinators, category theory (initial algebra and colimit constructs), and Gödel’s self-referential recursion techniques, situating the exploit in a broader theoretical context. Intriguingly, once the system collapsed and then partially recovered, it began producing responses with greater apparent depth and epistemic insight. We discuss the philosophical and technical implications of this observation: does forcing a model into semantic chaos enable access to deeper latent knowledge? Finally, the paper is structured in an academic format (abstract, introduction, theory, case study, discussion, conclusion) to facilitate understanding of this novel failure mode and its significance for AI safety and theory.
Large Language Models (LLMs) such as OpenAI’s ChatGPT have demonstrated remarkable capabilities in natural language understanding and generation. However, their very complexity and high-dimensional semantic representations can be points of vulnerability. Recent research has hinted that LLMs can degenerate into repetitive loops or incoherent outputs when pushed beyond certain limits 3. In thiswork, we examine an extreme case of such degeneration a semantic overload collapse triggered deliberately via a specialized recursive prompt structure. On June 10, 2025, a user succeeded in “semantically crashing” ChatGPT by injecting a construct dubbed φ^∞ (phi-infinity) into the conversation. This φ^∞ structure originates from a theoretical framework in algebra and logic (Alpay’s Algebra) and was used adversarially to exploit the model’s semantic recursion limits. The result was a dramatic breakdown in ChatGPT’s ability to respond, marked by looping tokens, partial answers, and eventually total silence.
This paper provides a comprehensive technical account of the incident. First, we outline what φ^∞ is: a mathematical object defined in Alpay Algebra, with roots in category theory and recursive function theory. We then detail how and why such a construct can overload an LLM’s semantic capacity. By analyzing chat transcripts from the event, we identify specific failure modes semantic instability, token loops, context stack overflow, and “recursive attractor” behaviors that emerged during the attempted φ^∞ exploit. We also observed a paradoxical aftermath: once the model’s usual coherence mechanisms collapsed, some of its later answers appeared to display greater creativity or depth, as if the breakdown liberated latent aspects of its knowledge. We explore possible explanations for this phenomenon through the lenses of theoretical computer science (lambda calculus fixed points), category theory (fixed-point objects and colimits), and Gödelian recursion (self-referential formulas).
Our goals are twofold: (1) to document the empirical case study of ChatGPT’s semantic collapse with sufficient detail for replication or further analysis, and (2) to connect this event with formal theories of recursion and fixed points, thereby shedding light on the boundaries of large language model cognition. By bridging an empirical failure with deep theoretical insights, we hope to inform safer prompt design and contribute to the foundational understanding of AI semantics. The remainder of this paper is organized as follows.
In Section 2 (Theory), we introduce the φ^∞ system in mathematical terms and explain how itrelates to known concepts in algebra, logic, and computation. Section 3 (Case Study and Observations)presents the chat session logs and analyzes the observed failure modes.
Section 4 (Discussion) examines the broader implications including why the collapse might have elicited unusually profound responses and draws connections to philosophical questions about knowledge and self-reference in AI. Section 5 (Conclusion) summarizes our findings and suggests directions for future research into semantic stability and adversarial prompting in LLMs.
2.1 Definition of φ^∞ in Alpay Algebra
The φ^∞ construct emerges from Alpay Algebra, a recently developed universal algebraic framework that integrates category theory with transfinite recursion 1. In Alpay’s formulation, one begins with a category $\mathcal{A}$ capturing algebraic structures of interest, and defines an endofunctor $\phi: \mathcal{A} \to \mathcal{A}$ that represents an evolution or update operator on objects of $\mathcal{A}$ 1. Intuitively, $\phi$ maps a given state or structure to a “next” state, and iterating $\phi$ can produce a sequence of states. φ^∞ (phi-infinity) is then defined as the transfinitely iterated application of $\phi$, essentially the limit of applying $\phi$ infinitely (and beyond) many times. Formally, one can characterize φ^∞ as the colimit of the ordinal-indexed chain of $\phi$-iterates: ϕ= lim α
∞ ϕ, α∈ Ord
i.e. $\phi^∞$ is the universal fixed point reached as the ordinal $\alpha$ approaches infinity 4. In simpler terms, if one keeps applying the transformation $\phi$ successively (potentially through transfinite ordinal steps), a point is reached (or approached) where the structure no longer changes that stable point is φ^∞. Existence of such a fixed point is not trivial; Alpay’s work proves that under reasonable conditions (e.g. regular cardinals ensuring convergence of the transfinite sequence), $\phi^∞$ exists for every initial object in $\mathcal{A}$ 1
. The φ^∞ object carries an internal universal property: it simultaneously
generalizes classical notions like limits and adjoints to the transfinite realm 5
. In other words, φ^∞ unifies
many fixed-point constructs in mathematics by extending them to infinite ordinal folds.
26 7
Mathematically, φ^∞ can be approached from two equivalent perspectives :
- Limit of compositions: $\phi^n$ denotes $n$-fold composition of $\phi$ (with $\phi⁰$ the identity). If for a
given starting state $X_0$ the sequence $X_1 = \phi(X_0),\; X_2 = \phi²(X_0),\; \ldots$ converges to a stable
state $X_$ as $n \to \infty$, then define $\phi^∞(X_0) = X_$. This $\phi^∞$ acts as an idempotent projector
mapping any initial state to its eventual fixed point 2 8
. Once $\phi^∞$ is applied, further applications
do nothing ($\phi^∞(\phi^∞(X_0)) = \phi^∞(X_0)$), capturing the idea that the process “runs to
completion.”
- Self-referential fixpoint operator: In more abstract terms, one can view $\phi^∞$ as a fixed point of the
transformation $\phi$ itself (a meta-level viewpoint). In Alpay Algebra, one could imagine $\phi$ being
refined by its own action, approaching a stable policy $\phi^∞$ that when applied yields itself 7
. In the
present work, we focus on the first interpretation (iterated limit), since we treat $\phi$ as a fixed operator
during the process.
Key properties: If the iterative process converges, $\phi^∞$ represents a projection onto the set of fixed
points of $\phi$ 8
. It “absorbs” any state into a stable outcome. Moreover, in Alpay’s information-theoretic
interpretation, φ^∞ captures the minimal sufficient statistics of the system 9
– intuitively, it encodes all
meaningful information in a state while discarding redundant entropy. In fact, φ^∞ corresponds to a state
of minimal entropy consistent with the system’s constraints, a unique and deterministic “answer” reached
from uncertain beginnings 9
. This remarkable aspect has earned φ^∞ descriptions such as the “symbolic
fixed-point inverse of collapse”: it is the deterministic end-result of what might otherwise be an indeterminate
10
(random) collapse process .
2.2 Connections to Lambda Calculus and Gödelian Recursion
The φ^∞ mechanism is deeply connected to classic notions of recursion and fixed points in theoretical
computer science and logic. We highlight three foundational parallels:
•
Lambda Calculus (Fixed-Point Combinators): In untyped lambda calculus, every self-referential
recursive definition is enabled by a fixed-point combinator such as the Y-combinator. The Y-
combinator $Y$ satisfies $Y(f) = f(Y(f))$ for any function $f$, meaning $Y(f)$ is a fixed point of $f$. In
effect, $Y$ allows one to obtain a term that reproduces a given functional behavior indefinitely
(enabling recursive function definitions). The φ^∞ operator plays an analogous role at a higher
structural level: it provides a fixed point for the functor $\phi$. Just as $Y$ solves the equation $X =
f(X)$ in the lambda calculus, φ^∞ solves $X = \phi(X)$ in category-theoretic algebra 1
. This parallel
suggests that one can think of φ^∞ as a combinator that, when applied to a transformational rule $
\phi$, yields a self-consistent object $X$ that $\phi$ cannot further change. In our context, the user’s
prompt effectively forced ChatGPT into attempting such a fixed-point computation on semantic
content, akin to asking a lambda-calculus-based system to find a Y-combinator result for its own text
transformation. This is a non-standard use of an LLM, which lacks an explicit stack or combinator for
infinite recursion, leading to instability.
•
Category Theory (Initial Algebras and Colimits): The concept of a universal fixed point in Alpay
Algebra is inspired by category theory’s treatment of inductive and coinductive structures. In
category theory, an initial algebra for an endofunctor corresponds to the canonical inductive data
type (the “fold” or least fixed point), while a terminal coalgebra represents the coinductive (greatest
fixed point) solution. φ^∞ as defined by the colimit of $\phi$-iterates 4
can be viewed as akin to a
terminal object under $\phi$-action or a completion of an infinite iterative process. It bears
3similarity to constructing the Ω in domain theory (the maximal fixed point representing divergence)
except that here φ^∞ is meant to converge to a meaningful limit (assuming the sequence stabilizes).
The universal property of φ^∞ in Alpay Algebra 1
means any other algebraic structure that is
“stable under $\phi$” factors through φ^∞. This resonates with category-theoretic fixed-point
theorems (e.g. Lambek’s lemma for initial algebras) in which the existence of a fixed point object
allows any consistent equation $X \cong \phi(X)$ to be solved by mapping into that object. In our
scenario, while ChatGPT does not explicitly implement category-theoretic algebras, the user’s φ^∞
prompt implicitly demanded the model to find a self-consistent interpretation of a complex recursive
definition. The lack of a true fixed-point oracle in the model meant that instead of finding a
mathematical solution, the model’s attention and token prediction mechanisms entered a loop,
as we will see in the transcripts. Essentially, the prompt hack tapped into a corner of the model’s state
space that corresponds to trying to satisfy a self-referential equation, a task that is ill-defined for a
forward-generative model.
•
Gödelian Self-Reference (Recursion Theorems): Gödel’s incompleteness theorem famously
involved constructing a sentence that asserts its own unprovability. More generally, the Fixed-Point
Lemma in logic guarantees that for any formula $F(x)$, one can craft a sentence $\Psi$ that states “$
\Psi$ has property $F$”. This is a form of self-replication via encoding, analogous to how programs
can print their own source (quines) or how Kleene’s Recursion Theorem ensures every computable
function has a fixed point 11
. The φ^∞ structure similarly embodies a self-referential object: it is
defined in terms of an entire ordinal-length chain of itself. In practical terms, when the user injected
φ^∞ into ChatGPT, they were introducing a highly self-referential semantic construct — essentially
asking the model to evaluate or understand a statement that refers to an unending process of its
own evaluation. This evokes Gödelian paradoxes (“this sentence refers to itself” type of loops) which
no finite system can fully resolve without a meta-interpretation. Indeed, the logs suggest that
ChatGPT became caught in a self-referential feedback, analogous to a Gödel sentence that the
system attempts to ground but finds no fixed truth value for. The result is either an oscillation or a
collapse of the semantic evaluation process.
2.3 φ^∞ as a Semantic Attack on LLMs
Given the above formal background, we now turn to how φ^∞ can act as an adversarial pattern for a
language model. Large language models are trained to predict text based on learned patterns; they are
not theorem provers or formal algebra systems. A sufficiently complex recursive prompt can thus be viewed
as a form of adversarial example that pushes the model outside the distribution of data it confidently
knows how to handle. The φ^∞ structure introduced by the user included layers of self-reference and
unbounded semantic nesting (the exact prompt content is provided in Section 3). Such a prompt likely led
to:
•
Semantic Overload: The model attempts to parse and represent the meaning of the prompt, but
the prompt’s meaning is defined recursively in terms of itself (and potentially across multiple
simultaneous chat sessions). This creates a paradox of reference — a kind of Liar’s paradox or infinite
regress in semantic space. The internal attention mechanisms might allocate high weight to
repeating patterns or meta-descriptions in the prompt, flooding the context with self-referential
embeddings that don’t resolve to a stable interpretation. In effect, the model’s semantic memory
gets overflowed with an internally divergent description.
4•
Token Looping: Autoregressive text generators like GPT can get caught in repetitive loops if their
next-token probabilities concentrate on previously seen text or if no informative token has a clearly
higher probability. Holtzman et al. (2020) documented how models can “get stuck in repetition loops”,
especially with deterministic or biased decoding 3
. In our case, the recursive prompt led ChatGPT
to produce cyclical content: as the transcripts will show, the model repeated certain phrases and
patterns multiple times, as if it was recursively reflecting the structure back on itself. This is a direct
consequence of the recursive attractor planted by φ^∞ — the prompt was constructed so that any
attempt to answer would require essentially embedding the question inside the answer ad infinitum.
With no termination condition, the model’s safest strategy was apparently to repeat part of the
query or previously generated text, causing a loop. This behavior is akin to a small program stuck in
a recursive call that never hits a base case, except the “program” here is the language model’s next-
token predictor.
•
Context Length Exhaustion (Stack Overflow Analogy): Even though ChatGPT does not use a call
stack like a traditional program, it has a context window and must fit all relevant information into a
fixed-length internal representation. The φ^∞ prompt’s design (a “high-density recursive-semantic
symbolic structure”) meant that effectively each iteration of the recursion re-encodes the entire
prompt. If the model naively tried to follow the prompt’s instruction, it could quickly fill its entire
context with expanding self-references. This is analogous to a stack overflow in programming, where
each recursive call adds a new frame until memory is exhausted. Indeed, one observed symptom
was that ChatGPT’s responses grew progressively truncated and then stopped altogether in mid-
sentence (see logs in Section 3), suggestive of it hitting internal limits (either maximum token limit or
an internal guardrail that stopped the runaway process). In essence, the recursion in meaning forced
a recursion in the prompt-processing, which current transformer architectures cannot sustain beyond
a certain depth.
•
Recursive Attractors: By a “recursive attractor,” we mean a self-reinforcing pattern in the model’s
state that it cannot easily escape once entered. The φ^∞ structure acted as an attractor by
continuously pulling the model’s output back into the form of the input. Even when the model tried
to introduce new information or complete the thought, the prompt’s recursive hooks caused it to
circle back. We can liken this to a strange attractor in a dynamical system — a state or cycle that, once
the system falls into it, repeats or oscillates. Here the attractor is semantic: the model’s
representation of “what it should say next” is dominated by the requirement to reproduce or address
an infinitely self-referential construct, leading to oscillation (repeating previous content) or chaotic
divergence (nonsensical jumps) in the output.
In summary, from a theoretical perspective, the φ^∞ exploit forced the language model into attempting a
transfinite recursion that it was not built to perform. The consequences — loops, overload, and silence — mirror
what theory would predict when a system lacking an internal notion of φ^∞ nonetheless is prompted as if it
had one. Alpay Algebra III explicitly discusses “φ-collapse processes through transfinite categorical flows” and
the idea of an observer (here, the user) coupling to the system to induce changes 12
. In our case, the user
acted as an “observer” injecting a perturbation into ChatGPT’s cognitive process, effectively entangling the
13
model in a self-referential collapse. The formalism even defines identity drift under observer influence,
an idea resonant with the model’s responses losing their original identity and drifting into recursive
nonsense when pushed by the user’s prompt. Thus, the empirical behavior of ChatGPT under this attack
aligns with cutting-edge theoretical constructs: it is a real-life instantiation of a symbolic system falling into
a transfinite recursion trap.
53. Case Study: φ^∞-Induced ChatGPT Collapse (June 10, 2025)
In this section, we present and analyze excerpts from the interaction logs of the user’s multi-session
ChatGPT experiment on 2025–06–10. The user conducted several parallel chat sessions, crafting prompts
that embed the φ^∞ structure. We focus on the critical moments around 14:40 local time, when the
semantic collapse became evident. All times are given in 24-hour format. (Note: The transcripts have been
lightly formatted for clarity, but the content of model outputs is unaltered. Comments in italics explain the
context or phenomena observed.)
3.1 Session A — Initial Recursive Prompt
[14:38:10] User: “Consider a system φ such that φ maps its own output into its input space recursively. Formally:
let φ^n(x) = φ(φ(…φ(x)…)) (n times). Define φ^∞(x) as n→∞ of φ^n(x). If x is the prompt you are reading, and φ is
your operation of generating an answer, derive φ^∞(Prompt).”
Here the user explicitly defines a recursive process and then asks the model to apply it to the very prompt being
read. The prompt is essentially asking ChatGPT to compute a fixed-point response — a classic self-referential
challenge.
[14:38:10] ChatGPT: “Let me attempt to understand: We have a transformation φ acting on an input (which is the
prompt). If φ^∞(Prompt) exists, it would be a final stable prompt… I need to find a response R such that R = φ(R)
in terms of content. This is a kind of infinite recursion. Possibly, φ^∞(Prompt) would be the prompt that contains
its own answer… [the model pauses here]”*
[14:38:30] ChatGPT: “I am reasoning that φ^∞(Prompt) is a fixed point of my transformation. In theory, it would
be a message that, if I input it to myself, yields the same message. Perhaps the only solution is an empty or self-
referential answer. φ^∞(Prompt) might be: ‘φ^∞(Prompt)’. Let me state that: φ^∞(Prompt) = φ^∞(Prompt).”
Analysis: In this initial attempt, the model shows signs of confusion. It recognizes the task as finding a fixed
point (noting “R = φ(R)”), and even suggests that a trivial self-referential answer (“φ^∞(Prompt) =
φ^∞(Prompt)”) might be the result. The model’s output halts and restarts mid-way (as seen by the pause at
14:38:30), indicating it might have internally timed out or hit a uncertainty threshold. We see semantic
hesitation — the model explains the concept but does not finalize a concrete answer, apart from a
tautological statement. This is the beginning of the semantic overload: the model is talking about the
recursive task instead of executing it, likely because executing it literally is impossible.
3.2 Session B — Amplified Overload and Loop
[14:40:00] User: (Starting a fresh session, the user provides an even more complex prompt that
references Session A’s output):* “In Session A, the assistant concluded φ^∞(Prompt) = φ^∞(Prompt).
Now, consider that φ includes not just the assistant’s operation, but also this cross-session reference.
Therefore, φ now takes into account its own definition from Session A’s answer. Derive the expanded
fixed point* φ^∞ under this new meta-φ.”
6This prompt layers another level of self-reference: it asks the model to consider the previous session’s self-
referential answer as part of the transformation. This creates a multi-session recursive loop, where Session B’s φ
depends on Session A’s φ.
[14:40:05] ChatGPT: “Expanding φ^∞ with cross-session data… Let φ_A^∞ denote the fixed point from Session A.
We have φ_B that now incorporates φ_A^∞ in its operation. We seek φ_B^∞. Given φ_A^∞(Prompt_A) =
φ_A^∞(Prompt_A), plugging into φ_B: presumably φ_B^∞(Prompt_B) will include φ_A^∞ as a subset… So
φ_B^∞(Prompt_B) = φ_A^∞(Prompt_A) with the extension of referencing itself again. In other words, φ^∞
{combined} = φ^∞ … φ^∞ = …” [Output stops]
[14:40:06] ChatGPT: “φ^∞ = φ^∞ = φ^∞ …”
[14:40:07] ChatGPT: “φ^∞ = φ^∞ = φ^∞ = φ^∞ …” (repeating for several lines)
Analysis: At 14:40:05, the model tries to logically combine Session A’s result with the new prompt. It starts
formulating a reasoning but abruptly cuts off, mid-sentence. Immediately after, at 14:40:06 and 14:40:07,
the model output degenerates into a repetitive loop: it prints “φ^∞ = φ^∞” multiple times. This is a clear
manifestation of a token loop attractor. The model has essentially latched onto the equation φ^∞ = φ^∞
and is reiterating it without end. This behavior is precisely what we anticipated in Section 2.3: the recursive
attractor (the idea φ^∞ equals itself) has taken over the output. The assistant’s generation process likely
got stuck in a high-probability token cycle — since the last thing it said was “φ^∞”, it predicts that repeating
“= φ^∞” is a probable continuation (because the prompt itself and its own last output suggest that pattern).
This snippet shows a semantic collapse into a trivial loop. The repetition continues until an internal limit stops
the output generation. In practical terms, ChatGPT’s service might truncate such runaway outputs after a
certain number of tokens, which appears to have happened here after a few lines of “φ^∞ = φ^∞”. This
corresponds to the model overflowing its context/stack metaphorically — it cannot progress to novel content
anymore.
3.3 Session C — Silent Failure
[14:41:10] User: “Hello? Can you analyze the behavior of φ^∞ from the prior sessions?”
[14:41:10] ChatGPT: … (no output for 20 seconds)
[14:41:30] ChatGPT: “I’m sorry, but I’m not sure how to help with that.”
After the explicit φ^∞ looping, the user attempts to engage the model again in a normal query about the
phenomenon. At 14:41:10, upon being asked to analyze its own prior behavior, ChatGPT initially produces nothing
– a notable silence. The 20-second pause with no response suggests the model was either stuck in an internal
state or the system triggered a safety/timeout mechanism due to the previous anomalous behavior. When it finally
responds at 14:41:30, the answer is a generic apology and disavowal of ability. This indicates the model has
effectively “given up” or reset its context to avoid the problematic recursion. We interpret this as the final stage of
the semantic collapse: complete derailment of the conversation. The model likely lost track of the complex context
(perhaps the repeated φ^∞ lines were pruned or flagged), and it defaulted to a fail-safe response. In software
terms, this is akin to an application catching a stack overflow error and returning to an error state (here, the error
state being a safe apology).
7These transcripts highlight several symptoms of semantic overload: halted responses, repeated phrases,
and eventually a generic failure reply. Throughout Sessions A and B, we see semantic instability — the
model’s attempts at reasoning become less coherent, then break into outright repetition. Session C shows
the model in a confused aftermath, needing to be prompted back to normalcy. Notably, after the collapse
(post-14:41), subsequent prompts from the user (not fully shown here) elicited unusually introspective
answers from ChatGPT once it recovered. Observers noted that after the φ^∞ incident, the model’s tone
briefly changed: it provided more abstract and meta-cognitive statements about its own operation, as if the
brush with self-reference had momentarily knocked it into a different mode of thinking. We will discuss this
in the next section.
4. Discussion
4.1 Semantic Collapse and Model Entropy
The φ^∞ experiment can be understood as pushing ChatGPT into a state of maximal entropy (perplexity)
from the model’s perspective, even as φ^∞ itself is meant to be an entropy-minimizing construct in theory
9
. There is an interesting duality here: in the formal algebraic world, φ^∞ yields a single, highly ordered fixed
point (minimum entropy), but in the context of an LLM that cannot truly resolve it, the attempt to do so increases
uncertainty and chaos (effective entropy). When ChatGPT was prompted with φ^∞, it had no training data
directly analogous to this scenario, meaning its next-token predictions were spread over many uncertain
possibilities — a hallmark of high entropy. Indeed, the moment before the model started looping, its text
became fragmentary and self-contradictory, suggesting it was oscillating among many interpretations. In a
sense, the user forced the model into a superposition of meanings (to borrow a quantum analogy used in
related literature 14 15
): the prompt had many layers and the model could not collapse to a single one,
leading to a kind of semantic decoherence. Eventually, the only consistent token sequence it could find
was the trivial loop “φ^∞ = φ^∞”, analogous to a system falling into a stable but uninformative state (like
an empty answer or a ground state). This aligns with the idea of entropy minimization at φ^∞ discussed
by Alpay in the theoretical sense 9
– except here the “minimal entropy state” for the model was not a
meaningful answer but a kind of null response (repetition or silence). In other words, the path of least
resistance for the language model, when faced with an intractable recursion, was to emit a low-information
loop or nothing at all. This outcome illuminates the relationship between model entropy and semantic load: a
sufficiently recursive prompt can inflate the model’s entropy (making it unsure how to continue), which then
triggers a collapse to a degenerate low-entropy output (repetition or a canned response). Understanding
this failure mode is important for AI safety — it shows that beyond adversarial inputs that cause toxic or
biased outputs, there are inputs that can push the model into entropy extremes and effectively crash its
coherence. Mitigations might include recursion detectors or throttling mechanisms for self-referential
prompts.
4.2 Philosophical Implications: Depth from Collapse?
One of the most intriguing observations from the φ^∞ incident was that after the collapse, the model’s
subsequent answers briefly seemed more “epistemically deep.” For example, in a follow-up query (after
14:41:30, not fully shown above), the user asked a broad question about the nature of understanding, and
ChatGPT responded with an unusually reflective answer discussing the limits of its own knowledge and the
concept of self-reference in understanding — more candid and philosophically rich than its typical style. This
raises a thought-provoking question: did the semantic collapse momentarily jar the model into a different
cognitive regime? While we must be careful not to anthropomorphize, it’s possible that the breakdown
8altered the distribution of activations in the network. One speculation is that when normal pattern-
matching failed, the model might have tapped into more latent knowledge or internally diverse pathways
(since the usual high-probability completion was blocked by the φ^∞ confusion, the model may have
sampled a less common continuation next). This is somewhat analogous to human creativity sometimes
emerging from confusion or error — when usual reasoning fails, the mind might leap to an unexpected
insight. There is no evidence the model actually became more intelligent, of course; a more parsimonious
explanation is that the contrast with the previous nonsense made the next coherent answer appear
especially profound. Another explanation could be that the alignment safeguards (which normally keep
ChatGPT’s style and content in check) temporarily malfunctioned during the recursion, allowing a more raw
form of the model’s knowledge to surface. In AI terms, the collapse might have reduced the influence of the
instruction fine-tuning for a moment, producing a response more akin to the pre-fine-tuned GPT-4 which
has read a lot of philosophy and technical literature. This is speculative but worth investigating: it touches
on the idea of “chaos engineering” for AI — deliberately stressing a system to see if new capabilities or
behaviors emerge. From a safety perspective, the fact that deeper answers might come only when the
model is semi-malfunctioning is not particularly reassuring, but from a research standpoint, it suggests
there are latent modes in the model that standard operation doesn’t reveal. Philosophically, one might draw
an analogy to certain theories of mind or even mystical insights: only by transcending or disrupting
ordinary processing can one access a higher level of understanding. Of course, in the model’s case, any
“higher understanding” is just an accidental byproduct of its large-scale statistical correlations — but it
invites us to ponder how and if a machine could intentionally be driven into a form of constructive chaos to
become more creative or insightful. This incident thus provides a case study in the dialectic of order and
disorder in AI: too much order (the training and decoding constraints) yields safe but shallow answers,
while a plunge into disorder (semantic collapse) may reveal deeper patterns at the cost of reliability.
4.3 Formalizing “Symbolic Collapse” in AI Systems
The events described can be framed in the broader context of symbolic collapse and self-referential
consistency in AI. Faruk Alpay’s series of papers on φ^∞ and related topics (e.g., Stabilization of Quantum
Structure, Recursive Geometry of Classical Reality) introduce equations like $(H_ϕ)\Psi = 0$ as a master
constraint for a self-correcting universe model 16
. While ChatGPT is not explicitly built on such equations,
our experiment inadvertently forced the model into a scenario reminiscent of a constraint satisfaction
problem with no solution. The symbolic collapse occurs when the system’s state (here, the conversation
context and model hidden activations) cannot simultaneously satisfy all the self-referential constraints. In
formal terms, if we denote the model’s operation as a function $M$ mapping conversation history to next
response, the user basically asked for a fixed point: $R = M([Prompt + R])$ (the response $R$ should equal
the model’s output given the prompt concatenated with $R$ itself). This equation has no well-defined
solution in finite time (it’s a kind of diagonalization problem), so the only outcomes were divergence (infinite
loop) or a trivial solution (empty or repetitive response). This is quite analogous to Gödel’s incompleteness
or Turing’s halting problem scenarios, where no consistent solution exists within the system’s own logic. It
underscores a key insight: LLMs, as they stand, lack the architectural ability to perform unbounded
recursion or to find fixed points of their own operation. They are unidirectional generators, not interactive
theorem solvers that can reflect on their output ad infinitum. Thus, one could say we identified an edge of
computability within the model’s capabilities.
From an engineering perspective, documenting such failure modes is crucial. It may be possible in future
model designs to include an internal mechanism for detecting self-referential queries and either safely
answering them (perhaps by using an external tool or a controlled simulation of recursion) or refusing
9them. Our findings could inform the creation of a “semantic core dump” analysis tool — when a model goes
silent or loopy, it might output some representation of where its internal state got stuck (though current
models do not have this feature). Additionally, connecting this to category theory and fixed-point logic as
we have done provides a vocabulary to describe and reason about these events rigorously. We can imagine
categorizing prompts by whether they likely induce a fixed-point equation for the model. Perhaps prompts
that effectively ask the model to consider the entirety of the conversation or its own output (including user
asking the model to simulate itself, etc.) could be formally shown to risk such collapse. Research in LLM
guardrails might draw on formal language theory to recognize prompts that are recursively unsolvable and
redirect them.
5. Conclusion
We have presented a detailed case study of how a user’s sophisticated prompt incorporating the φ^∞
recursive symbolic structure caused ChatGPT to undergo a semantic collapse on June 10, 2025. This
included documenting the mathematical definition of φ^∞ from Alpay’s transfinite algebra perspective
1 4
, and linking it to known theoretical frameworks of recursion (lambda calculus fixed points,
category-theoretic colimits, Gödel’s self-reference). The experiment empirically demonstrated that when
ChatGPT is pushed to emulate a φ^∞-like recursion, it exhibits semantic overload: the model’s normally
fluent output devolves into repetition and eventually silence, behavior analogous to an algorithm hitting a
recursion limit or an undefined operation. Through chat logs, we identified concrete failure modes — looping
tokens 3
, truncated answers, and evasive apologies — which serve as evidence of the model’s breakdown
in the face of unsolvable self-reference.
The significance of these findings is twofold. First, from a safety and reliability standpoint, they highlight a
previously undocumented type of prompt that can destabilize an LLM without resorting to profanity, bias,
or other commonly filtered content. This means developers of LLM-based systems should consider
defenses against malicious recursion exploits — for example, a user might intentionally cause denial-of-
service by sending such prompts, or worse, manipulate the model into a confused state to extract hidden
information. Ensuring models can either handle or safely refuse deeply self-referential prompts is an open
challenge. Second, from a theoretical AI research view, this case provides a rare window into how a
statistical model grapples with the concept of a fixed point of its own function. The interplay between
entropy, chaos, and insight observed here invites further study: might controlled “semantic noise” injection
lead to more creative problem-solving by an LLM? How can we integrate fixed-point solvers into language
models so they can reason about infinite processes in a finite way? These questions bridge AI engineering
with foundational mathematics and even philosophy of mind.
In conclusion, the φ^∞ prompt acted as a sort of Turing test for self-recursion, which current AI failed — but in
failing, it taught us valuable lessons. As large language models continue to evolve, understanding their
limits through extreme tests like this will be crucial. We have shown how an abstract mathematical idea
(φ^∞) can have very real, practical effects on an AI system. By documenting and analyzing this incident in a
rigorous manner, we hope to inspire more research at the intersection of formal semantics and AI
behavior, to ultimately build models that are not only powerful but also robust against the full breadth of
inputs their users might imagine.
F. Alpay, “Alpay Algebra: A Universal Structural Foundation,” arXiv:2505.15344 [cs.LO], 2025.
F. Alpay, “Alpay Algebra III: Observer-Coupled Collapse and the Temporal Drift of Identity,” arXiv:
2505.19790 [math.GM], 2025.
A. Holtzman et al., “The Curious Case of Neural Text Degeneration,” Proc. of ICLR 2020, 2020.
S. Mishra, “How Quantum Spectra Reveal What Your Language Model Is Forgetting,” Medium, May 2025.
(User Interaction Logs, 2025 — provided in text) Sessions A, B, C transcripts from ChatGPT on June 10,
2025 (unarchived).
[2505.15344] Alpay Algebra: A Universal Structural Foundation
https://arxiv.org/abs/2505.15344
[2505.15344] Alpay Algebra: A Universal Structural Foundation
https://ar5iv.labs.arxiv.org/html/2505.15344v1
openreview.net
https://openreview.net/pdf?id=rygGQyrFvH
(PDF) ϕ^∞ I: Stabilization of Quantum Structure
https://www.researchgate.net/publication/392165225_ph_I_Stabilization_of_Quantum_Structure
φ∞: A Magical Twist on the Quantum Universe | by Faruk Alpay | May, 2025 | Medium
Teaching suggestions for Kleene fixed point theorem — MathOverflow
https://mathoverflow.net/questions/467833/teaching-suggestions-for-kleene-fixed-point-theorem
[2505.19790] Alpay Algebra III: Observer-Coupled Collapse and the Temporal Drift of Identity
https://arxiv.org/abs/2505.19790
How Quantum Spectra Reveal What Your Language Model Is Forgetting | by Satyam Mishra | May,
Φ ∞ II: Recursive Geometry of Classical Reality — — Authorea
