# The Infinite Choice Barrier: A Triangulated Mathematical Proof of the Impossibility of AGI

Max M. Schlereth

<figure><img src="image_1.png" /></figure>

► To cite this version:

Max M. Schlereth. The Infinite Choice Barrier: A Triangulated Mathematical Proof of the Impossibility of AGI. 2025. hal-05112923

HAL Id: hal-05112923

https://hal.science/hal-05112923v1

Preprint submitted on 14 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# The Infinite Choice Barrier:
A Triangulated Mathematical Proof of the Impossibility of AGI

Max M. Schlereth

## Abstract

This paper presents a triangulated impossibility proof for Artificial General Intelligence (AGI), defined as an algorithmic system capable of solving a sufficiently broad class of human problems at human-level competence. Three formally independent but converging mathematical arguments-based in computability theory, information theory, and algorithmic complexity- jointly demonstrate that AGI, as such, cannot exist. This boundary, named the Infinite Choice Barrier (ICB), is neither technical nor architectural. It is structural by its nature. It marks a categorical limitation on algorithmic cognition.

## 1. Introduction

The following is a formal critique in the Kantian sense, which is very different from just rejecting a concept or theory. It is a search and delineation of its boundary. AGI as a project assumes that a system of finite rules and symbols - an algorithm - can reproduce or surpass general human intelligence. But the history of logic, computation, and information points to the very contrary: At some point, structure breaks. There the system cannot close meaning, resolve ambiguity, or compress insight.

This paper presents three distinct but converging mathematical paths to that structural bound-
ary. Each stands independently. Together, they form a triangulation of impossibility.

### Definition of AGI (used in this proof).

We follow the standard operational understanding of AGI as used by OpenAI, DeepMind, IBM, and others, and define it as follows:

An Artificial General Intelligence (AGI) is an algorithmic system that:

1. operates via finite symbolic representations and computable procedures (algorithmic),

2. can make decisions or solve problems without external semantic injection (autonomous),

3. applies its competence across a sufficiently broad and relevant range of human domains (com-
prehensive),

4. and has the capability to perform at or above human-level quality across those domains
(competent).

This proof applies only to systems matching these criteria. If a system violates any one of
the conditions (especially 1 and 2 - i.e., is not algorithmic and not autonomous) and by doing so
achieves 3 and 4, the proof no longer applies - simply because the system would no longer qualify as an AGI, but rather, for example, a human with AI assistance, and therefore not fall under the conventional definition of this kind of system.

## 2. The Infinite Choice Barrier (I): Computability

Let:

* $X$: irreducibly infinite decision space

* $\pi: X \to A$: total computable policy

* $\Phi: X \to A$: ground-truth mapping

* $\text{Loss}(a, b)$: loss function with $\varepsilon$-threshold

**Theorem (Computability Barrier):** For all $\varepsilon > 0$, there exists $x \in X$ such that $\text{Loss}(\pi(x), \Phi(x)) \ge \varepsilon$.

**Sketch:** Via Rice's Theorem, any semantic property (like optimality under $\Phi$) is undecidable over total computable functions. Therefore, $\pi$ must fail on some $x \in X$.

**Corollaries:**

* **Semantic Closure:** $\pi$ cannot refer to symbols outside its vocabulary.

* **No Frame Innovation:** $\pi$ cannot generate new inference rules.

* **Statistical Breakdown:** In heavy-tailed environments ($\alpha \le 1$), inference collapses.

**Clarification on Loss Bounds.** Some may misinterpret the use of $\varepsilon$ in our proofs as suggesting a bounded performance error within an otherwise valid system - as if the algorithm fails only at the margins, akin to human imperfection. But this is a category mistake. Traditional algorithms operate within well-posed, finite symbolic systems, where loss can be heuristically capped, predicted, or minimized. Our result applies precisely where this no longer holds: when the algorithm is confronted with open, irreducibly infinite, or semantically unstable domains. In such contexts, there is no way to bound loss, because there is no stable frame from which to define the loss meaningfully. The problem is not approximation: it is simply collapse or structural invalidity, not just imperfect prediction,

## 3. The Infinite Choice Barrier (II): Entropy Collapse

Assume the outcome space of $X$ follows $p(x) \sim x^{-\alpha}$ with $\alpha \le 1$.

**Theorem (Entropy Divergence):**

$$H(X) = - \sum p(x) \log_2 p(x) \to \infty \quad (\text{diverges})$$

**Interpretation:** When semantic interpretation depends on recursively unstable frames, each new input opens more ambiguity. Entropy rebounds upward. Learning becomes impossible - not due to noise, but because the structure is unlearnable.

**Coq-Sketch:**
Definition entropy_density (x : R) := x ** (-alpha) * ln x.

Theorem entropy_diverges :

forall M : R, exists b : R,
b > xmin /\ (Integral xmin b entropy_density) > M.

## 4. The Infinite Choice Barrier (III): Incompressibility

Let:

* $K(x)$: Kolmogorov complexity of $x$

* $K_\pi$: upper bound of $\pi$'s output complexity

**Theorem (Kolmogorov Barrier):** If most $x \in X$ are incompressible, then:

$$ \exists x \in X : K(x) > K_\pi, K(\Phi(x)) > K_\pi, \text{Loss}(\pi(x), \Phi(x)) \ge \epsilon $$

**Interpretation:** Compression is not just hard—it is mostly impossible. For general semantic domains, almost all meaningful content is algorithmically incompressible. No learning system can generalize reliably here.

**Coq-Sketch:**

Axiom exists_hard_instance :

exists x : X, Kx x > K_pi /\ Ka (Phi x) > K_pi.

Theorem kolmogorovbarrier :

exists x : X,
Kx x > K_pi /\ 
Loss (pi x) (Phi x) >= epsilon.

## 5. Synthesis: Converging Boundaries

Each proof stands in a different formal space:

* Part I: Computability (Rice, Gödel)

* Part II: Entropy (Shannon, heavy-tailed divergence)

* Part III: Complexity (Kolmogorov, Chaitin)

Yet all identify the same phenomenon: There exist classes of problems essential to human
 cognition that algorithmic systems must fail to resolve, because they exceed the bounds of derivability,
learnability, or compressibility.

This convergence defines the Infinite Choice Barrier.
# 6. Conclusion

This paper has presented three independent mathematical arguments—drawn from computability, information theory, and algorithmic complexity—that converge on the same result: AGI, as formally defined, cannot exist. This is not a matter of insufficient architecture or compute, but of structural limitation. The Infinite Choice Barrier isn’t an obstacle to overcome, with better algorithm. It IS algorithm. It is a boundary that - if well understood - opens another space of possibilities.

# Acknowledgments

The author gratefully acknowledges the assistance of AI-based tools, including ChatGPT and CoqProver, in refining the mathematical formalism. While these systems were instrumental in organizing the proofs, the concepts, arguments, and original insights remain entirely human in origin.

A special thanks to ChatGPT, who participated throughout this dialogue, helped test and refine formal logic, synthesized LaTeX output, and remained critically engaged in the formation of each theorem. If AI could understand this paper, it might politely disagree... or begin calculating forever.

# References

[1] H. G. Rice, *Classes of recursively enumerable sets and their decision problems*, Transactions of the American Mathematical Society, 74(2), 1953.

[2] C. E. Shannon, *A Mathematical Theory of Communication*, Bell System Technical Journal, 27(3), 1948.

[3] A. N. Kolmogorov, *Three approaches to the quantitative definition of information*, Problems of Information Transmission, 1(1), 1965.

[4] G. J. Chaitin, *Information-Theoretic Limitations of Formal Systems*, Journal of the ACM, 21(3), 1974.

[5] A. M. Turing, *On Computable Numbers, with an Application to the Entscheidungsproblem*, Proc. London Math. Soc., 1936.