The ARC Principle: Formalisation and Preliminary Validation of Recursive Capability Scaling
Michael Darius Eastwood · Independent AI alignment researcher
Published
This paper formalises and preliminarily tests the ARC Principle (Artificial Recursive Creation), first proposed in Infinite Architects (Eastwood, 2026): that capability in intelligent systems scales super-linearly with recursive depth. The principle is expressed mathematically as U = I x R^alpha, where effective capability (U) scales with base intelligence (I) multiplied by recursive depth (R) raised to an empiricall
Michael Darius Eastwood
Michael Darius Eastwood, independent researcher, London: originator of the embedded-correction alignment thesis (manuscript 8 December 2024, SHA-256 anchored: f0d1f38f).
First published 2026-01-17 · Updated 2026-03-20
Abstract
This paper formalises and preliminarily tests the ARC Principle (Artificial Recursive Creation), first proposed in Infinite Architects (Eastwood, 2026): that capability in intelligent systems scales super-linearly with recursive depth. The principle is expressed mathematically as U = I x R^alpha, where effective capability (U) scales with base intelligence (I) multiplied by recursive depth (R) raised to an empiricall
Preliminary Evidence for Super-Linear Capability Amplification Through Sequential Self-Reference
The ARC Principle: $U = I \times R^{\alpha}$
Michael Darius Eastwood
Author, Infinite Architects: Intelligence, Recursion, and the Creation of Everything (2026)
London, United Kingdom | OSF: 10.17605/OSF.IO/6C5XB | ISBN 978-1806056200 (ISBN-10: 1806056208)
This paper formalises and preliminarily tests the ARC Principle (Artificial Recursive Creation), first proposed in Infinite Architects (Eastwood, 2026): that capability in intelligent systems scales super-linearly with recursive depth. The principle is expressed mathematically as $U = I \times R^{\alpha}$, where effective capability ($U$) scales with base intelligence ($I$) multiplied by recursive depth ($R$) raised to an empirically determined power $\alpha$.
Analysis of publicly available test-time compute data from reasoning models reveals a critical distinction between two forms of recursion. Parallel recursion (majority voting across independent samples) yields sub-linear scaling with $\alpha \approx 0.1$ to $0.3$. Sequential recursion (chain-of-thought reasoning where each step builds on previous steps) yields super-linear scaling with $\alpha \approx 1.3$.
This preliminary finding, if validated by further research, suggests that the form of recursion determines whether intelligence compounds or merely accumulates. We propose that $\alpha = 2$ represents an asymptotic theoretical limit, analogous to the speed of light in special relativity: a ceiling that optimising systems approach but may never reach.
The scaling laws governing artificial intelligence have been extensively studied. Kaplan et al. (2020) established power-law relationships between model performance and parameters, while Hoffmann et al. (2022) refined these with compute-optimal training prescriptions. These laws govern what to scale but do not address why scaling produces intelligent behaviour.
The emergence of reasoning models in 2024 and 2025 introduced a new variable: test-time compute. OpenAI's o1 (September 2024) and DeepSeek's R1 (January 2025) allocate computational resources at inference time to reason before responding, producing substantial capability improvements on reasoning benchmarks.
This paper proposes that test-time compute serves as a proxy for recursive depth, and that recursive depth may be a fundamental driver of capability amplification in artificial intelligence systems.
1.2 The ARC Principle
The ARC Principle (Artificial Recursive Creation), first articulated in Infinite Architects (Eastwood, 2026), proposes:
$$U = I \times R^{\alpha}$$
Capability scales with intelligence multiplied by recursive depth raised to a power
Where:
$U$ = Effective capability (measurable via benchmark performance)
$I$ = Base intelligence (single-pass processing capacity without recursive reasoning)
$R$ = Recursive depth (number of self-referential processing iterations)
The principle's core claim: recursion does not merely add to capability; it multiplies it according to a power law.
1.3 Scope and Claims
This paper makes the following claims, each with explicit epistemic status:
Claim
Status
Evidence Level
$U = I \times R^{\alpha}$ is a useful framework for AI systems
PROPOSED
Theoretical
Parallel recursion yields $\alpha < 1$ in AI benchmarks
PRELIMINARY
Limited data (o1)
Sequential recursion yields $\alpha > 1$ in AI benchmarks
PRELIMINARY
Limited data (DeepSeek-R1)
$\alpha = 2$ is the theoretical limit
HYPOTHESISED
Theoretical only
The form of recursion matters
SUPPORTED
Consistent with both datasets
What this paper does NOT claim:
That $U = I \times R^2$ applies to cosmological or universal scales (that application remains speculative)
That $\alpha = 1.3$ is definitively established (more data points are needed)
That the principle has been independently replicated (it has not)
We present a principle with preliminary supporting evidence and invite rigorous testing.
2. Theoretical Framework
2.1 Defining Recursion
Recursion is self-reference: a process whose output becomes its input. It is distinct from mere iteration (repeating the same operation) because each cycle operates on the transformed results of previous cycles.
2.2 Two Forms of Recursion
Parallel Recursion (Weak): Multiple independent solutions generated simultaneously. No information transfer between branches. Example: Generating N samples and selecting by majority vote. Expected scaling: Diminishing returns as redundancy increases.
Sequential Recursion (Strong): Each processing step builds explicitly on previous steps. Errors can be detected and corrected iteratively. Example: Chain-of-thought reasoning with self-reflection. Expected scaling: Compounding returns as depth enables self-correction.
The ARC Principle predicts that sequential recursion should produce higher $\alpha$ values than parallel recursion.
Figure 1 | ARC Principle Architecture. U = I × g(R) with α = 1/(1−β). β coupling feedback loop. Three regimes: sequential (α_seq≈0.49), parallel (α_par≈0), ARC Bound (α≤2). Cauchy Unification: α=d/(d+1) physical; α=1/(1−β) intelligence. Previous α≈2.24 RETRACTED (Paper IX §7). 13 falsification criteria. Source: Papers I, II the six-model study, III, VII, IX · evidence spine C-1, C-2 · OSF 10.17605/OSF.IO/6C5XB.
2.3 The Quadratic Limit Hypothesis
We hypothesise that $\alpha = 2$ represents a theoretical maximum. Bennett, Bernstein, Brassard, and Vazirani (1997) proved that Grover's quantum search achieves exactly quadratic speedup and that this is optimal for unstructured search. If recursive intelligence operates analogously to amplitude amplification, quadratic scaling may represent a fundamental computational limit.
3. Empirical Analysis
3.1 Data Sources
OpenAI o1 System Card (September 2024). Benchmark: AIME 2024 (American Invitational Mathematics Examination). Variable: Number of samples (majority voting). Source: openai.com/index/openai-o1-system-card.
$\alpha$ decreases as recursive architectures mature
Not met
F3
The relationship is additive rather than multiplicative
Not met
F4
More extensive datasets show $\alpha < 1$ for sequential reasoning
Untested
5. Limitations
Scientific integrity requires explicit acknowledgement of limitations:
Limited Data Points: The DeepSeek-R1 analysis relies on only two data points. Additional measurements would strengthen or refute the finding.
Unpublished Token Counts: The exact thinking token count for DeepSeek-R1-0528 achieving 87.5% accuracy is estimated, not published.
Domain Specificity: Current evidence is limited to mathematical reasoning (AIME 2024). Generalisation to other domains remains untested.
5.1 Falsifiability — what would refute this
The ARC Principle — that capability scales super-linearly with recursive depth, $U = I \times R^{\alpha}$, with the sequential/parallel distinction determining the regime — is refutable. It would be overturned by any of the following:
No super-linear depth scaling. If capability scaled only linearly or sub-linearly with recursive depth $R$ across reasoning models (i.e. $\alpha \le 1$ where the principle predicts $\alpha > 1$ for sequential recursion), the central super-linearity claim fails.
Exponent–composition mismatch. If the measured exponent systematically failed to match the predicted $\alpha = 1/(1-\beta)$ from the composition parameter $\beta$, the mechanistic identity — not merely the fitted curve — is falsified.
Regime non-distinction. If sequential and parallel recursion produced the same scaling regime (no super-linear vs sub-linear split), the paper’s central distinction collapses.
Better-fitting alternative. If a functional form outside the ARC family fit the test-time compute data materially better across models, ARC would not be the operative law.
Domain-independence failure. If $U = I \times R^{\alpha}$ held for reasoning models but robustly failed in other recursive domains meeting the premise, the domain-independent claim would have to narrow to “a property of current reasoning models”.
6. Implications
6.1 For AI Development
If the ARC Principle holds, recursive depth constitutes a third scaling axis alongside parameters and data. Investment in recursive architectures may yield better returns than scaling model size alone.
6.2 For AI Safety
If recursion amplifies not only capability but also embedded values, then well-aligned initial values should strengthen through recursive self-improvement. Misaligned values would also compound, making early alignment critical.
6.3 For Scientific Understanding
The ARC Principle connects to several established frameworks including Kaplan et al. (2020) scaling laws, Integrated Information Theory (Tononi, 2008), and Grover's quantum search optimality proof (Bennett et al., 1997).
7. Conclusion
We have formalised the ARC Principle and presented preliminary evidence:
The form of recursion determines whether capability compounds
In plain terms: 'Thinking about thinking makes you smarter. Not linearly smarter, but disproportionately smarter, if the thinking is sequential rather than parallel.'
The principle stands. The research continues.
Acknowledgments
Data analysis and manuscript preparation were assisted by AI systems (Claude, Anthropic). The intellectual framework, hypothesis formulation, and interpretive conclusions are the author's own.
References
Bennett, C. H., Bernstein, E., Brassard, G., & Vazirani, U. (1997). Strengths and weaknesses of quantum computing. SIAM Journal on Computing, 26(5), 1510-1523.
DeepSeek AI. (2025). DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning. arXiv:2501.12948.
Eastwood, M. D. (2026). Infinite Architects: Intelligence, Recursion, and the Creation of Everything. Independent publication.
Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 212-219.
Hoffmann, J., Borgeaud, S., Mensch, A., et al. (2022). Training Compute-Optimal Large Language Models. arXiv:2203.15556.
Kaplan, J., McCandlish, S., Henighan, T., et al. (2020). Scaling Laws for Neural Language Models. arXiv:2001.08361.
Lloyd, S. (2002). Computational capacity of the universe. Physical Review Letters, 88(23), 237901.
OpenAI. (2024). OpenAI o1 System Card. openai.com/index/openai-o1-system-card.
Tononi, G. (2008). Consciousness as Integrated Information. The Biological Bulletin, 215(3), 216-242.
Wei, J., et al. (2022). Chain-of-Thought Prompting Elicits Reasoning in Large Language Models. NeurIPS 2022.
Reproducibility
The complete research toolkit is available on GitHub:
Priority established: Infinite Architects, published 2 January 2026 (print; ebook 6 January). Copyright 2026 Michael Darius Eastwood.
Declaration of AI-Assisted Human Authorship
The author of this work is Michael Darius Eastwood, a human being. Every core concept, hypothesis, experimental design, claim and conclusion in this paper originates from human ideation. No part of this manuscript is a wholly generated artificial-intelligence output.
Artificial-intelligence tools (Anthropic's Claude family and other large-language-model assistants) were used as instruments under continuous human direction, in the way a word processor, calculator or research assistant is used: for editing and prose refinement, literature search and summarisation (manually verified against primary sources), document structure, formatting, brainstorming against author-defined questions, and the acceleration of drafting to author-defined outlines and instructions. All selection, coordination, arrangement and final editorial judgment are the author's. Every substantive output was reviewed, tested or verified by the author, who takes full responsibility for the accuracy and integrity of the final text. The tools increased the speed of the work; they were never relied upon as its source.
United Kingdom. In accordance with the Copyright, Designs and Patents Act 1988, the author undertook the arrangements necessary for the creation of this work and asserts full human authorship and moral rights: this is a human-authored work produced with computer assistance, not a computer-generated work. United States. Consistent with United States Copyright Office guidance on works containing AI-generated material, the human contribution (conception, selection, coordination, arrangement and final expression) is asserted as sufficient for full human authorship. Inventions. Any novel technical contribution described in this work was conceived by the human author; no artificial-intelligence system autonomously invented anything presented here.
