Michael Darius Eastwood · Independent AI alignment researcher
Published
The founding paper of the programme argues something simple that most scaling debates miss: it is not how much extra thinking a system does that matters, it is the shape of that thinking. Two systems can burn the same amount of test-time compute and get wildly different returns, depending on whether the extra effort is spent generating many independent guesses or building one careful chain of reasoning on top of another.
Standard scaling laws tell you what to grow (parameters, data, compute) to make an AI system more capable. They do not explain why letting a model think longer at inference time sometimes helps enormously and sometimes barely at all. Paper I asks a more mechanical question. If we treat recursive depth as a variable of its own, what functional form does capability take as depth increases, and does that form depend on the kind of recursion the system is doing?
The paper formalises the ARC Principle as U = I × R to the power of alpha, where U is measured capability, I is base intelligence at a single pass, R is recursive depth, and alpha is an exponent that has to be measured. The seed idea, U = I × R, appears in the 8 December 2024 manuscript; Paper I promotes R to R to the alpha so that the shape of the returns curve becomes a testable quantity. Two forms of recursion are distinguished. Parallel recursion samples many independent attempts and picks one; sequential recursion lets each step build on the previous one, so errors can be caught and corrected. Preliminary analysis of published data from OpenAI o1 (parallel, majority voting) gave alpha around 0.1 to 0.3, sub-linear and diminishing. DeepSeek R1 (sequential chain of thought) gave alpha around 1.34, super-linear. The paper also floats a Grover-inspired conjecture that alpha equal to 2 might be an upper bound for fixed architectures.
Paper I is explicitly preliminary. Its evidence base is thin: a handful of published data points, two systems, one benchmark (AIME 2024). It does not prove the equation and does not claim replication. The bigger unresolved question, the exact value of alpha for sequential reasoning across architectures, is passed straight through to Paper II, which then had to retract its own first estimate. The quadratic ceiling is a conjecture, not a derivation. Domain generalisation is untested. The most honest reading is that Paper I is a hypothesis with directional support, not a settled result.
Paper I is the seed. Paper II tries to measure alpha in a controlled setting; the earlier unblinded single-model fit of alpha approximately 2.24 was retracted, corrected to approximately 0.49 under blinding on current frozen systems, well below the quadratic conjecture. Paper VII derives the family of power-law solutions from Cauchy’s functional equation, giving the equation a formal home. Paper III lifts the same framework onto alignment scaling. Paper X later supersedes the fixed-exponent framing of U = I × R to the alpha as the operative safety criterion and installs the co-scaling stability condition as the current operative claim. The equation itself and the ARC Bound alpha at most 2 remain live hypotheses whose real test on genuine self-improving systems is the beta over k programme. Read as a set, the papers move from “here is a shape” to “here is a shape with a constrained empirical interpretation on frozen systems and a narrower operative safety claim”.
The full paper HTML and PDF are on OSF under 10.17605/OSF.IO/6C5XB. The replication toolkit lives at github.com/MichaelDariusEastwood/arc-principle-validation. The two datasets Paper I analyses are public: the OpenAI o1 system card and the DeepSeek R1 technical report (arXiv:2501.12948). The interesting checks are structural: pick any pair of published test-time compute curves and fit U = I × R to the alpha to them. If sequential and parallel yield indistinguishable alphas, the paper’s central claim weakens.
Related notes: the ARC Principle as the book presents it for the accessible-reader U equals I times R squared framing; Claim 2 explained for the compact evidence-and-caveats summary; Claim 10 explained for the sequential-versus-parallel result and the Sharma and Chopra concurrent work; Claim 12 explained for the 2.24-to-0.49 retraction narrative. New to Michael Darius Eastwood or the ARC/Eden research programme: /start-here.html for the five-minute map.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.