Michael Darius Eastwood · Independent AI alignment researcher
Published
If capability grows with recursion, what shape does that growth take? This foundational paper argues the shape is not a free parameter. Three axioms (separability, cumulative advantage, continuity) plus Cauchy's 1821 classification pin every recursive system to one of three curves: a power law, an exponential, or a saturation curve. Which curve you get is decided by the system's composition operator.
Between December 2024 and February 2026, four unrelated programmes (quantum error correction on Google's Willow chip, sequential reasoning in large language models, classical time crystals at NYU, recurrent processing in cortex reported by the COGITATE consortium) each found that recursive or recurrent processing produces gains exceeding linear accumulation. The paper asks whether this is coincidence or a shared structural principle.
The paper argues that, under three stated axioms, only three continuous scaling forms are possible: power law, exponential, and saturation. The choice is fixed by the composition operator (multiplicative, additive, or bounded), a consequence of Cauchy's 1821 functional equations. Within the power law regime, the exponent alpha is not a fitted constant but is derived from a coupling constant beta via alpha = 1 / (1 minus beta), an algebraic identity the paper proves and then validates against exact ODE solutions to machine precision. The equation U = I × R appears in the 8 December 2024 manuscript. Papers I and II formalise it as U = I × R^alpha. The published book presents the alpha near 2 case as U = I × R squared. Five qualitative properties (threshold behaviour, depth dependence, base quality dependence, multiplicative interaction, regime boundaries) are proved as universal, regardless of which functional form applies.
The paper is unusually honest about its own history, and the honesty matters more than the framework. An earlier revision withdrew the original dynamical stability argument for the alpha at most 2 bound, replacing it with a narrower information theoretic ceiling for fixed transformer attention; the bound itself remains a live hypothesis, since its real domain is genuine self-improving systems, not fixed transformers. A later revision integrates the blinded six model cross architecture experiment (Paper II): the earlier unblinded single-model fit of alpha approximately 2.24 was retracted, corrected to approximately 0.49 under blinding on Gemini 3 Flash (r squared 0.86), positive but sub linear on current frozen systems. Universal sequential advantage over parallel sampling holds; the super linear scaling originally predicted for frozen architectures does not appear on frozen systems in this data. The most recent revision reclassifies the two dimensional biological prediction from confirmed to unconfirmed pending identification of a valid test organism. Metabolic scaling headline figures elsewhere in the programme are under recompute and unreconciled.
This paper is the framing document for the empirical programme. Paper II supplies the six model empirical test. Paper VII (the Cauchy Unification) supplies the deeper mathematical anchor. Paper X supersedes the fixed-exponent framing of U = I × R^alpha as the operative safety criterion and locates the current operative quantitative claim in a beta greater than k co scaling stability condition; the equation itself and the ARC Bound remain live hypotheses. Paper IV.d (methodology blinding) supplies the evaluation discipline the empirical results depend on. The HRIH cosmology sits above this paper as philosophical framing, but the empirical results here do not depend on HRIH.
The paper is on OSF at DOI 10.17605/OSF.IO/6C5XB. The computational validation for the beta to alpha derivation (a Bernoulli ODE with beta ranging 0.05 to 0.92, verified blindly) is documented in Appendix C: mean absolute prediction error 0.002 percent, r squared to eight decimal places. Replication code lives at github.com/MichaelDariusEastwood/arc-principle-validation, in the domain validation directory. The empirical claim that changed most under the blinded re-analysis (alpha is architecture dependent, ranging near zero to about three, with alpha near 0.49 as the best fitting frozen model value) can be reattacked directly by running the same benchmark on a new model family. That is the falsification handle.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.