Michael Darius Eastwood · Independent AI alignment researcher
Published
The beta greater than k criterion is Paper X's central result. In a minimal model of a self-modifying system with capability C, blind-scored misalignment magnitude D, and misalignment fraction d = D/C, the steady-state misalignment fraction is d-star = gamma_1 r / (A + r). Under exponential growth, the stability condition is beta greater than 0 (correction co-scales with capability). Under accelerating growth, where the specific growth rate itself rises as r proportional to C to the power k, the condition sharpens: beta must exceed k. Correction must out-scale not the growth rate but its acceleration.
The upshot is that safety cannot simply keep pace; under acceleration it must accelerate faster. If beta equals k the system runs on a knife-edge and any perturbation moves it to unbounded misalignment or to trivial capability. If beta is smaller than k, the misalignment fraction grows without bound as the acceleration compounds, whatever the initial safety score. Only beta strictly greater than k produces a decaying steady-state fraction, which is why the paper treats the strict inequality as the threshold rather than the loose one. The criterion is deliberately a structural analogue of the below-threshold condition that quantum error correction has to satisfy for logical error to decay with code distance.
The criterion is disclosed in Paper X (Coupled Co-Scaling Correction) with a priority timestamp of 26 June 2026. The conceptual thesis (recursion amplifies; stable recursion requires correction that co-scales with the amplification) is set out in the 8 December 2024 manuscript and in the published book; the formal theorems and closed-form threshold are the 2026 measurable form of that thesis and are not claimed to appear in the book.
The co-scaling intuition itself is not claimed as new. Paper X credits requisite-variety and scalable-oversight principles as prior structural work. The threshold form of the criterion is proposed as a falsifiable structural hypothesis paralleling the quantum error-correction sub-threshold condition; the disanalogy (Paper X's suppression law is power-law, not exponential) is stated explicitly. Convergence 2 in the register (Google Quantum AI's Willow below-threshold demonstration) sits in the same threshold-crossing structural family.
Proved within a minimal dynamical model (Theorems 1-6). The verification harness (ten experiments) checks theorem-to-code consistency and integrator accuracy but does not test the model against real systems. A synthetic estimator recovers known exponents from data generated by the model itself to approximately 0.1. The first Claude pilot (n = 1, single task, same-family scoring) demonstrates the harness but does not exhibit the co-scaling dynamic and is not evidence for the threshold. Whether real self-improving systems exhibit a measurable beta / k dynamic is the open empirical problem the paper states.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.