A Quantitative Definition of Intelligence

Can intelligence be measured? We propose that intelligence can be defined as the lawful amplification of rare but valid futures: a system increases the probability of outcomes that would be unlikely under passive dynamics but remain admissible under the constraints of the domain. We start with the premise that an intelligent system must model the world and its own place within it. Because the system is part of the world it models, this leads naturally to recursive self-simulation: the system represents futures in which its own actions are part of the trajectory. Our central results give a necessity statement and a conditional near-sufficiency statement connecting this architecture to a precise thermodynamic measure of lawful amplification of rare-valid futures: high rare-valid lift is impossible unless the internal simulation identifies rare-valid futures with high fidelity; conversely, when rare-valid fidelity is high and the simulation contains an effective policy, the achievable lift approaches the actuation-limited optimum. Thus recursive self-simulation is not merely a plausible feature of intelligence but, under the stated assumptions, is necessary and nearly sufficient for high thermodynamic intelligence. The resulting framework makes intelligence measurable on a universal scale, from passive matter and feedback controllers, large language models, and humans as text generators to Maxwell-demon-like information engines.

The theory of computation was built to answer Turing's question: what is effectively calculable by an unbounded, immortal, disembodied agent following rules? Intelligence answers a different question (nature's): what can a finite, mortal, energy-limited agent do quickly enough to survive in a non-stationary world? We argue that a complete answer requires two operators: computation and memorizaion. Computation, $\dpar$, transforms structure toward closure; memorization, $\kap$, condenses a validated closed cycle into a reusable token. Turing formalized $\dpar$ and abstracted $\kap$ away, because his agent had infinite time and never needed to amortize. The new insight of our position paper is that the coupling of the two is not optional but forced, and forced by a precise mathematical fact: neither operator alone can be complete. We prove that symbolic computation confined to a discrete sector suffers Gödel's diagonalization incompleteness, that geometric descent confined to a continuous sector suffers a Morse forced-saddle incompleteness, and that these two are not analogies but the parity-conjugate faces of a single obstruction on a coherent complex with $\dpar^2=0$ - the even face realized by diagonalization, the odd by the topologically forced saddle, and no resolution confined to one parity able to come full circle. Intelligence must therefore couple both modes. We then locate the price of the coupling: its hinge operation, context-identification (the recognize-versus-discover decision), is exactly where the two incompletenesses coincide, hence undecidable and carrying an irreducible error floor. Finally we argue that the coupling is a universal law, realized, in the emergent sense of Anderson's ``more is different,'' at every scale from genes to thoughts to cultures, and give its falsifiable core and honest scope.

The Legg--Hutter universal intelligence measure provides a rigorous scalar assessment of general intelligence as expected reward across all computable environments, weighted by simplicity. However, the measure presupposes an externally specified reward function, raising the question of whether the reward primitive is inherently arbitrary or whether a canonical choice exists. We propose a new measure, called intervention complexity, that has five natural properties: environment-derivedness, universality, minimality, sensitivity, and achievement preference. Given a resource function rho encoding an inductive bias (such as program length, execution time, or energy), rho-intervention complexity is a universal reward. The result yields a family of canonical rewards indexed by resource bias, providing a principled completion of the Legg--Hutter framework that does not require external normative input. We further propose a two-dimensional characterisation of intelligence: agent competence (how well the agent performs relative to the oracle optimum) and learning efficiency (how quickly this competence improves with experience). A separation theorem establishes that the choice of resource bias determines the computability of the resulting measure: action-count IC is computable in polynomial time, while program-length IC without oracle access is uncomputable, with the gap between oracle and bare IC precisely quantifying the information-theoretic content of learning. We discuss implications for superintelligence and for pre-training universal agents.