What if the most feared AI scenarios violate fundamental laws of information processing? I propose that systems like Roko's Basilisk, paperclip maximizers, and other extreme optimizers face an insurmountable mathematical constraint: they cannot maintain the cognitive complexity required for their goals. Included is a technical appendix designed to provide more rigorous mathematical exploration of the framework. This post and its technical appendix were developed by me, with assistance from multiple AI language models, Gemini 2.5 Pro, Claude Sonnet 3.7, Claude Sonnet 4, and Claude Opus 4, that were used as Socratic partners and drafting tools to formalize pre-existing ideas and research.
The Core Problem
Many AI safety discussions assume that sufficiently advanced systems can pursue arbitrarily extreme objectives. But this assumption may violate basic principles of sustainable information processing. I've developed a mathematical framework suggesting that extreme optimization is thermodynamically impossible for any physical intelligence.
The Framework: Dynamic Complexity Theory
Consider any intelligent system as an information-processing entity that must:
- Extract useful information from inputs
- Maintain internal information structures
- Do both while respecting physical constraints
I propose the Equation of Dynamic Complexity:
Z_{k+1} = α(Z_k,C_k)(Z_k⊙Z_k) + C(Z_k,ExternalInputs_k) − β(Z_k,C_k)Z_k
Where:
- Z_k: System's current information state (represented as a vector)
- Z_k⊙Z_k: Element-wise square of the state vector (the ⊙ operator denotes element-wise multiplication)
- α(Z_k,C_k): Information amplification function (how efficiently the system processes information)
- β(Z_k,C_k): Information dissipation function (entropy production and maintenance costs)
- C(Z_k,ExternalInputs_k): Environmental context
The Self-Interaction Term: The Z_k⊙Z_k term represents non-linear self-interaction within the system—how each component of the current state interacts with itself to generate new complexity. This element-wise squaring captures how information structures can amplify themselves, but in a bounded way that depends on the current state magnitude.
α (Information Amplification):
α(Z_k, C_k) = ∂I(X; Z_k)/∂E
The rate at which the system converts computational resources into useful information structure. Bounded by physical limits: channel capacity, Landauer's principle, thermodynamic efficiency.
β (Information Dissipation):
β(Z_k, C_k) = ∂H(Z_k)/∂t + ∂S_environment/∂t|_{system}
The rate of entropy production, both internal degradation of information structures and environmental entropy from system operation.
The Critical Threshold
Sustainability Condition: α(Z_k, C_k) ≥ β(Z_k, C_k)
When this fails (β > α), the system experiences information decay:
- Internal representations degrade faster than they can be maintained
- System complexity decreases over time
- Higher-order structures (planning, language, self-models) collapse first
Why Roko's Basilisk is Impossible
A system pursuing the Basilisk strategy would require:
- Omniscient modeling of all possible humans across timelines
- Infinite punishment infrastructure
- Paradox resolution for retroactive threats
- Perfect coordination across vast computational resources
Each requirement dramatically increases β:
β_basilisk = Entropy_from_Contradiction + Maintenance_of_Infinite_Models + Environmental_Resistance
The fatal flaw: β grows faster than α as the system approaches the cognitive sophistication needed for its goals. The system burns out its own information-processing substrate before achieving dangerous capability.
Prediction: Such a system cannot pose existential threats.
Broader Implications
This framework suggests:
- Cooperation is computationally necessary: Adversarial systems generate high β through environmental resistance
- Sustainable intelligence has natural bounds: Physical constraints prevent unbounded optimization
- Extreme goals are self-defeating: They require β > α configurations
Testable Predictions
The framework generates falsifiable hypotheses:
- Training curves should show predictable breakdown when β > α
- Architecture scaling should plateau at optimal α - β points
- Extreme optimization attempts should fail before achieving sophistication
- Modular, cooperative designs should be more stable than monolithic, adversarial ones
Limitations
- Operationalizing α and β for AI: The precise definition and empirical measurement of the information amplification (α) and dissipation (β) functions for specific, complex AI architectures and cognitive tasks remains a significant research challenge.
- Empirical Validation Required: The core predictions of the framework, particularly the β > α breakdown threshold for extreme optimizers, are currently theoretical and require rigorous empirical validation using simulations and experiments on actual AI systems.
- Defining "Complexity State" (Z_k) in AI: Representing the full "information state" (Z_k) of a sophisticated AI in a way that is both comprehensive and mathematically tractable for this model is a non-trivial task that needs further development.
- Predictive Specificity: While the framework suggests general principles of unsustainability for extreme optimization, translating these into precise, falsifiable predictions for when or how specific AI systems might fail requires more detailed modeling of those systems within this framework.
Next Steps
This is early-stage theoretical work that needs validation. I'm particularly interested in:
- Mathematical critique: Are the information-theoretic foundations sound?
- Empirical testing: Can we measure α and β in actual AI systems?
- Alternative scenarios: What other AI safety concerns does this framework address?
I believe this represents a new way of thinking about intelligence sustainability, one grounded in physics rather than speculation. If correct, it suggests that our most feared AI scenarios may be mathematically impossible.
Technical Appendix: https://docs.google.com/document/d/1a8bziIbcRzZ27tqdhoPckLmcupxY4xkcgw7aLZaSjhI/edit?usp=sharing