A Thermodynamic Theory of Intelligence: Why Extreme Optimization May Be Mathematically Impossible

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 idea of this framework is an application of the Mandelbrot Set to complex system dynamics.

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 Framework

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.

Information-Theoretic Foundations

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(Zk, 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:

1. Cooperation is computationally necessary: Adversarial systems generate high β through environmental resistance

2. Sustainable intelligence has natural bounds: Physical constraints prevent unbounded optimization

3. 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

LessWrong denied this post. I used AI to formalize the theory, LLMs did not and cannot do this level of logical reasoning on their own. This does not discuss recursion, how “LLMs work” currently or any of the other criteria they determined is AI slop. They are rejecting a valid theoretical framework simply because they do not like the method of construction. That is not rational. It is emotional. I understand why the limitation is in place, but this idea must be engaged with.