Prediction, Models, and Reality

SEPP provides a single, unifying principle that clarifies the relationship between any formal model and the reality it purports to describe. It allows us to move beyond the simple statement "all models are wrong" and provides a formal, quantitative reason why they are wrong and how they are useful.

A model is a formal system, $F_{model}$, that is, by design, simpler than the real-world system, $S_{reality}$, it represents. That is, $K(F_{model}) \ll K(S_{reality})$. SEPP's core inequality then becomes a fundamental law of modeling:

$\mathrm{Exp}(F_{model}) \le K(F_{model}) + c \ll K(S_{reality})$

This simple inequality has profound consequences that cut across all scientific disciplines:

1. Incompleteness is a Law, Not a Flaw: A model is not "wrong" in the sense of being a failed attempt at a perfect description. SEPP proves that incompleteness is a necessary, mathematical consequence of the act of modeling itself. To simplify is to limit expressive power. Therefore, any model is guaranteed to have blind spots and to fail at its boundaries.

2. Prediction is Compression: A successful prediction is an act of informational compression. A model makes a prediction when its simple formal rules can generate a description of a future state that is simpler than simulating the entire universe to see what happens. SEPP explains why prediction is so hard: the model's low complexity gives it a limited budget of expressive power to certify future states. For simple, low-entropy systems (like planetary orbits), this is possible. For complex, high-entropy systems (like the stock market, the weather, or human society), the informational complexity of the future state often exceeds the expressive power of any simple model we can construct.

3. The Role of Heuristics and Emergence: SEPP explains why high-level sciences (like biology, psychology, and sociology) are not reducible to physics. The simple axioms of physics ($F_{physics}$) have insufficient expressive power to describe the high-entropy emergent phenomena of these more complex domains. A new, more complex "effective theory" or heuristic model ($F_{biology}$) is not just a convenient shorthand; it is a necessary new formal system. It sacrifices foundational simplicity to gain the expressive power needed to describe the emergent regularities that are computationally inaccessible from the level below. Science is therefore necessarily a layered hierarchy of models, with each layer representing a different optimal trade-off on the SEPP spectrum.