A New Mathematical Framework Takes the Simulation Hypothesis Beyond Speculation

The simulation hypothesis—the idea that our universe might be an artificial construct running on some advanced computational system—has fascinated scientists, philosophers, and the general public for years. From science fiction to serious academic debates, the question “Are we living in a simulation?” has often been discussed using intuition, probability arguments, or philosophical thought experiments. What has been largely missing, however, is a clear mathematical definition of what it actually means for one universe to simulate another.

That gap is exactly what a new research paper by David H. Wolpert, a professor at the Santa Fe Institute, aims to address. Published in the journal Journal of Physics: Complexity, the study introduces the first rigorous mathematical framework for defining simulation at the level of entire universes. Once those definitions are put in place, many popular assumptions about simulations turn out to be far less solid than previously thought.

Why the Simulation Debate Needed a Mathematical Reset

Much of the discussion around the simulation hypothesis has relied on informal reasoning. Famous arguments—such as the claim that simulated realities must be less powerful than the “base” reality running them—are often taken for granted without being carefully examined.

Wolpert’s work starts from a simple observation: you cannot meaningfully analyze simulations without first defining what a simulation is. Without that foundation, claims about hierarchies of realities, degraded simulations, or inevitable limits remain speculative. By introducing formal definitions, the debate shifts from philosophy-heavy conjecture to mathematics and computer science, where precise conclusions can be drawn.

Treating Universes as Computers

At the heart of the new framework is a major conceptual shift. Instead of viewing universes purely as physical systems with unknowable internal structures, Wolpert treats universes as computational entities. In this view, the evolution of a universe can be understood as computation, governed by rules that map one state to another.

This approach relies on the physical Church–Turing thesis, a principle suggesting that any physical process that can be observed can, in principle, be reproduced by a computational process. If that assumption holds, then questions about simulation become questions about what computations are possible, rather than metaphysical speculation about reality versus illusion.

By grounding the analysis in computation, Wolpert brings powerful tools from theoretical computer science into a debate that has rarely used them in a formal way.

A Surprising Role for Recursive Computation

One of the most striking aspects of the framework comes from applying a classic result in computer science known as Kleene’s second recursion theorem. In simple terms, this theorem shows that a computer program can be written in such a way that it contains an exact description of itself and can reproduce its own behavior.

When Wolpert extends this idea from programs to entire universes, the implications become surprisingly strange. Under the right conditions, a universe that can accurately simulate another universe could itself be fully reproduced inside that simulation. In other words, the simulated universe may be capable of simulating the simulator.

This result challenges the common idea that there must always be a clear hierarchy, with a “higher” reality running a “lower” one. Mathematically, such a hierarchy is not required at all.

Rethinking Simulation Hierarchies

A popular assumption in simulation discussions is that each deeper level of simulation must be computationally weaker than the one above it. This idea is often used to argue that infinite chains of simulations are impossible, because resources would eventually run out or errors would accumulate.

Wolpert’s framework shows that this assumption does not follow from mathematics. According to the model, simulations do not have to degrade. A simulated universe can, in principle, be just as computationally powerful as the universe simulating it. As a result, infinite chains of simulations remain mathematically consistent.

Even more intriguingly, the framework allows for the possibility of closed loops of simulations, where Universe A simulates Universe B, and Universe B simulates Universe A. In such cases, the idea of a single “base reality” becomes difficult—or even meaningless—to define.

What This Means for Identity and Existence

Once simulation is defined mathematically, it raises deep questions about personal identity. If universes can simulate each other without loss of fidelity, then multiple versions of the same observer could exist across different simulated realities. From a mathematical standpoint, these versions may be indistinguishable.

This challenges traditional philosophical ideas about individuality and originality. Instead of asking which version is “real,” the framework suggests that multiple instances can all be real in the same formal sense. The familiar intuition that there must be one original and many copies may simply not apply.

What the Framework Does Not Claim

Despite its ambitious scope, the research is careful about what it does not attempt to do. The framework does not provide experimental tests to determine whether our universe is a simulation. It does not offer predictions that could be checked with current or future technology. And it does not claim that simulations are physically feasible given known constraints such as energy, entropy, or quantum limits.

Instead, the contribution is conceptual. By clearly defining what simulation means, the framework provides a foundation on which future work—both philosophical and scientific—can build.

How This Fits into the Broader Simulation Debate

The simulation hypothesis has a long intellectual history, from ancient philosophical thought experiments to modern probabilistic arguments. In recent decades, it has gained renewed attention due to advances in computing, artificial intelligence, and theoretical physics.

What sets Wolpert’s work apart is its insistence on formal structure. Rather than asking how likely simulations are, or whether advanced civilizations would choose to run them, the paper focuses on what is mathematically possible. That shift changes the nature of the debate and opens up new directions for inquiry.

Why This Work Matters

By introducing a rigorous mathematical framework, this research shows that many widely held beliefs about simulations are not inevitable truths but assumptions that may not hold up under formal analysis. Concepts like hierarchical realities, weaker simulated worlds, and finite simulation depth are revealed to be optional rather than necessary.

Perhaps most importantly, the work demonstrates that once simulation is carefully defined, the landscape of possibilities becomes far richer and stranger than previously imagined. The question is no longer just whether we live in a simulation, but what kinds of simulation structures are even logically coherent.

Research paper:
David H. Wolpert, What computer science has to say about the simulation hypothesis, Journal of Physics: Complexity (2025)
https://doi.org/10.1088/2632-072x/ae1e50