Equivalence Classes

The Equivalence Class Structure of Physical Reality

Alice as an Equivalence Class

The same Alice DNA simulation can be implemented in massively many different algorithms and optimization levels. Each different implementation, once optimized, results different execution trace.

However, each static execution trace yield precisely the same Alice, just like $2+2+2$ and $2*3$ yield the same output, based on different algorithms (execution traces).

Alice is what is invariant across that set of traces.

The Probability Metric of Emergence

Framing the emergence of a conscious entity through combinatorial probability is a way to quantify reality:

$$ \text{P}(\text{Alice}) = \frac{\vert{}\text{Alice Equivalence Class}\vert{}}{2^n} $$

Given a state space of $n$ bits, the likelihood of Alice coming into existence depends entirely on how robust or spacious her equivalence class is.

• If a pattern requires hyper-specific, fragile constraints to function, its equivalence class is tiny, making its spontaneous emergence statistically rare.
• If a pattern is robust—meaning thousands of minor variations in low-level physics or code still yield the exact same conscious experience—its equivalence class expands, making it a natural attractor in the state space.