The Equivalence Class Structure of Physical Reality - juhameskanen/abstract GitHub Wiki

The Equivalence Class Structure of Physical Reality

Alice as an Equivalence Class

The first thought experiment of The Four Axioms (the optimization argument) establishes a result stronger than substrate independence alone.

Substrate independence says $2+2-4$ holds for both apples and bananas. In other words, the carrier doesn't matter, neurons vs silicon produce the same Alice. The equivalence class result goes further: the entire computational history producing the execution trace doesn't matter either. This stems from the fact that the same Alice DNA simulation can be implemented in massively many different algorithms and optimization levels. 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).

Each distinct optimized program produces a distinct execution trace, yet all of them are the same Alice. Alice is what is invariant across that set of traces. Real-world Alice is simply one more member of this equivalence class — not privileged over the simulated ones.

Observer Equivalence Class

Let $C_O$ be the complexity measure governing the framework's compression principle. Alice is the equivalence class $[A]$ of all execution traces $T_1, T_2, \ldots$ that instantiate her genome/dynamics, under the relation

$$ T_i \sim T_j \iff \text{the } C_O\text{-optimal compressed representation of } T_i \text{ equals that of } T_j. $$

That is, $T_i$ and $T_j$ encode the same Alice if they differ only in the components of the trace that the optimal code discards as distortion (substrate details, thermal/molecular noise, algorithmic path taken to compute the same result) — not in the components that survive compression (the causal/functional structure that constitutes her as an observer).

Fermions as Equivalence Classes

The same structure applies to particles. Many different bitstring configurations yield the same fermion — the fermion is the equivalence class, not any particular bitstring representative.

This is already implicit in the exact theorems of Paper VIII. The norms $|F| = 1/\sqrt{n}$ and $|B| = \sqrt{(n-1)/n}$ depend only on $n$ and are completely independent of the phases $\phi_k$. Arbitrary phase choices produce different bitstrings but identical physical fermion structure. The winding number $m$ partitions the class into physically distinct subclasses.

The density matrix $\rho$ is the natural representative object in both cases — it is phase-invariant by construction, and is therefore the mathematical object that is the equivalence class.

The hierarchy is clean:

Alice: equivalence class of execution traces
Fermion: equivalence class of bitstring configurations under the codec
$\rho$: the mathematical object that is the equivalence class in both cases

The Born rule probability measure lives on the equivalence class, not on individual representatives.

D-ψ-G as the Foundational Inversion

Bits are not fundamental. They are Born-rule sampled discrete descriptions of compression artifacts ($\psi$). This is the D-ψ-G trinity in action:

ψ is the primitive — the equivalence class, the compression artifact, the fermion
D is the Born-rule sampled discrete description of ψ — bits are what you get when you look, not what fundamentally exists
G is the geometric layer that emerges when ψ is integrated over scales large enough that discrete sampling averages out

The direction of explanation runs $G \leftarrow \psi \leftarrow D$, not the other way around. Standard physics builds upward from D, which might be why quantizing gravity is hard — we are trying to reconstruct ψ from its samples and then derive G, which is exactly backwards.

The compression principle operates at the ψ level. Bits are what an internal observer samples from it. Geometry is what large-scale ψ structure looks like from inside.

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