The Observer Wavefunction

The Observer's Wavefunction and the Walk Through Time

##No Outside

In a static, timeless universe there is no medium through which information could travel from an ``external world'' to an observer. No light ray crosses from the outside in, and no signal reaches the brain. Whatever Alice perceives as the world outside her is already part of her own description. Information that is not part of that description is, by construction, irrelevant to her. The universe, as far as Alice is concerned, is the description of Alice.

We call this description the observer wavefunction $\Psi_A$. It is a complex-valued spectral description: the modes (frequencies, amplitudes, phases) that jointly specify Alice living through her entire life, from birth to death. In our framework, $\Psi_A$ is the wavefunction of the universe.

Frames and Decompositions

The wavefunction $\Psi_A$ can be decomposed into a sequence of frames

$$ \Psi_A ;\longrightarrow; {\psi_1, \psi_2, \dots, \psi_N}, $$

where each $\psi_t$ is a sub-wavefunction describing one spatial configuration, one ``moment''. Nothing moves from $\psi_t$ to $\psi_{t+1}$. Both are already contained in $\Psi_A$. What we call time is the ordering in which Alice traverses the frames.

The decomposition is not unique. The same $\Psi_A$ admits many ways of being split into frames, and each split defines a different walk. The wavefunction therefore contains all possible walks through all of its possible decompositions.

The Most Probable Walk

Not all walks carry equal weight. By the exponential law of Section~\ref{chap:the-filter}, a description of total cost $m$ has measure $2^{-m}$. A walk is a description of a whole history, so its weight is set by the cost of the whole walk. Extending a history by one frame costs $\Delta m_t$ additional bits, so the continuation has relative weight $2^{-\Delta m_t}$. Walks along which each frame is well predicted by the accumulated description are cheap and dominate the measure. The walk Alice actually experiences is the most probable one, and the lawfulness of her world (smooth, predictable, wave-like) is what that dominance looks like from the inside.

Why the History Ends

The frames $\psi_t$ are not equally complex. Early frames are simple, with few active modes. Along the walk, the number of modes $M(t)$ in $\psi_t$ increases, and so does the spectral cost

$$ C(\psi_t) ;=; \sum_{k=1}^{M(t)} \bigl( b_{\omega_k} + b_{\phi_k} \bigr), $$

where $b_{\omega_k}$ and $b_{\phi_k}$ are the bits needed to specify the frequency and phase of mode $k$ at the resolution required by the observer filter. Higher frequencies cost more bits, so the cost grows faster than the mode count alone.

Early on, the modes already present predict the next frame well, so extending the walk by one frame is cheap. As $M(t)$ grows and the added modes become less predictable from the existing ones, each extension costs more, and the relative weight $2^{-\Delta m_t}$ of continuing falls. Past some point no continuation retains appreciable measure and the walk ends. Death is then a statistical event: the exhaustion of observer-compatible continuations. Since $n$ is the length of the complete description, this also gives a finite $n$ without further assumption.

Assumptions

The measure is taken over whole histories, not individual frames.

• $\Delta m_t$ first decreases and then increases. This is a premise, not a derivation, and can be tested by computing $\Delta m_t$ frame by frame for a toy observer.

• Incompressible content accumulates faster than the model absorbs it.

• All resolution-dependent quantities are conditional on the observer filter.

• The spectral codec is an empirical input, supported by observation.

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