The Wavefunction as Compression

The Wavefunction as Compression

Introduction

The Four Axioms page established the universe must be static, timeless, informational, substrate independent and finite, where simulation and the computer running the simulation are not causally related by two different perspectives of the same information. Black hole Singularities as Zero Entropy States applies the principle to a black hole and shows that singularities map to zero entropy states.
Emergence of Spacetime studies the time reverted case and concludes that expansion from that singularity is the geometric reading of entropy increase, with emergent micro-structures following filter-independent hump-like distributions Emergence of Spacetime. The page Information Theoretic Emergence of Spacetime and Matter completes the picture and derives curved spacetime from the entropy and micro-structures only.

One feature of the observed universe is not explained by this framework: the wave-like behaviour of the microcosm. Particles exhibit interference, superposition, and discrete measurement outcomes. The wavefunction is complex-valued. Dynamics are unitary. Why?

This page proposes a hypothesis and investigates its consequences.

The Compression Hypothesis

The Wavefunction Compression Principle (WCP). The cosmos waves because we are observing compressed structure. The quantum wavefunction is the universe's data-compression codec.

The MPEG Analogy

Consider a conventional MPEG-compressed digital movie. Each frame is composed of discrete pixels. If observers existed inside such a movie, made of pixels and perceiving only pixel-level interactions, what would they observe as their laws of physics?

They would find that their constituent pixels follow mysterious, deterministic, wave-like patterns. To a pixel-physicist, these transitions would appear fundamental. In reality, they are the mathematical output of the Discrete Cosine Transform (DCT) --- the compression basis that MPEG codecs use to represent raw pixel data in a maximally compact form.

The parallel with quantum mechanics is direct. The quantum wavefunction distributes probability amplitudes across basis states in Hilbert space exactly as a spatial Fourier transform distributes image information across frequency coefficients. Complex-valued amplitudes are exceptionally efficient at encoding relational information: they produce precisely the spatial smoothness, structural continuity, and predictive regularity that stable observers require.

The wave-like behaviour of the microcosm is not a primitive feature of reality but the signature of compressed information as perceived from within. An observer composed of compressed structures will perceive their world as wave-governed for the same reason that a pixel-observer would perceive DCT dynamics: the compression codec is the physics.

The Dithering Analogy and the Born Rule

The Born rule --- the identification of measurement probabilities with squared wavefunction amplitudes --- has a natural interpretation under this hypothesis.

Consider a rendering engine producing a sphere with intended shading intensity $0.85$, on hardware restricted to discrete outputs of $0.8$ or $0.9$. The engine employs dithering: distributing $0.8$ and $0.9$ across adjacent pixels with relative frequencies $0.15$ and $0.85$ respectively, so that the perceived average matches the target.

A quantum superposition

$$ |\psi\rangle = \alpha|0.8\rangle + \beta|0.9\rangle $$

is the universe's complex-valued implementation of this same principle. The Born rule $P = |\alpha|^2$ is the probability rule of an optimally compressed continuous description rendered through a discrete measurement apparatus. The squaring arises because amplitudes are complex-valued: the information-theoretic weight of a mode is proportional to the power in that mode, which is the squared amplitude.

The Born rule can be derived from the spectral complexity axioms alone; Born rule is what optimal compression looks like when a continuous description is projected to finite set of information.

Fermions, Bosons and Pauli Exclusion

Fermions play the role of pixels - real particles, sampled from the wavefunction through Born rule.

We argue that Bosons don't exist as fundamental objects. They are explanatory fictions — the story we tell about why fermions (pixels) move the way they do, when we don't know they're being sampled from a wavefunction.

The actual causal chain is:

$$ \text{wavefunction (codec)} \xrightarrow{\text{Born rule sampling}} \text{fermion configurations (pixel frames)} $$

But an observer who only sees the pixel outputs, not the codec, invents a story:

$$ \text{fermion} \xrightarrow{\text{"virtual boson exchange"}} \text{fermion} $$

The boson is the compression residual made into a noun. It's not wrong — it's a valid effective description — but it's not fundamental.

The analogy is exact: in MPEG, macroblocks don't "exchange" anything. The DCT coefficients just are what they are. But if one only watched the decoded pixels and tried to explain their correlations without knowing about DCT, one would invent something that looks exactly like boson exchange.

This explains natively why fermions suffer from Pauli exclusion whereas bosons not.

Spectral Complexity

From Kolmogorov to Spectral Complexity

Kolmogorov complexity provides the theoretical foundation for minimal description length but is fundamentally uncomputable, discrete, and discontinuous. It cannot smoothly support stable prediction, continuous spacetime geometry, or the gradual evolutionary reasoning of an internal observer.

The observational evidence suggests that physical systems already possess a natural mode of information representation: decomposition into spectral modes. This motivates a continuous, computable replacement.

In the case of complex-valued wavefunction, the only algorithm we need is 'cos()' - the simplest continuous periodic function. Its Kolmogorov complexity is vanishingly small compared to full informational cost of the wavefunction. Most cost comes from the data - the modes, frequencies and phases.

Spectral Complexity

The spectral complexity $C_s(\Psi)$ of a state $\Psi$ is the total informational cost required to uniquely specify the amplitudes, frequencies, and phases of the spectral modes composing $\Psi$:

$$ C_s(\Psi) = C_{\mathrm{base}} + \sum_{i=1}^{N} \left[ C(A_i) + (1+\kappa),\frac{\omega_i}{\Delta\omega} \right] $$

where $C_{\mathrm{base}}$ is the $\mathcal{O}(1)$ cost of the underlying trigonometric subroutines, $C(A_i)$ is the bit depth required to encode amplitude $A_i$, and $\omega_i/\Delta\omega$ is the linear resource cost of resolving mode $i$ at frequency $\omega_i$ relative to the system's minimum frequency resolution $\Delta\omega$.

Frequency and phase are Nyquist-bandwidth costs: representing a mode of frequency $\omega_i$ requires $O(\omega_i/\Delta\omega)$ samples to avoid aliasing, and pinning down its phase to the precision the reference basis can actually resolve costs the same order. Because $\Delta\omega$ is already set by the system's coherence extent ($\Delta\omega \sim 2\pi/L$) via the Fourier uncertainty relation, the phase cost doesn't introduce a new length scale — it rescales the coefficient on the term already present. $\kappa$ is an $\mathcal{O}(1)$ constant carrying the relative weight of phase-resolution against frequency-counting cost; there is no a priori reason to expect $\kappa=1$, and it should be treated as an open parameter of the model, fixed either by an independent argument or by matching to a known physical constant (see below).

Three properties of this measure are essential.

• Computability. Unlike Kolmogorov complexity, $C_s$ is directly computable from the spectral decomposition of any state.

• Continuity. $C_s$ assigns a smooth, continuous cost gradient across neighbouring states. Small perturbations in frequency or amplitude produce small changes in complexity.

• Linear frequency-and-phase scaling. The dominant cost term $(1+\kappa),\omega_i/\Delta\omega$ scales linearly with frequency. This mirrors the physical relation $E=\hbar\omega$: energy scales linearly with frequency. Under Solomonoff suppression $P \propto 2^{-C_s}$, this linear cost produces exponential suppression of high-frequency modes:

$$ P \propto 2^{-(1+\kappa),\omega/\Delta\omega} = e^{-(1+\kappa)(\ln 2),\omega/\Delta\omega}, $$

which is a Boltzmann distribution with $\beta = (1+\kappa)\ln 2 / \Delta\omega$.

The identification

$$ \hbar = \frac{\Delta\omega}{(1+\kappa),\ln 2} $$

connects the minimum frequency resolution of the spectral complexity measure to Planck's constant. As before, this is not a derivation but a precise identification that warrants further investigation — now with the phase contribution folded into the same suppression rate rather than omitted from it.

Numerical Results

Emergence of Smooth Spacetime

Applying the spectral complexity cost function as a selection weight to the bitstring evolution of Paper~II --- replacing uniform sampling with $C_s$-weighted sampling --- immediately suppresses the high-entropy white noise of the unweighted ensemble. The dominant configurations are those with low spectral complexity: smoothly varying, wave-governed spatial profiles. Symmetric wave packets and lattice-like structures emerge as the most probable outcomes without being hard-coded.

Emergence of Inertia and Interference

Two specifically quantum-mechanical phenomena emerge from spectral compression in numerical simulation, without any imposed equations of motion.

• Inertia. A localized wave packet moving through the spectral field maintains its velocity without external forcing. The minimum $C_s$ continuation of a moving packet is the packet continuing to move: any deflection increases the spectral cost by introducing new frequency components. Resistance to deflection --- inertia --- is the geometric consequence of spectral economy.

• Interference. When two wave packets overlap, the minimum $C_s$ description of the combined state is not the sum of two independent descriptions but a single spectral decomposition of the superposition. The cross-terms --- interference fringes --- are cheaper to encode than two separate packets because they share spectral modes. Interference emerges as the compression-optimal description of overlapping structures.

Why Waves?

JPEG, MPEG, audio codecs, use of Fourier and other numerous real world applications are a large, working existence proof that spectral bases are what compression picks for natural data. We don't need to re-derive it, we can cite it as evidence.

Natural signals have approximately translation-invariant statistics (neighbouring pixels and samples correlate), and the transform that compacts that structure into a few coefficients is the Fourier-type basis. The Karhunen-Loève transform, which is the optimal decorrelating basis, approaches the DCT for highly correlated stationary sources. So the codecs are using waves because that is what minimizes description length for that kind of data.

Two caveats:

It is not always sinusoids. JPEG2000 uses wavelets, and the best basis depends on the data's structure. That supports our point that the basis is selected by what the observer's data looks like.

Real versus complex. Video codecs use real coefficients with no interference between them. They show that spectral compression is natural and cheap, but they say nothing about why the micro-cosmos has complex amplitudes and Born-rule probabilities. That narrower question is the part that still needs its own argument. Why complex?

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