3 Spectral Complexity

3.1 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.

Definition 1 (Spectral Complexity). The spectral complexity C_s(Ψ) of a state Ψ is the total informational cost required to uniquely specify the amplitudes, frequencies, and phases of the spectral modes composing Ψ:

where C_base is the 𝒪(1) cost of the underlying trigonometric subroutines, C(A_i) is the bit depth required to encode amplitude A_i, and ω_i∕Δω is the linear resource cost of resolving mode i at frequency ω_i relative to the system’s minimum frequency resolution Δω.

Frequency and phase are Nyquist-bandwidth costs: representing a mode of frequency ω_i requires O(ω_i∕Δω) samples to avoid aliasing, and pinning down its phase to the precision the reference basis can actually resolve costs the same order. Because Δω is already set by the system’s coherence extent (Δω ∼ 2π∕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. κ is an 𝒪(1) constant carrying the relative weight of phase-resolution against frequency-counting cost; there is no a priori reason to expect κ = 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 + κ)ω_i∕Δω scales linearly with frequency. This mirrors the physical relation E = ℏω: energy scales linearly with frequency. Under Solomonoff suppression P ∝ 2^−C_s, this linear cost produces exponential suppression of high-frequency modes:

which is a Boltzmann distribution with β = ln2∕Δω.

The identification

connects the minimum frequency resolution of the spectral complexity measure to Planck’s constant.

3.2 Resolution of the Boltzmann Brain Problem

Under the Solomonoff prior P(s) ∝ 2^−L(s) applied to spectral complexity, the probability of a configuration is exponentially suppressed by its spectral cost. Chaotic, high-entropy configurations — Boltzmann brains, random fluctuations, disordered universes — have high spectral complexity: they require many high-frequency modes with large amplitudes to describe. Their probability under C_s is therefore exponentially suppressed relative to smooth, law-like, compressible configurations.

This resolves the Boltzmann brain problem without additional axioms. The selection of structured observers is not a fine-tuning accident. It is a direct consequence of the measure: smooth, predictable data compresses better than chaos, and the spectral complexity measure weights configurations by their compressibility. We do not find ourselves in a chaotic universe because chaotic universes are exponentially expensive to describe.