Fourier Basis and Spectral Complexity

3 Fourier Basis and Spectral Complexity

3.1 Natural basis

Paths are parameterised as finite Fourier series in the natural eigenbasis for this boundary value problem — functions that satisfy $a(0) = 0$ and $a˙(τmax) = 0$ :

( ) (2k-−-1)πτ- fk(τ ) = sin 2τ , k = 1,2,...,Kmax, max

with $Kmax = 12$ modes per field. The mode $k = 1$ reproduces the Hawking solution exactly; higher modes add higher-frequency corrections. Both $a(τ)$ and $ϕ (τ)$ use the same basis with independent coefficients $Ak$ and $Bk$ :

K∑max K∑max a(τ) = Akfk (τ), ϕ(τ) = Bkfk(τ). k=1 k=1

3.2 Spectral complexity in the natural basis

The natural frequencies of the basis modes are

(2k − 1)π ωk = --2τ-----, max

with uniform spacing $Δ ω = π∕τmax$ . Spectral complexity is computed directly from the Fourier coefficients, not from a discrete Fourier transform of the discretised path. A DFT of a non-periodic signal (as every path on $[0,τmax]$ is, relative to the DFT’s assumed periodicity) suffers spectral leakage — a single sine wave fills dozens of DFT bins, artificially inflating $Cs$ . Working in the problem’s own basis is both physically correct and numerically honest.

A mode $k$ is declared active if its power fraction exceeds a threshold $𝜖 = 10− 4$ :

A2k B2k ∑--A2-> 𝜖 (for the a field), ∑--B2- > 𝜖 (for ϕ). j j j j

Spectral complexity is then the total frequency cost of all active modes across both fields:

∑ -ωk- ∑ ωk-- Cs = Δ ω + Δ ω, k∈𝒜a k∈𝒜ϕ

where the ratio $ω ∕Δ ω = (2k − 1)∕2 k$ takes values $1, 3, 5,... 2 2 2$ by construction. The minimum- $C s$ path is therefore the one that achieves the required boundary conditions using the lowest-frequency modes — which is precisely $k = 1$ , the Hawking solution.

This measure is both smooth and computable in practice. Unlike uncomputable Kolmogorov complexity [kolmogorov1965] over Turing machines, the dominant cost arises from the sum of frequencies of retained modes. This makes $Cs$ a natural, differentiable proxy for algorithmic compressibility in the physical setting.

Under Solomonoff-like induction the prior becomes $P (ψ) ∝ 2−Cs(ψ )$ , yielding exponential suppression of high-frequency (rough) configurations — precisely analogous to the $−SE e$ weighting in the Euclidean path integral.

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