In signal processing, the Nyquist frequency \(f_\mathrm{Nyquist} = f_s/2\) is the highest frequency that can be resolved given a sampling rate \(f_s\). Frequencies above \(f_\mathrm{Nyquist}\) are aliased and indistinguishable.
In the Abstract Universe:
Sampling Rate: The rate of bit-flips along observer-compatible paths in configuration space \(\mathcal{C}\).
Planck Limit: The highest frequency resolvable in the spectral wavefunction \(\Psi(\gamma)\).
Consequence: Structures smaller than the Planck length or evolving faster than the Planck time cannot be resolved by \(\Psi\); they are mathematically unrepresentable and effectively non-existent for any observer.
Let \(\gamma = (s_1, s_2, \dots)\) be an observer path through \(\mathcal{C}\), with each \(s_i\) an \(n\)-bit configuration.
Define a spectral encoding length \(\mathcal{L}(\gamma)\). Updates to \(\mathcal{L}\) occur only when significant motifs (observer-relevant patterns) change along the path.
\[t_{i+1} - t_i = \begin{cases} 1, & \text{if a motif update occurs} \\ 0, & \text{otherwise} \end{cases}\]
Thus, time emerges discretely, with “ticks” corresponding to spectral transitions along observer-compatible paths.
Let a “mass” be a localized cluster of reusable microstructure motifs in configuration space. The number of configurations at distance \(r\) that can reference these motifs grows with the surface area of a 3D sphere:
\[A(r) = 4 \pi r^2\]
Consequently, the density of spectral-compatible configurations falls as:
\[\rho(r) \propto \frac{1}{r^2} \, L(d)\]
where \(L(d)\) is the log-normal distribution of motif distances \(d\).
Observer paths minimize spectral cost \(\mathcal{L}(\gamma)\) and are biased toward regions of high \(\rho(r)\). In the continuum limit, the variational principle
\[\delta \int_\gamma \mathcal{L}(\text{state}) \, ds = 0\]
reproduces classical geodesics in 3D space, giving rise to an effective inverse-square law of interaction.
Summing over all observer-compatible paths weighted by their spectral amplitude:
\[\langle \mathcal{O} \rangle = \sum_{\gamma \in \mathcal{T}_{\mathrm{obs}}} \mathcal{O}(\gamma) \, 2^{-\mathcal{L}(\gamma)}\]
reproduces a path-integral formalism. In this framework:
Everettian view: All observer-compatible paths exist simultaneously; probabilities emerge from spectral weighting.
Wheeler–DeWitt view: Configuration space is timeless; time is induced along the observer path \(\gamma\).
Classical limit: Low-\(\mathcal{L}\) paths dominate, yielding smooth geodesics in agreement with General Relativity.
The Planck scale, discrete time, and gravity emerge naturally from:
The maximal resolvable spectral frequency (Nyquist-Planck analogy).
Discrete spectral updates along observer paths.
Statistical bias of observer paths toward high-density microstructure regions, producing effective inverse-square laws.
In this view, gravitation is not a fundamental force but a statistical-geometric phenomenon rooted in the spectral structure of observer-compatible paths in configuration space.