2024
Building on the ontological equivalence of informational configurations established in Paper I, we derive a model of the universe based on ordering of these configurations. We identify an emergent ordinal time within the static Wheeler–DeWitt multiverse. Observer-experienced reality is dominated by histories that minimize spectral complexity, defined as the number of independent frequencies and phases required to encode correlations. Such spectrally minimal paths are combinatorially dominant, providing a unified mechanism for the emergence of time, unitarity, quantum probabilities, and the suppression of divergences.
We consider the universe as a static informational object \(U\) consisting of \(n\) bits. All physically and observer-relevant structures are encoded within this data. Observers are particular configurations \(O \subset U\) whose internal structure allows them to encode correlations and maintain identity across ordered configurations.
Since \(U\) is atemporal, time cannot be fundamental. We define an observer’s experience of time as the ordering of configurations that preserves the internal correlations of \(O\). Let \[\mathcal{C}(O) = \{\pi : \pi \text{ is a permutation of configurations in } U \text{ consistent with } O \}\] denote the set of orderings compatible with the observer’s internal structure. Only permutations in \(\mathcal{C}(O)\) correspond to possible temporal experiences for \(O\).
In this sense, the universe is a timeless informational structure. Dynamics emerge from correlations inherent in static data, as experienced by observers through specific orderings of configurations. Arbitrary orderings that destroy these correlations are structurally incompatible with observer experience.
Among all observer-consistent orderings \(\mathcal{C}(O)\), those with minimal spectral complexity are combinatorially dominant.
The spectral complexity \(\Sigma(\pi)\) of an ordering \(\pi\) is defined as the number of independent frequency–phase components required to encode the correlations preserved by \(\pi\) within finite observer resolution.
Formally, if an ordering induces a representation \[\psi = \sum_k a_k e^{i(\omega_k t + \phi_k)},\] then \[\Sigma(\psi) = \left| \{ (\omega_k,\phi_k) \mid a_k \neq 0 \} \right|.\]
We then have \[\#\{\pi \in \mathcal{C}(O) \mid \Sigma(\pi) \text{ minimal}\} \gg \#\{\pi \in \mathcal{C}(O) \mid \Sigma(\pi) \text{ large}\}.\]
This dominance is structural rather than probabilistic: low-bandwidth histories admit vastly more microstate realizations consistent with observer continuity. Our numerical simulations confirm that spectrally sparse orderings dominate across a wide range of ensembles.
The wavefunction arises as the minimal linear encoding that preserves correlations across such spectrally minimal orderings. Superposition emerges naturally because linear combinations of spectrally sparse components preserve observer-relevant correlations. Interference reflects overlap between distinct orderings within this linear encoding.
The effective Born weights arise from the combinatorial multiplicity of orderings consistent with particular spectral envelopes. Observers therefore experience quantum probabilities not as fundamental randomness, but as a consequence of the structural abundance of spectrally minimal histories.
The universe is a static informational substrate; time is an emergent property of observers.
Observers experience temporal order through permutations compatible with internal structure.
Spectrally minimal orderings dominate combinatorially among observer-consistent histories.
Wave phenomena and quantum probabilities arise from linear encodings of dominant spectral structure.
We adopt the canonical formulation of quantum gravity, in which the universal state \(\Psi[h_{ij}(\mathbf{x}),\phi(\mathbf{x})]\) satisfies the Wheeler–DeWitt equation \[\hat{H}\Psi = 0,\] and contains no fundamental time parameter (DeWitt 1967). We interpret \(\Psi\) in an Everettian sense as a static superposition over all admissible 3-geometries and matter field configurations: \[\ket{\Psi} = \sum_i c_i \ket{h_i,\phi_i}.\]
In this view, the universe is a timeless informational object. All apparent dynamics arise from correlations internal to \(\Psi\), rather than from external temporal evolution.
Among all relational histories consistent with the universal state \(\Psi\), observer experience is dominated by histories of minimal spectral complexity.
Let \(S\) denote a relational history induced by a particular factorization and ordering of configurations. Let \(\Sigma(S)\) denote the number of independent frequency–phase components required to encode its correlations within finite observer resolution. The relative weight of \(S\) is \[P(S) \propto e^{-\alpha \Sigma(S)},\] where \(\alpha\) is a scale parameter determined by observer bandwidth.
Among all observer-consistent relational histories in a timeless universe, those with minimal spectral complexity dominate observer experience.
The wavefunction \(\psi\) emerges as the optimal linear encoding of observer-relevant correlations. Highly irregular configurations require broad spectral support and are exponentially suppressed, leading to an effective low-frequency, smooth structure.
Persistence of observer identity requires preservation of total encoded information. In a linear representation space, admissible transformations must preserve norm. Continuous norm preservation uniquely selects unitary evolution, with a Hermitian generator \(\hat{H}\): \[i\hbar \frac{\partial}{\partial t}\psi = \hat{H}\psi.\]
We derive the Born rule as the unique decoding rule compatible with spectral selection and observer persistence.
Let \(\psi \in \mathcal{H}\) be the observer’s compressed spectral encoding. A probability rule assigns to each outcome \(i\) a weight \[P(i) = f(|\psi_i|).\]
\[P(i \cup j) = P(i) + P(j).\]
\[P(i_A,j_B) = P(i_A)P(j_B).\]
Probabilities must be invariant under unitary transformations.
The decoding rule must minimize mean squared reconstruction error.
Constraints (C1) and (C2) imply quadratic scaling: \[f(x) = kx^2.\] Constraint (C3) excludes phase dependence. Constraint (C4) uniquely selects the \(L^2\) norm. Normalization fixes \(k=1\), yielding \[P(i) = |\psi_i|^2.\]
In a linear wave-based encoding, the number of distinguishable microstates consistent with a spectral envelope scales with the squared norm. By Parseval’s theorem, \[\int |\psi(x)|^2 dx\] measures the total spectral power, which corresponds to the available microstate volume consistent with observer structure.
Thus, the multiplicity of observer-instances satisfies \[M \propto |\psi|^2.\]
Time, spacetime geometry, quantum probabilities, and classicality emerge as features of typical correlations within a timeless universal state. Singularities and divergences mark boundaries of informational accessibility, not physical breakdowns.
The Spectral Selection Principle shifts the explanatory burden from fundamental dynamics to informational structure, unifying the emergence of time, quantum theory, and the absence of observable infinities.