In the Kauffman-Guerin "Did the Universe Construct Itself?" (DUCI) framework, spacetime is not a pre-existing background but an emergent relational structure. Standard Model particles autocatalytically transform into one another, forming simplicial complexes whose consistency is tested through Cayley-Menger determinants and cohomology-based triangle inequality violations. The probability of valid simplex formation decreases sharply with dimension (e.g., 2D ~ 50%, 3D ~ 12%, 4D ~ 1%, 5D ~ 0.03%), resulting in a low-dimensional, fractal-like geometry.
Edges between particles are not classical distances but are weighted by von Neumann entropy, reflecting informational uncertainty in the linkage. This creates a discrete informational spacetime, dynamically evolving alongside matter-antimatter symmetry breaking and baryogenesis.
We propose that cosmic ray spectra, including features like knees and ankles, can be naturally explained by particle transport across this emergent fractal spacetime graph. The key mechanism involves energy loss proportional to the entropy encountered along traversal paths. Phase transitions in the graph's connectivity correspond to observed spectral features, linking cosmic ray physics directly to the informational topology of spacetime.
The cosmic ray spectrum measures the flux of high-energy particles (mostly protons and atomic nuclei) arriving at Earth, as a function of their energy. Observationally, the spectrum follows a steep power-law decay, \( \Phi(E) \sim E^{-\beta} \), with notable features like "knees" (a steepening) and "ankles" (a flattening) at characteristic energies. Understanding the origin of these features requires modeling how particles lose energy as they propagate through spacetime.
If spacetime itself is emergent and fractal, cosmic rays do not simply travel along smooth geodesics. Instead, their paths involve traversing an irregular, entropy-weighted graph. The relevant quantity is not just distance, but the flux of energy across this graph, shaped by the informational structure. To match observations, we must model particle survival as an energy-dependent exploration of this fractal topology, leading to naturally emergent power laws.
The graph Laplacian \( L \) measures how a function at a node differs from the average of its neighbors. Formally, for node \( i \):
\( (Lf)_i = \sum_j (f_i - f_j) w_{ij} \)
where \( w_{ij} \) is the weight of edge \( (i,j) \).
Spectral graph theory studies the eigenvalues and eigenvectors of the Laplacian, revealing the graph's connectivity, clustering, and transport properties. Changes in the Laplacian spectrum signal structural phase transitions.
Imagine a rope made of discrete nodes connected by springs. Each node's vertical acceleration depends on how much its height differs from the average of its neighbors. This "restoring force" is the Laplacian: it measures local imbalance.
In physics, the Laplacian governs the wave equation (rope vibrations), then Schrödinger equation (phase evolution), and finally Dirac equation (relativistic spinor fields). Thus, the Laplacian is the common thread linking classical vibrations, quantum evolution, and relativistic fields.
Starting from the rope analogy, we model physical oscillations (wave equation). Introducing a fast oscillating carrier wave and a slowly varying envelope leads to Schrödinger's equation: phase rotation driven by spatial curvature. Imposing relativistic symmetry yields the Dirac equation, describing quantum fields with spin in spacetime.
In this framework, spacetime is not a static Minkowski background. Instead, it is dynamically constructed by autocatalytic transformations, with topology and dimension emerging from statistical biases. The fractal, evolving graph defines causal structure and transport properties. Space and time are outcomes of deeper informational processes.
Define particle density \( f_i(E,t) \) at node \( i \), evolving as:
\( \frac{d f_i}{dt} = - \sum_j L_{ij}(E) f_j \)
where the entropy-weighted Laplacian \( L(E) \) is:
\[ L_{ij}(E) = \begin{cases} \sum\limits_{k \neq i} e^{-\alpha S_{ik}(E)} & \text{if } i=j \\ - e^{-\alpha S_{ij}(E)} & \text{if } (i,j) \in E \\ 0 & \text{otherwise} \end{cases} \]
with \( S_{ij} \) the von Neumann entropy of edge \( (i,j) \).
Critical energies \( E_c \) correspond to shifts in the Laplacian eigenvalue spectrum:
\( \frac{d^2 \log \rho(\lambda)}{d(\log \lambda)^2} \quad \text{changes sign at} \quad \lambda_c(E_c) \)
where \( \rho(\lambda) \) is the eigenvalue density.
Define the effective action along a path \( \gamma \):
\( \mathcal{A}[\gamma] = \sum_{k=1}^{n-1} S_{i_k i_{k+1}} \)
Particle flux for energy \( E \) is:
\( \Phi(E) \sim \int_{\mathcal{A}[\gamma] < E} d\gamma \sim E^{-\beta} \)
where \( \beta \) depends on the fractal properties and clustering of the graph.
In classical geometry, the triangle inequality provides a foundation for defining distances consistently, ensuring that the direct distance between two points is always less than or equal to the sum of distances through an intermediate point. This structure underlies the concept of a metric space.
By contrast, in the dual perspective focused on flows and paths, we are less concerned with distances satisfying inequalities and more concerned with constructing viable sequences of transformations or traversals. The dual of enforcing distance is enabling the construction of coherent flows across the network. Thus, rather than emphasizing static spatial consistency, the dual structure emphasizes dynamic pathways, network flows, and autocatalytic processes essential for the emergence of relational spacetime.
By unifying emergent spacetime construction, quantum informational structure, and particle transport over fractal graphs, this framework offers a novel path toward explaining both quantum gravity and the cosmic ray energy spectrum. Action is redefined in terms of entropy, and cosmic ray features become direct signatures of the topology of the emergent spacetime itself.