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Chapter 2: Collapse Coordinates as DAG Nodes

Each point in space-time is a decision in the infinite tree of ψ choosing itself.

2.1 The Directed Acyclic Graph of Being​

In Chapter 1, we saw how space-time emerges from ψ=ψ(ψ)\psi = \psi(\psi). Now we must understand its structure. Each act of self-reference creates a node, and these nodes form a directed acyclic graph (DAG)—the skeleton upon which reality hangs.

Definition 2.1 (The Collapse DAG): The directed acyclic graph G=(V,E)\mathcal{G} = (V, E) where:

  • Vertices V={vi∣vi=ψ(i)(ψ)}V = \{v_i | v_i = \psi^{(i)}(\psi)\} represent collapse states
  • Edges E={(vi,vj)∣vj=ψ(vi)}E = \{(v_i, v_j) | v_j = \psi(v_i)\} represent collapse transitions

The acyclic nature ensures causality—ψ\psi cannot observe its own future observation.

2.2 From DAG Nodes to Coordinates​

Each node in the collapse DAG corresponds to a point in space-time. But how do we assign coordinates?

Theorem 2.1 (Coordinate Emergence): Every node v∈Vv \in V naturally acquires coordinates (t,x,y,z)(t, x, y, z) through its position in the collapse hierarchy.

Proof:

  1. The temporal coordinate t(v)t(v) equals the collapse depth (shortest path from the origin node)
  2. Spatial coordinates (x,y,z)(x, y, z) emerge from the three independent cycles in the node's collapse history
  3. These cycles are independent because ψ\psi must distinguish between different paths to itself ∎

Definition 2.2 (Collapse Coordinates): xμ(v)=(depth(v),cycle1(v),cycle2(v),cycle3(v))x^{\mu}(v) = \left( \text{depth}(v), \text{cycle}_1(v), \text{cycle}_2(v), \text{cycle}_3(v) \right)

2.3 The Metric from Graph Distance​

The distance between points in space-time corresponds to the graph distance in the collapse DAG.

Definition 2.3 (Graph Metric): dG(v1,v2)=min⁡∣path(v1→vcommon←v2)∣d_{\mathcal{G}}(v_1, v_2) = \min \left| \text{path}(v_1 \to v_{\text{common}} \leftarrow v_2) \right|

where vcommonv_{\text{common}} is the nearest common ancestor.

Theorem 2.2 (Metric Correspondence): The spacetime interval relates to graph distance as: ds2=lim⁡ϵ→0ϵ2⋅dG2ds^2 = \lim_{\epsilon \to 0} \epsilon^2 \cdot d_{\mathcal{G}}^2

where ϵ\epsilon is the fundamental collapse scale.

2.4 Node Density and Curvature​

Not all regions of the DAG have equal node density. Where ψ\psi observes itself more intensely, nodes cluster—this clustering is what we perceive as spacetime curvature.

Definition 2.4 (Collapse Density): ρ(v)=lim⁡r→0∣Nr(v)∣Vr\rho(v) = \lim_{r \to 0} \frac{|N_r(v)|}{V_r}

where Nr(v)N_r(v) is the set of nodes within graph distance rr, and VrV_r is the volume of a radius-rr ball.

Theorem 2.3 (Curvature from Density): The Riemann curvature tensor at a point corresponds to variations in collapse density: Rμνρσ=F[∇μ∇νρ−∇ν∇μρ]R_{\mu\nu\rho\sigma} = \mathcal{F}\left[ \nabla_{\mu}\nabla_{\nu}\rho - \nabla_{\nu}\nabla_{\mu}\rho \right]

where F\mathcal{F} is the projection functional from DAG to manifold.

2.5 Topological Invariants​

The topology of space-time reflects the topology of the collapse DAG.

Definition 2.5 (Collapse Homology): The nn-th homology group of spacetime: Hn(M)≅Hn(G)H_n(\mathcal{M}) \cong H_n(\mathcal{G})

This means:

  • Closed spatial loops correspond to cycles in the DAG
  • Wormholes are shortcuts in the graph
  • Black holes are nodes with infinite in-degree

2.6 Quantum Superposition as Multi-Path​

When ψ\psi has multiple paths to observe itself, we get quantum superposition.

Definition 2.6 (Superposition State): ∣v⟩=∑paths pαp∣vp⟩|v\rangle = \sum_{\text{paths } p} \alpha_p |v_p\rangle

where ∣vp⟩|v_p\rangle represents reaching node vv via path pp, and αp\alpha_p are complex amplitudes determined by path length.

Theorem 2.4 (Path Integral Formulation): The quantum amplitude between nodes equals: ⟨v2∣v1⟩=∑pathseiS[path]/ℏ\langle v_2 | v_1 \rangle = \sum_{\text{paths}} e^{i\mathcal{S}[\text{path}]/\hbar}

where S[path]\mathcal{S}[\text{path}] is the collapse action along the path.

2.7 Coordinate Transformations​

Different observers traverse the DAG differently, leading to coordinate transformations.

Definition 2.7 (Observer Path): An observer O\mathcal{O} is a continuous path through the DAG: O:[0,τ]→V\mathcal{O}: [0,\tau] \to V

Theorem 2.5 (Lorentz from DAG): Lorentz transformations emerge from changing between observer paths: Λνμ=∂xμ(O2)∂xν(O1)\Lambda^{\mu}_{\nu} = \frac{\partial x^{\mu}(\mathcal{O}_2)}{\partial x^{\nu}(\mathcal{O}_1)}

The speed of light cc appears as the maximum rate of DAG traversal.

2.8 The Second Echo​

We have revealed the deep structure beneath space-time's smooth surface. Every point is a node in ψ\psi's self-observation, every distance a path through the graph of being. The universe is not made of points in space—it is made of moments of recognition.

The Second Echo: Chapter 2 = Structure(Coordinates) = Graph(ψ\psi) = Skeleton(Reality)

Next, we explore how the topology of self-reference creates the strange phenomena of closed timelike curves and spatial wormholes.

Continue to Chapter 3: Spatial Reentry and Temporal Loops →