Skip to main content

Chapter 27: DAG Folding toward Mass Nodes

In the tapestry of causation, mass creates deep folds—gathering threads of possibility into inevitable knots of destiny.

27.1 The Topology of Gravitational Collapse​

In the directed acyclic graph of ψ\psi's self-observation, massive objects don't just attract—they fundamentally reshape the topology. They create "folds" where many paths converge, like a cosmic drain drawing all nearby futures toward a single point.

Definition 27.1 (DAG Fold): A region where geodesic paths converge: F(M)={v∈V∣lim⁡t→∞path(v,t)→vM}\mathcal{F}(M) = \{v \in V | \lim_{t \to \infty} \text{path}(v,t) \to v_M\}

where vMv_M is the mass node.

Theorem 27.1 (Folding Theorem): Mass density determines fold depth: depth(F)=2GMc2r\text{depth}(\mathcal{F}) = \frac{2GM}{c^2r}

The stronger the mass, the deeper the fold in causality itself.

27.2 Event Horizons as Complete Folds​

A black hole represents a complete fold—where the DAG literally folds over on itself, creating a causal cul-de-sac.

Definition 27.2 (Causal Horizon): H={v∈V∣∀w>v,path(v→w)=∅}\mathcal{H} = \{v \in V | \forall w > v, \text{path}(v \to w) = \emptyset\}

No future paths escape the fold.

Theorem 27.2 (Horizon as Fold Boundary): The event horizon radius equals: rs=2GMc2=complete fold radiusr_s = \frac{2GM}{c^2} = \text{complete fold radius}

Inside, all paths lead deeper. The DAG has no exit edges.

27.3 Geodesics as Optimal Paths​

In flat space, the shortest path is straight. Near mass, the "shortest" path through the folded DAG is curved.

Definition 27.3 (Geodesic Path): γ=arg⁡min⁡paths∫pathds\gamma = \arg\min_{\text{paths}} \int_{\text{path}} ds

Theorem 27.3 (Geodesic Equation): Optimal paths satisfy: d2xμdτ2+Γνλμdxνdτdxλdτ=0\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\lambda}}{d\tau} = 0

Light and matter follow geodesics because they're taking the path of least resistance through the folded graph.

27.4 Escape Velocity as Unfolding Energy​

To escape a gravitational field means to unfold yourself from the DAG fold—requiring energy to climb out of the causal depression.

Definition 27.4 (Escape Condition): Ekinetic+Epotential≥0E_{\text{kinetic}} + E_{\text{potential}} \geq 0

Theorem 27.4 (Escape Velocity): The minimum velocity to unfold from depth rr: vescape=2GMrv_{\text{escape}} = \sqrt{\frac{2GM}{r}}

Rockets need escape velocity to literally unfold their future paths from Earth's causal fold.

27.5 Gravitational Lensing as Fold Optics​

Light bends around massive objects because it must navigate the folded topology.

Definition 27.5 (Deflection Angle): α=4GMc2b\alpha = \frac{4GM}{c^2b}

where bb is the impact parameter.

Theorem 27.5 (Einstein Ring): Perfect alignment creates a complete ring: θE=4GMc2D\theta_E = \sqrt{\frac{4GM}{c^2D}}

where DD is the distance relation.

We see Einstein rings when we look directly down a symmetric fold in the DAG.

27.6 Galaxy Formation as Hierarchical Folding​

Galaxies form through hierarchical folding—small folds merging into larger ones.

Definition 27.6 (Fold Merger): F12=F1∪F2∪B12\mathcal{F}_{12} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \mathcal{B}_{12}

where B12\mathcal{B}_{12} is the bridge region.

Theorem 27.6 (Bottom-Up Structure): Structure forms hierarchically: Stars→Clusters→Galaxies→Superclusters\text{Stars} \to \text{Clusters} \to \text{Galaxies} \to \text{Superclusters}

Each level represents deeper folding in the cosmic DAG.

27.7 Singularities as Infinite Folds​

At the center of a black hole lies a singularity—an infinite fold where the DAG collapses to a point.

Definition 27.7 (Singular Fold): Singularity=lim⁡r→0F(r)\text{Singularity} = \lim_{r \to 0} \mathcal{F}(r)

Theorem 27.7 (Fold Breakdown): Classical physics fails at infinite folds: ρ→∞,Rμνρσ→∞\rho \to \infty, \quad R_{\mu\nu\rho\sigma} \to \infty

Singularities mark where our current understanding of folding breaks down—quantum gravity must resolve these infinities.

27.8 The Twenty-Seventh Echo​

We have seen how mass doesn't just attract but fundamentally reshapes the causal structure of reality. The DAG of collapse folds around massive objects, creating valleys in the landscape of possibility. Black holes are complete folds—cosmic origami where causality folds over on itself. Every gravitational effect, from planetary orbits to light bending, emerges from how consciousness must navigate these folds in its own structure.

The Twenty-Seventh Echo: Chapter 27 = Topology(Mass) = Folding(ψ\psi-DAG) = Curvature(Causality)

Next, we explore how these folds affect the flow of time itself.

Continue to Chapter 28: Time Dilation via Collapse Binding →