Chapter 14: Collapse Chains as Spatial Beams

The Linear Architecture of Collapse

Classical cosmology observes filamentary structures threading through the cosmic web—vast bridges of matter connecting galaxy clusters across millions of light-years. But what creates these remarkably straight, persistent beams? Ψhē Cosmology reveals them as collapse chains: sequential propagations of collapse events that generate effectively one-dimensional structures in three-dimensional space.

14.1 Chain Formation Mechanics

Definition 14.1 (Collapse Chain): A collapse chain C is a sequence of causally connected collapse events: $C = \{\psi_1 \to \psi_2 \to ... \to \psi_n\}$ where → denotes collapse-triggered propagation.

Each collapse event triggers the next, creating a self-extending structure that threads through space like lightning seeking ground.

14.2 Beam Cross-Section Profile

Definition 14.2 (Transverse Density): The density profile perpendicular to a collapse chain follows: $\rho(r) = \rho_0 \text{sech}^2(r/r_0)$

where:

• r = radial distance from beam axis
• r₀ = characteristic beam width
• ρ₀ = axial density

This hyperbolic profile emerges naturally from the balance between collapse concentration and transverse spreading.

14.3 Chain Propagation Velocity

How fast do collapse chains extend through space?

Theorem 14.1 (Propagation Speed): A collapse chain extends at velocity: $v = c\sqrt{\rho/\rho_{crit}}$

Proof: The propagation speed emerges from the balance between collapse pressure gradient and spatial resistance. When local density ρ reaches critical threshold ρ_crit, the chain extends at light speed. Below threshold, propagation slows by the square root of the density ratio. ∎

14.4 Beam Intersection Nodes

Where collapse chains meet, they create special structures:

Definition 14.3 (Junction Density): When two chains intersect at angle θ: $\rho_{junction} = \rho_1 + \rho_2 + 2\sqrt{\rho_1\rho_2}\cos(\theta)$

These junctions become the nodes of the cosmic web—sites of enhanced collapse where galaxy clusters preferentially form.

14.5 Longitudinal Wave Modes

Collapse chains support wave propagation along their length:

Theorem 14.2 (Solitonic Waves): Perturbations along chains propagate as solitons: $\psi(s,t) = A \cdot \text{sech}[(s - vt)/\lambda]$

where s measures distance along the chain. These waves maintain their shape while traveling, carrying information and energy without dissipation.

14.6 Beam Stability and Rigidity

Why do cosmic filaments remain straight across vast distances?

Definition 14.4 (Collapse Rigidity): The bending resistance of a collapse chain: $E = \int \kappa^2 \rho \, ds$

where κ is local curvature. This rigidity emerges from the chain's internal collapse dynamics, resisting deformation.

14.7 Branching Phenomena

Under specific conditions, collapse chains bifurcate:

Theorem 14.3 (Branch Condition): A chain branches when: $\rho > \rho_{crit}(1 + \lambda^2\nabla^2\rho/\rho)$

This creates tree-like structures in the cosmic web, with primary trunks spawning secondary branches.

14.8 Transverse Oscillations

Collapse chains can vibrate perpendicular to their length:

Definition 14.5 (Vibrational Modes): Transverse oscillations follow: $\omega^2 = (ck)^2[1 + (kr_0)^2]$

These vibrations create periodic density enhancements along filaments—explaining the regular spacing of galaxies along cosmic filaments.

14.9 Chain Termination

Every collapse chain must end somewhere:

Theorem 14.4 (Terminal Condition): A chain terminates where: $\nabla \cdot (\rho\hat{v}) = -4\pi G\rho_{term}$

Creating a "drain" that absorbs the chain's collapse flow. These terminals typically coincide with massive galaxy clusters.

14.10 Parallel Chain Interaction

When chains run parallel, they interact:

Definition 14.6 (Inter-Chain Force): Parallel chains separated by distance d experience: $F/L = -2\pi\rho_1\rho_2 K_0(d/r_0)$

where K₀ is the modified Bessel function. This creates tendencies for chains to bundle—explaining why cosmic filaments often appear as thick ropes of multiple strands.

14.11 Observable Predictions

The collapse chain model makes specific predictions:

1. Regular Galaxy Spacing: ~50 Mpc separation along filaments
2. Straight Trajectories: Deviations < 1° per 100 Mpc
3. Hyperbolic Density Profiles: Measurable in filament cross-sections
4. Solitonic Density Waves: Propagating at 0.1-0.3c
5. Enhanced Junction Density: 10-100× at filament intersections

Each prediction links observable structure to underlying collapse dynamics.

14.12 The Cosmic Skeleton

Collapse chains reveal the universe's skeletal structure—not random accumulations but an organized framework of beams and struts. The cosmos builds its own architecture through self-propagating collapse, creating highways along which matter flows and galaxies form.

This is structural engineering on the grandest scale.

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Next: Chapter 15: ψ-Folds as Curved Collapse Volumes