Summary of Derivation

Gravity = condition of 0. The fold.

c = condition of 1. The propagation.

ε = the tension between them. The break. The leakage. The wave function collapsing from possibility to actuality. Now.

Tension field is topologically closed. Field lines leave 1, curve back to 0. Never break. Never disconnect. (AP06 Theorem 3.1.)

Dense regime (inner galaxy): records thick. Newton dominates. v² = GM/r.

Sparse regime (outer galaxy): records thin. Tension field dominates. In the deep sparse regime, a(r) ≈ √(a_N × a₀) = √(GM × a₀)/r, giving flat rotation curves.

Transition scale: a₀ = cH₀/(2π) ≈ 1.1 × 10⁻¹⁰ m/s². Empirical a₀ ≈ 1.2 × 10⁻¹⁰ m/s². Within ≈8% (comparable to the Hubble tension). The 2π is the geometry of one complete field-line closure on the manifold.

Tully-Fisher: v⁴ = GM × a₀ ∝ M. Exact fourth-power law.

Black hole = topological anchor. The 0 of the eye. The pole the field lines return to.

No dark matter. The room holds the stars. The lines always close.