The Derivation Chain

AP06 Theorem 3.1 → field lines must close (leakage nonzero).

AP17 → tension field between 0 and 1.

AP18 → acceleration floor a₀ = αcH₀/(2π), with Lemma 1 (monoid additivity).

AP08 → substrate homogeneity.

Lemma (Energy–Measure Bridge) (Axioms B + R, AP18 Lemma 1, AP05/AP08 Noether) → E = kμ → energy proportional to measure.

Proposition 1 (Lemma + AP08 + AP15 + AP20) → E = Tℓ, T = λ → stretching energy proportional to length.

Axiom S + expansion (R) → field lines stretch but cannot break → vacuum under tension.

Proposition 2 (Steiner tree on a metric manifold) → tension lines bundle into shared corridors → filaments.

Filament intersections → density spike → fold (Axiom C) → direct vacuum (primordial supermassive black hole).

Direct vacuum → deep anchor → gas capture → galaxy.