The Three Premises

Three premises. Each derived from the axioms. Together they give you the floor.

2.1 — Premise 1: Closure (Axiom S)

The two sectors ℒ and 𝒫 are connected by the involution σ (Axiom S). The tension field of ε between 0 and 1 has field lines that leave 1 (propagation, matter, the visible) and return to 0 (fold, collapse, the dark). Every field line must close.

This is not a postulate. It follows from Axiom S (the sectors are connected) and AP06 Theorem 3.1 (leakage is nonzero for finite c and finite coupling). A field line is the manifold expression of the σ-correspondence: for each element in ℒ, the involution requires a corresponding structure in 𝒫. If a field line did not close, it would represent a disconnection between the sectors. A disconnection violates σ. Therefore every field line closes. You cannot break the connection.

Closure is topological. It does not depend on the strength of the field. You cannot weaken it away. A weak field line — far from the source, gently curving — still closes. It must. The topology requires it.

2.2 — Premise 2: Finite Extent (Axioms R + C, via EH)

The monoid (ℳ, ·) accumulates records (Axiom R). Records are irreversible; the monoid grows monotonically. Under EH (proved in AP20: AS = manifold identity), the monoid admits embedding into a smooth manifold M. The manifold is the accumulated record. Cosmologically, the manifold is expanding; the expansion rate is the Hubble parameter H₀.

Propagation is bounded by c (Axiom C). The maximum distance a signal can traverse in one Hubble time 1/H₀ is Rₕ = c/H₀. This is the Hubble radius — the boundary of causal coherence. Rₕ here is the recession-speed boundary the closure argument needs, not the cosmological event horizon; nothing downstream depends on the distinction. Beyond it, you lose the ability to close the loop: the manifold is expanding faster than c, the records being written faster than propagation can maintain coherence.

A field line cannot close beyond Rₕ. A field line that extends past Rₕ enters a region where the expansion velocity v_exp = H₀r exceeds c. There the field line is stretched faster than c can communicate the curvature needed to bring it back. The line cannot curve. It cannot close. But closure is required (Premise 1). Therefore no field line extends beyond Rₕ. The maximum extent of any closing field line is Rₕ = c/H₀. And that boundary gives you the scale.

2.3 — Premise 3: Propagation at c (Axiom C)

The tension field propagates at c, the rate set by the constraint. This follows from AP15 (The Connection): the connection Aμ carries the causal structure (Axiom C). The tension field is the substrate’s coherence — the pre-state’s unity expressed on the manifold — and its propagation speed is the propagation limit of the manifold: c.