Claim Summary · Conditionality Footer

Claim Summary

DERIVED. The floor a₀ = αcH₀/(2π) (Theorem, section 4). The functional form a₀ ∝ cH₀ and the exact coefficient α/(2π) ≈ 0.1662. The dipolar topology from {S, B, C} via Poincaré–Hopf (section 3.3). The coherent fraction 1/(2π) from the loop geometry (3.4). The linear scaling from Lemma 1 (3.5). The correction factor α = 2 ln(sec(½) + tan(½)) ≈ 1.0445 from the S² dipole profile (Propositions 2–3, 3.6). Parameter-free.

PREDICTION. H₀ = 2πa₀/(αc) = 74.3 ± 1.2 km/s/Mpc — the Hubble constant from galactic rotation curves (section 7).

EMPIRICAL. The numerical match at H₀ = 74 (within measurement uncertainty of the observed a₀); the present a₀/(cH₀) ≈ 0.176 (at H₀ = 70) pending H₀ convergence (section 5).

CLOSED. Local debts D1 (the 2π coefficient) and D1′ (the correction factor α). KS-45.2 (the dipole topology).

FALSIFICATION REGISTER. KS-45 (2π geometric factor, EMPIRICAL), KS-45.1 (Hubble prediction, EMPIRICAL), KS-45.2 (dipole topology — CLOSED). Updated: KS-39 (a₀ value, EMPIRICAL, AP17). KS-42 not closed by AP18 (closed in Rosin V7); KS-43 not addressed by AP18.

Conditionality Footer

Dependencies

AP06 (the Leakage Theorem), AP15 (the connection carries the causal structure), AP08/AP10 (the manifold and its S² link), AP20 (the Embedding Hypothesis proved; AS = manifold). Refines AP17 (The Room).

Dependents

AP21 (uses Lemma 1 and inherits a₀) and the broader Notebook VI cosmology chapters. Rosin V7 grounds the tension field AP18 uses by deriving its field equation.

Kill switches closed by this paper

KS-45.2 (the dipole topology, via Poincaré–Hopf). Local debts D1 and D1′ closed. KS-45 upgraded from LIVE — HARD to LIVE — EMPIRICAL.

Kill switches that remain live

KS-45 (2π geometric factor) and KS-45.1 (Hubble prediction), both LIVE — EMPIRICAL pending H₀ convergence. Inherited and unchanged: KS-39 (sharpened), KS-43 (not addressed).

Structural debts owed

None. All AP18 local debts (D1, D1′) are closed; the result is parameter-free.

Items held open without claiming debt status

The empirical confrontation on a₀/(cH₀) and H₀, pending the resolution of the Hubble tension (KS-45, KS-45.1).