Claim Summary · Conditionality Footer

Claim Summary

DERIVED (conditional on the tension-field topology). Flat rotation curves from the tension field. The transition scale a₀ = cH₀/(2π), within ≈8% of the empirical value (section 4.3). Tully-Fisher v⁴ ∝ M, an exact fourth-power law (section 4.4). The 2π factor from the closure geometry (section 4.3, Step 2).

STRUCTURAL. Tension-field topology: field lines close (Axiom S, AP06 Theorem 3.1). The central black hole as topological anchor (section 6). The galaxy as the eye — the 0–1 topology (section 3). The dense→sparse regime transition (section 4).

CORRECTION ON THE RECORD. The first-draft Λ-sign error: v²(r) = GM(r)/r − Λc²r²/3 (Λ is repulsive; it is not the flattening mechanism). The tension field of ε replaces it (section 8).

CONJECTURAL / UNTESTED. Tension-field closure at galactic scales as an empirical fact; the ≈8% residual in a₀ (KS-39); the interpolation function (KS-43); the central-anchor prediction (KS-44); cluster dynamics (KS-40); structure formation (KS-41). The Big Bang extrapolation (5.3) and the M•–σ derivation.

FALSIFICATION REGISTER. KS-3 (isotropic acceleration, EMPIRICAL), KS-39 (a₀ value, EMPIRICAL), KS-40 (clusters, EMPIRICAL), KS-41 (structure formation, EMPIRICAL), KS-42 (tension-field equation — CLOSED, Rosin V7), KS-43 (interpolation function, EMPIRICAL), KS-44 (galaxies without a central anchor, EMPIRICAL).

Conditionality Footer

Dependencies

AP06 (the Leakage Theorem; the condensate as condition), AP08 (Einstein’s field equations; the FRW metric), AP10 (N = 3, D = 4), AP20 (AS = manifold; EH and QRA proved), Edition 04 (The Lock — the substrate stiffness scale).

Dependents

AP18 (refines a₀ exactly), AP21 (structure formation), and the broader Notebook VI cosmology chapters. Rosin V7 grounds AP17’s tension field by deriving its field equation.

Kill switches closed by this paper

None closed by AP17 itself. KS-3 is upgraded from LIVE — HARD to LIVE — EMPIRICAL. KS-42 (the tension-field equation), which AP17 names as its principal theoretical gap, is subsequently CLOSED in Rosin V7.

Kill switches that remain live

KS-3, KS-39, KS-40, KS-41, KS-43, KS-44 — all LIVE — EMPIRICAL.

Corrections

Corrects the first draft of AP17 (the Λ-sign error in the rotation-velocity equation). Λ is repulsive and steepens rotation curves; it is not the flattening mechanism. The tension field of ε replaces it.

Items held open without claiming debt status

The Big Bang extrapolation (5.3); the M•–σ quantitative derivation; the a₀ ≈8% residual (a precision target, KS-39).