An ABC/WV Closure Theorem for the Quantum Traction Theory Vacuum Sector: One Clock for the Cosmological Constant, the Galaxy Acceleration Knee, and the Hubble Ladder

Quantum Traction Theory (QTT) forces three observationally distinct quantities — the cosmological constant ρ_Λ, the galactic acceleration knee a₀, and the entire laboratory Hubble ladder H₀^(P) — to be three projections of one Absolute-Background-Clock (ABC).

This paper establishes the closure theorem from which that single-clock structure follows, and exhibits all three of its observational faces.

QTT supplies two upstream descriptions of the homogeneous vacuum branch from the same axiomatic ledger. The ABC route states that a homogeneous branch coasts, H_Λ^ABC T_Λ^ABC = 1. The White-Void (WV) route states that the same branch is a counted creation source in the endurance-current divergence, 3 (H_Λ^ABC)² = C_WV,Λ/̅t. Demanding that both descriptions name a single homogeneous branch yields the closure theorem C_WV,Λ = 3 ̅t (H_Λ^ABC)² = 3 ̅t / (T_Λ^ABC)². The closure depends only on QTT axioms A1, A3, A6, A7 and eliminates one independent vacuum amplitude.

Vacuum face. The closure-consistent FLRW shadow returns u_Λ^obs/u_P = 3 / [8π Π_Λ² (N_Λ^ABC)²] with Π_Λ = cos(7π/48). With Planck 2018 inputs, the closure delivers u_Λ^obs = 5.251 × 10^-10 J/m³ and u_Λ^obs/u_P = 1.133 × 10^-123, matching the observed dark-energy density to within input precision.

Galaxy face. The same ABC clock that sets H_Λ^ABC sets the MOND acceleration knee through a_0,T = c H_T / (2π) and a_0,P^QTT(z) = c H_P(z) / (2π). Three independent forward paths return a₀^SPARC ≈ 1.04 – 1.10 × 10^-10 m/s², all inside the McGaugh 2016 RAR-knee band. The MOND scale is not free; it is the same clock as ρ_Λ.

Hubble ladder face (new in v2.5). The structural identity δ_cre⁽⁰⁾ = θ₀ · τ_drift · Ω_b^abs = (π/8) · 3 · (1/18) = π/48, fusing the fixed time tilt, the three spatial creation channels of A3, and the cosmic baryon invariant 1/18, locks the Time-Drift angle without any fit. This forces a parameter-free four-rung Hubble ladder: H_T0 = 63.49 (ABC base); H₀^early = H_T0/cos(π/8) = 68.72 (tilt-only / CMB-like); H₀^hom = H_T0/cos(7π/48) = 70.79 (homogeneous late / TRGB-like); H₀^sat = H_T0/cos²(π/8) = 74.39 (saturation ceiling) — all in km/s/Mpc. Planck 67.36 ± 0.54 sits near rung 2; DESI BAO+SH-light 68.4 at rung 2; CCHP TRGB 70.39 ± 1.94 at rung 3 (0.3σ); SH₀ES Cepheid 73.04 ± 1.04 sits inside the saturation interval (70.79, 74.39]. The Hubble tension dissolves as a probe-projection effect of one absolute clock. The endpoint ratio H₀^sat/H₀^early = sec(π/8) = 1.0824 matches the SH₀ES/Planck observed ratio 73.04/67.36 = 1.0843 at 0.18% as a structural identity.

Sharp falsifiers. F1: closure failure if any substrate count of C_WV,Λ violates 3 ̅t (H_Λ^ABC)². F6: galaxy knee clock-link failure if β_QTT=1 is rejected at ≥5σ in a same-probe high-z RAR sample. F8 (new in v2.5): Hubble ladder collapse if TRGB and Cepheid converge to a common H₀ at <1% on a shared SN Ia sample, or if SH₀ES drifts outside (70.79, 74.39].

Renewal Dust hint (new in v2.5). Appendix E of this paper points forward to QTT's third dark-sector mechanism: Renewal Dust (RD), a purely gravitational, non-Standard-Model-coupled residual whose structural form is T^μν_RD = ρ_RD u^μ u^ν with L^int_RD = 0. The Einstein equation in QTT then reads G_μν + Λ g_μν = (8πG/c⁴)(T^SM_μν + T^RD_μν). RD gravitates but has no SM coupling, structurally consistent with WIMP/axion direct-detection nulls. Renewal Dust is NOT a claim of this paper. Its quantitative phenomenology — substrate-counting derivation of ρ_RD, cosmological scaling, structure formation, S₈ tension — is the subject of a forthcoming companion paper.

Corpus-level numerical lattice. The closure sits inside an already-locked QTT lattice that passes independent checks with no per-paper fitted constants: t₀ = T₀ cos(7π/48) = 13.81 Gyr versus Planck 13.797 ± 0.023 Gyr (0.65σ); Δm²₃₁/Δm²₂₁ = 4π² cos²(π/8) = 33.697 versus NuFIT 6.0 33.55 ± 0.89 (0.17σ); Ω_b^lab = 0.0498 versus Planck 0.0493 ± 0.0006 (0.9σ); CCHP TRGB at QTT rung 3 within 0.3σ.

Version 2.5 changes vs v2.2. Title now lists three corollaries instead of two. Abstract leads with the three-corollary thesis. New Section 11 (§10 in v2.2) titled \"The QTT Hubble ladder corollary\" derives the δ_cre⁽⁰⁾ = (π/8)·3·(1/18) = π/48 structural lock as Theorem 2 and the four-rung ladder as Theorem 3, with explicit observational anchors and the SH₀ES bracket. Master matrix Table A extended with four Hubble ladder rungs as theorem-tagged rows. Falsifiers extended with F8 (Hubble ladder collapse). Operational status table extended with two ladder rows. Conclusion rewritten with five claims and three corollaries. Relation to existing approaches expanded to address the Hubble tension. New Appendix E \"Renewal Dust\" added as a forward-looking structural pointer with explicit \"What this paper does not claim\" discipline. All v2.2 corrections preserved (closure-consistent u_Λ language, three SPARC paths, MUSE-DARK β uncertainty, status-tagged matrix, G row caveat). .