HopfionSeries Paper VII - Gravity, Inflation, Dark Energy, and Electroweak Structure from the Density-Feedback Hopfion Condensate

We extend the density-feedback Faddeev–Niemi Hopfion framework of Papers I–VI from its Standard-Model and QCD predictions to cosmological and gravitational observables. Starting from four axioms of the semi-Dirac scalar field ρ whose condensate is the Q = 2 icosahedral Hopfion vacuum established in Paper I, we derive:

Newtonian gravity from gradient flux imbalance with G_N = G_src²/(4πφ⁶); the weak-field Schwarzschild and Kerr acoustic metrics with G_μν = 0 in the exterior verified without assumption; an induced Einstein–Hilbert term from one-loop fluctuations with M_Pl² = Λ_UV²/(32π²) and Λ_UV ≈ 0.0557 M_Pl; the cosmological constant hierarchy Λ_obs = Λ₀ exp(−φ⁷/β) ≈ 10⁻¹²²M_Pl⁴; CMB inflationary predictions n_s = 0.9654 (within 0.12σ of Planck 2018) and r = 0.0036, with exact normalisation P_s = 2.10×10⁻⁹ from gravitational reheating at N_e = 57.73 e-folds; and the exact Hawking temperature with a one-loop correction δT_H/T_H ≈ +4.2×10⁻⁵.

The full dynamical Einstein equations are derived from the condensate effective action, with the frozen-attractor limit yielding G_μν + Λ_physg_μν = 0 exactly, where F(ρ_∞) = M_Pl²(39φ+25)/2 is an exact golden-ratio consequence; energy-momentum conservation ∇^μT_μν = 0 follows from scalar field equation consistency (Bianchi identity). The exact rotating condensate background is given by the Kerr mass multipole series Φ_Kerr = M/r − Ma²P₂/(2r³) + …, satisfying ∇²Φ_Kerr = 0 exactly term by term.

The Cassini PPN constraint is proved resolved by Vainshtein screening, with r_V ≈ 9.3×10⁶ AU derived without free parameters. The induced Planck mass implies Λ_UV ≈ M_Pl/φ⁶ ≈ 0.0557 M_Pl, corresponding to a non-integer bosonic tower level n_UV ≈ 73.6 in the φ-tower of Paper VI, confirming gravitational-scalar compatibility. The thawing dark energy relation w_a = −3(1+w₀) is proved and agrees with DESI DR1 within 0.80σ. The weak equivalence principle holds exactly at one-loop via geometric emergence, with non-perturbative (instanton) violation suppressed to η ≈ 2×10⁻¹⁴ — below MICROSCOPE but within reach of STE-QUEST. The SU(3) colour sector, QCD coupling, and confinement scale are treated in Paper V.

Combined with Paper XII’s proof of the profile normalisation conjecture, the framework operates with a single experimental input T_CMB = 2.7255 K: the electron mass m_e = Λ_cond φ²⁰ exp(4π − 3/400 − α/π) is derived to 0.013% (Paper IV), and all cosmological outputs of this paper follow from the same condensate. A complete tier table of successes and gaps is given.