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 GN = Gsrc2/(4πφ6); the weak-field Schwarzschild and Kerr acoustic metrics with Gμν = 0 in the exterior verified without assumption; an induced Einstein–Hilbert term from one-loop fluctuations confirming a gravitational sector at the condensate scale, with the Planck mass fixed by the density-feedback flux and the ultraviolet cutoff ΛUV = MPl/φ6 ≈ 0.0557 MPl set three φ-tower steps below it;the cosmological constant Λobs = Λ0 exp(−φ7c/β*) ≈ 10−122MPl4 with normalisation c = φ3 + p0QH — a canonical leading term plus one Chern–Simons monodromy insertion (p0 = 1/V*2 = 1/φ2, the same S-matrix scalar carrying α−1) per unit of Hopf charge QH = 2, matching the observed value; CMB inflationary predictions ns = 0.9654 (within 0.12σ of Planck 2018) and r = 0.0036, with exact normalisation Ps = 2.10×10−9 from gravitational reheating at Ne = 57.73 e-folds; and the exact Hawking temperature with a one-loop correction δTH/TH ≈ +4.2×10−5. The full dynamical Einstein equations are derived from the condensate effective action, with the frozen-attractor limit yielding Gμν + Λphysgμν = 0 exactly, where F(ρ∞) = MPl2(39φ+25)/2 is an exact golden-ratio consequence; energy-momentum conservation ∇μTμν = 0 follows from scalar field equation consistency (Bianchi identity). The exact rotating condensate background is given by the Kerr mass multipole series ΦKerr = M/r − Ma2P2/(2r3) + …, satisfying ∇2ΦKerr = 0 exactly term by term. The Cassini PPN constraint is proved resolved by Vainshtein screening, with rV ≈ 9.3×106 AU derived without free parameters, and independently by chameleon screening at the framework’s own density-feedback coupling. The Planck mass is fixed by the density-feedback flux, with the condensate ultraviolet cutoff ΛUV = MPl/φ6 ≈ 0.0557 MPl three φ-tower steps below it, at the non-integer tower level nUV ≈ 73.6 in the φ-tower of Paper VI, confirming gravitational-scalar compatibility. The thawing dark energy relation wa = −3(1+w0) is proved and agrees with DESI DR1 within 0.80σ. Dark matter is identified as the invisible Frank elastic strain of the condensate director — a cold, pressureless, splay-unstable orientation mode that self-organises into a disclination, giving flat rotation curves with no fitted halo while tracking the baryons and the radial-acceleration relation, with a MOND-scale acceleration a0 ≈ cH0/(2π) set by the same condensate. 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 TCMB = 2.7255 K: the electron mass me = Λcond 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.