Exploring PT-Symmetric Quaternionic Spacetime as a Geometric Avenue Towards Connecting General Relativity and Quantum Mechanics

We present a four-dimensional, PT-symmetric quaternionic extension of General Relativity whose metric splits as Gμν = gμν(R) + Gμν(1) with a purely imaginary flux component Gμν(1) ≃ (φ/Mpl) Hμν. A single real scalar field φ controls three scale-dependent flux amplitudes ε(t), ε(r) and ε(ET) that account for (i) late-time cosmic acceleration, (ii) flat galaxy rotation curves, and (iii) the high-ET missed-energy tail seen at the LHC—without invoking extra dimensions or exotic matter.Cosmology. With  ε(t) = 2 tanh[γ ln(t/tpl)] a Pantheon–SN Ia fit yields  H0 = 73.6 +11.5/–12.0 km s–1 Mpc–1,  Ωm = 0.284 ± 0.014,  γ = 0.080 +0.045/–0.049, and χ2/dof = 0.996. Early-time values ε(t) ∼ 10–3 restore the Planck sound horizon and recast the “Hubble tension” as a geometric flux mismatch between epochs.Galaxies. Using  ε(r) = 2 tanh(r/rs) we confront all 175 SPARC rotation curves. An automated scan yields  ∑χ̃2Quat = 3.94 × 102 versus 6.81 × 102 (ΛCDM) and 2.98 × 103 (MOND); quaternionic dynamics provides the best AIC in 73 galaxies, ΛCDM in 92, MOND in 10. Frame-dragging around a 10^6 M⊙ Kerr black hole—evaluated with a Kretschmann-scaled, Planck-suppressed coupling—yields  ε(10 rg) ≃ 7 × 10–8 and a graviton-speed shift δcg/c ∼ 10–18, furnishing an internal strong-field consistency check below current detector sensitivity.High energy. The oscillatory law  ε(ET) = α (ET/E1) sin^2(ET/E1) fits the public CMS 13 TeV MET spectrum with  χ2Quat = 4.04 × 10^6 (48 dof), a ∼5× improvement over ΛCDM (1.91 × 10^7) and a ∼14× improvement over MOND (5.52 × 10^7). The fit selects  α = 6.39 ± 0.01,  E0 = 55.69 ± 0.01 GeV,  E1 = 250 GeV (fixed).Unified coupling. A curvature-driven coupling  λ ≃ κ R(g(R))/Mpl2 with κ ∼ 10–2 allows the same field equation  □φ = λ φ (φ2 – Mpl2) to generate the three flux profiles across thirty orders of magnitude in length and energy scale.Outlook. Forthcoming data from DESI and the HL-LHC will probe the cosmic and collider regimes, while next-generation gravitational-wave observatories will set deeper limits on the Kerr-sector prediction, providing decisive tests of this flux-based geometric bridge between spacetime curvature and quantum phenomena.