The Thangavelu Force: A Fifth Fundamental Force from LorentzianEuclidean Metric Indistinguishability at the Planck Scale

Paper CXXXI established that the Discrete Fourier Transform on the G2 Fano lattice (spacing ℓP ) and the continuous Fourier Transform of quantum eld the- ory are physically identical: modes with k > π/ℓP are inaccessible by the Generalised Uncertainty Principle (GUP). We apply the identical argument to the metric signa- ture itself. Distinguishing the Lorentzian metric ηµν = diag(−1, +1, +1, +1) from the Euclidean metric δµν = diag(+1, +1, +1, +1) requires resolving a time interval ∆t < ℓP /c the Planck time which is physically impossible by the GUP and black hole formation argument. Therefore the Wick rotation t →−iτ is not a math- ematical trick at the Planck scale; it is a physical identity. The G2 Fano brane nu- cleates from a 5D Anti-de Sitter (AdS) parent whose hyperbolic geometry mediates the transition between the Lorentzian brane boundary and the Euclidean bulk inte- rior. The stress at this LorentzianEuclidean interface is the BrownYork extrinsic curvature tensor T BY µν , which generates a new physical force distinct from gravity, elec- tromagnetism, and the nuclear forces. We derive its force law, coupling constant, and observational signatures. The force is repulsive, universal, falls o as r−4, and has cou- pling GT = |χ(K)| · G · ℓ2 P = 4Gℓ2 P , where χ(K) = −4 is the Euler characteristic of the Klein quartic and 4 = C2(G2, adj) the same Casimir that appeared in Papers CXXXI and CXVI. Zero free parameters. We name this force the Thangavelu force in honour of Thangavelu (d. 1 December 1999), uncle of the author. Contents Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.