Hypercomplex General Relativity (HGR): From Parallel Transport to Curvature Fields

Hypercomplex General Relativity (HGR): From Parallel Transport to Curvature Fields Overview Hypercomplex General Relativity (HGR), known in French as Relativité générale hypercomplexe (RGH), is a theoretical research programme developed by Laurent Besson. It investigates the gravitational role of hypercomplex internal geometry and its relationships with symplectic structures, Weyl transport and observable spacetime geometry. This edition presents a revised main manuscript, explicit effective models and reproducible calculations. It distinguishes geometric assumptions, formal results, phenomenological constructions and questions requiring further investigation. Historical development The programme comprises two historical stages: Analytic formulation: the 1990s–2025. According to the author’s chronology, this stage includes the early development of the programme, a manuscript typed in 1998, a revision undertaken in 2015 and subsequent analytic developments through 2025. Symplectic formulation: 2025–2026. The initial hypercomplex approach is reformulated using a fibred geometric structure, internal connections and symplectic data, with the construction of observable spacetime geometry as a central question. These periods describe the development of the work and are distinct from the publication dates of individual archived versions. The revised manuscript documents this history and identifies subsequent mathematical corrections without attributing them retrospectively to the earlier texts. Geometric starting point: two orders of parallel transport In General Relativity, comparing parallel transport in two different orders reveals the Riemann curvature tensor. HGR extends this operation to an object carrying a tangent-space direction, an internal quaternionic orientation and a Weyl scaling weight. For the specified product connection, the same commutator yields spacetime, internal and scale curvatures. Their appearance follows from the chosen transport structure. The selection of that structure, possible mixed components and its physical dynamics require additional geometric and variational input. Main contents and results The manuscript develops: A covariant effective reference action, metric and field equations, conservation identities, and a comparison of its source tensors with those of fields coupled to General Relativity. An FLRW reduction with the lapse constraint retained. The minimal isotropic Yang–Mills sector has positive radiation-like energy. In the minimal Einstein limit, the null energy condition excludes a spatially flat cosmological bounce. A canonical description of the internal gauge sector, including conjugate variables and the Gauss constraint, together with an observer-dependent, gauge-invariant classical field intensity. Quaternionic and Lie-algebra non-commutativity are distinguished from quantum operator non-commutativity. Separate effective or phenomenological modules for cosmological bounces, Higgs–Weyl couplings, octonionic extensions, and a parameterized closure relation for the present Hubble rate H0. A reduced Wheeler–DeWitt model, adiabatic corrections and numerical calculations for a quartic internal Hamiltonian. Applications to hypercomplex halo profiles, dark-matter-dominated galaxies, rotation curves, gravitational lensing, baryonic feedback, and weak or vanishing couplings between specified internal sectors. Detailed variations, dimensional corrections and selected Python calculations support independent examination and reproduction of the stated results. Scientific scope and open questions The effective action provides a definite calculational framework; its unique derivation from the proposed internal geometry remains open. Quantum composite observables require a specified regularization, renormalization prescription and state. Bounded response functions alone do not establish dynamical stability or ultraviolet completeness. A fundamental construction of the observable Lorentzian metric, healthy dynamical realizations of bounce scenarios, stable halo solutions and a complete perturbation analysis remain to be established. The H0 relation becomes predictive only when its inputs are fixed independently. Joint validation against cosmological observations and local gravitational tests is an outstanding task. This work is presented as a theoretical preprint for scientific examination and further development. Its results have the scope of the assumptions and models explicitly defined in the manuscript. Keywords: hypercomplex general relativity; parallel transport; curvature; quaternionic geometry; symplectic geometry; Weyl connection; effective gravity; cosmology; Wheeler–DeWitt equation; galactic halos.