Hypercomplex General Relativity (HGR), also referred to as Relativité Générale Hypercomplexe (HGR) in French, is a unified theoretical framework...
This deposit gathers and documents the complete body of work related to Hypercomplex General Relativity (HGR), an extension of General Relativity developed by Laurent Besson.
HGR introduces an internal hypercomplex structure within the spacetime metric, allowing a reformulation of gravity that preserves local consistency and the standard relativistic limit while avoiding Ostrogradsky-type instabilities.
The present record synthesizes successive versions of the model, including the initial formulation (2015), a comprehensive theoretical update (2025), and recent additions addressing critical discussions.
In a cosmological context, the model leads to a holographic-like behavior characterized by a late-time geometric component compatible with current observational constraints from the Cosmic Microwave Background and large-scale structure data.
This deposit includes the Lagrangian formulation, the field equations, and numerical implementations based on a modified version of standard cosmology codes, and is released within an open science framework to ensure long-term accessibility, reproducibility, and citation.
Keywords: general relativity, modified gravity, hypercomplex geometry, cosmology, theoretical physics.
The goal of this development is to refine the internal structure of Hypercomplex General Relativity and determine whether quantization of the internal sector is physically necessary or conceptually supportive.
\nThe previous classical formulation introduced a curvature-dependent threshold controlling local gravitational corrections. The present work reinterprets this threshold in terms of a canonical internal gauge structure.
\nThe hypercomplex connection is modeled as a non-abelian gauge field. A full Hamiltonian analysis is performed, including:
\n3+1 decomposition,
\nidentification of canonical variables,
\nGauss constraint implementation,
\nwell-defined phase space.
\nThe non-commutative structure emerges from the symplectic form and canonical commutation relations, rather than being assumed.
\nThis establishes that the internal HGR sector possesses a legitimate gauge-theoretic foundation.
\nA scalar observable is constructed from quadratic electric-like and magnetic-like internal invariants.
\nThis observable:
\nis locally gauge invariant,
\nis positive definite,
\ncan be promoted to an operator,
\ncontrols the effective gravitational correction threshold.
\nThe threshold becomes state-dependent via expectation values, introducing a statistical interpretation of rebound domains.
\nIn a semi-classical approximation, fluctuations of the internal observable modify the effective threshold.
\nThis implies that:
\ninternal quantum variance may broaden the population of rebound-triggering domains,
\nthe mechanism admits a probabilistic interpretation at high curvature.
\nHowever, the core rebound mechanism does not rely strictly on quantization.
\nA critical issue arises: without regulation, the internal scalar could grow unbounded at high curvature.
\nTo ensure stability, a saturation function is introduced:
\nthe internal response becomes bounded,
\ncorrections remain finite,
\nultraviolet divergence is avoided,
\nlocal activation remains possible.
\nThis mechanism stabilizes the theory without requiring full quantum gravity.
\nEstablished:
\nCanonical internal gauge structure.
\nWell-defined observable controlling corrections.
\nDynamically bounded internal response.
\nStabilized domain-triggered rebound mechanism.
\nNot yet addressed:
\nFull quantization of gravity.
\nComplete cosmological perturbation analysis.
\nWeak-field post-Newtonian constraints.
\nQuantization of the internal sector provides micro-structural grounding and theoretical elegance.
\nHowever, the macroscopic consistency of HGR does not depend on full gravitational quantization. The saturation mechanism alone is sufficient to ensure stability and physical viability.
\nThis stage transforms HGR from an intuitive hypercomplex extension into a symplectically structured, dynamically regulated gravitational framework.
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