Hypercomplex General Relativity (HGR): A symplectic and geometric reformulation of Weyl, hypercomplex numbers with late cosmological applications. Appendix: HGR: Beginning of quantization. Appendix: Dark matter galaxies or halos and rotation speed, dark matter-dominated galaxies, and hypercomplex halos in HGR. Weakless interaction between dark matter , H0 equation.

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.

Objective

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.

The 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.

1. Canonical Structure of the Internal Sector

The hypercomplex connection is modeled as a non-abelian gauge field. A full Hamiltonian analysis is performed, including:

• 3+1 decomposition,

• identification of canonical variables,

• Gauss constraint implementation,

• well-defined phase space.

The non-commutative structure emerges from the symplectic form and canonical commutation relations, rather than being assumed.

This establishes that the internal HGR sector possesses a legitimate gauge-theoretic foundation.

2. Construction of a Gauge-Invariant Internal Observable

A scalar observable is constructed from quadratic electric-like and magnetic-like internal invariants.

This observable:

• is locally gauge invariant,

• is positive definite,

• can be promoted to an operator,

• controls the effective gravitational correction threshold.

The threshold becomes state-dependent via expectation values, introducing a statistical interpretation of rebound domains.

3. Semi-Classical Regime

In a semi-classical approximation, fluctuations of the internal observable modify the effective threshold.

This implies that:

• internal quantum variance may broaden the population of rebound-triggering domains,

• the mechanism admits a probabilistic interpretation at high curvature.

However, the core rebound mechanism does not rely strictly on quantization.

4. Internal Saturation Mechanism

A critical issue arises: without regulation, the internal scalar could grow unbounded at high curvature.

To ensure stability, a saturation function is introduced:

• the internal response becomes bounded,

• corrections remain finite,

• ultraviolet divergence is avoided,

• local activation remains possible.

This mechanism stabilizes the theory without requiring full quantum gravity.

5. Conceptual Status at the End of This Stage

Established:

• Canonical internal gauge structure.

• Well-defined observable controlling corrections.

• Dynamically bounded internal response.

• Stabilized domain-triggered rebound mechanism.

Not yet addressed:

• Full quantization of gravity.

• Complete cosmological perturbation analysis.

• Weak-field post-Newtonian constraints.

Strategic Insight

Quantization of the internal sector provides micro-structural grounding and theoretical elegance.

However, the macroscopic consistency of HGR does not depend on full gravitational quantization. The saturation mechanism alone is sufficient to ensure stability and physical viability.

This stage transforms HGR from an intuitive hypercomplex extension into a symplectically structured, dynamically regulated gravitational framework.