Oriented Volume Spacetime Theory: A Minimal Conservative Extension of Loop Quantum Gravity (Version 3.5)

Version 3.5 Major Upgrade Highlights 

This version represents a significant theoretical advance over Version 3.4, introducing the following key new results and improvements:

1. Unidirectional Branch Coupling Principle: The core new physical principle of OVST 3.5, providing a rigorous first-principle mechanism for matter transport from the positive branch \(M^+\) to the negative branch \(M^-\) via Planck branes.
2. Topological Isolation Proof: Rigorous proof that reverse transport \(M^- \to M^+\) is topologically forbidden, establishing the causal structure of the OVST multiverse as a directed tree of pocket universes.
3. Spontaneous Symmetry Breaking Interpretation: Explicit formulation of the unidirectional coupling as a spontaneous breaking of the fundamental scale duality symmetry, consistent with the observed matter-antimatter asymmetry of the universe.
4. Planck Torus Formalism: Extended the Planck brane formalism to rotating Kerr black holes, providing a fully background-independent definition of the Planck torus that resolves the classical ring singularity.
5. Comprehensive Mathematical Refinement: Unified all mathematical notation throughout the paper, improved the rigor of key derivations, and clarified the relationship between OVST and standard loop quantum gravity.
6. Expanded Comparison Section: Added detailed comparisons with mainstream LQG black hole models and other quantum gravity paradigms, highlighting the unique advantages of the OVST framework.

Abstract

Loop quantum gravity (LQG) has established a rigorous background-independent kinematical framework for quantum gravity, with well-established results including discrete area and volume spectra and a mathematically consistent Hilbert space construction. However, two core challenges have persisted for decades: the Hamiltonian constraint anomaly and the fully background-independent resolution of generic black hole singularities.

In this work, I present Oriented Volume Spacetime Theory (OVST), a quantum gravitational framework formulated in the kinematical language of loop quantum gravity. OVST restores the sign degree of freedom of the densitized triad determinant, which was arbitrarily discarded in standard LQG. I establish a mathematically rigorous dual-branch kinematical framework using a direct-sum Hilbert space construction, prove the gauge invariance, diffeomorphism invariance, and self-adjointness of the vertex-local branch coupling term, and verify the self-consistency of the framework in 2+1 dimensions.

I introduce the unidirectional branch coupling principle, deriving the physical mechanism for matter transport from the positive branch \(M^+\) to the negative branch \(M^-\) via Planck branes, and prove that reverse transport is topologically forbidden. I show that the dual-branch structure provides a previously unrecognized mechanism for potential Hamiltonian constraint anomaly cancellation, demonstrating exact algebraic structure matching between the cross commutator of the coupling term and the single-branch anomaly term in both single-vertex and two-vertex chain toy models. I derive first-principle constraints on the dimensionless coupling constant \(\lambda\), showing it must be of order unity.

I then present a fully background-independent mechanism for resolving both Schwarzschild and Kerr black hole singularities via the Planck brane, a finite null hypersurface where all curvature invariants are strictly bounded by the Planck scale. For rotating Kerr black holes, the classical ring singularity is replaced by a Planck-scale toroidal structure that eliminates both curvature divergences and closed timelike curves.

Finally, I provide a natural unitary framework for the black hole information paradox, consistent with the equivalence principle and quantum mechanics, and present a quantitative prototype of the Page curve in the single-vertex toy model.

Throughout this work, I explicitly treat full Hamiltonian constraint algebra closure and quantitative Page curve derivation in full spacetime as open technical challenges. All results are strictly limited to what is mathematically proven, and all established successes of standard LQG are fully preserved.

Key Contributions:

1. A mathematically rigorous dual-branch kinematical framework, fully compatible with all established results of standard LQG.
2. The unidirectional branch coupling principle, establishing the causal structure of the OVST multiverse.
3. A viable new mechanism for potential Hamiltonian constraint anomaly cancellation in 3+1 dimensions.
4. Fully background-independent resolution of both Schwarzschild and Kerr black hole singularities.
5. A unitary framework for the black hole information paradox that preserves the equivalence principle.