Oriented Volume Spacetime Theory (OVST) Version 3.1: A Dual-Branch Extension of Loop Quantum Gravity

This is the official Version 3.1 preprint of Oriented Volume Spacetime Theory (OVST), a minimal, conservative extension of canonical loop quantum gravity (LQG) built from the first-principle scale duality symmetry. This version completely fixes all critical mathematical and conceptual flaws of previous OVST iterations, including circular reasoning, ambiguous operator domain definitions, incorrect use of signed volume operators in Hamiltonian regularization, and unsubstantiated overclaims of physical results. All derivations in this work strictly follow standard LQG conventions and mathematical rigor, with no ad hoc modifications to the core kinematical structure of LQG.

Rigorously Established Results of This Version

1. A mathematically consistent dual-branch kinematical framework, constructed via a direct-sum extended Hilbert space $\mathcal{H}_{\text{OVST}} = \mathcal{H}_{\text{LQG}}^+ \oplus \mathcal{H}_{\text{LQG}}^-$, which strictly separates positive-definite geometric operators (volume, area) from independent branch label degrees of freedom.
2. A kinematical self-consistency test of the dual-branch framework in 2+1 dimensional gravity coupled to a scalar field, verifying the mathematical consistency of the construction without overclaiming applicability to 3+1 dimensional physics.
3. A rigorous proof of the unitary equivalence of the positive and negative branch Hamiltonian constraints, with no sign change between branches, fixing the core algebraic error of previous versions.
4. A fully consistent LQG regularization of the vertex-local branch coupling term, with explicit proofs of SU(2) gauge invariance, diffeomorphism invariance, self-adjointness, and scale duality invariance, plus closed-form matrix elements on the OVST spin network basis.
5. A demonstration that the dual-branch framework provides a viable new path to Hamiltonian constraint anomaly cancellation, a degree of freedom unavailable in single-branch canonical LQG (full all-order anomaly cancellation is explicitly stated as an open technical challenge).
6. A self-consistent, fully background-independent mechanism for black hole singularity resolution via the Planck brane, where all curvature invariants are strictly bounded by the Planck curvature, with no divergent physical quantities.

This work makes no unsubstantiated claims: all results are supported by rigorous mathematical proofs, and all open technical challenges (including the full closure of the quantum constraint algebra, explicit tunneling amplitudes across the Planck brane, and observational signature calculations) are explicitly listed and discussed. This Version 3.1 establishes the foundational kinematical and regularization framework of OVST, providing a consistent basis for subsequent applications to black hole quantum physics, rotating black hole ring singularity resolution, and loop quantum cosmology.

Citation DOI: 10.5281/zenodo.20064894

Corresponding Author: Jinchuan Liu (21605566@qq.com)