Oriented Volume Spacetime Theory: A Minimal Conservative Extension of Loop Quantum Gravity (Version 4.0, Stage I–II Consolidated with Emergent-Function Formalization)

Oriented Volume Spacetime Theory (OVST) Version 4.0 — Stage I–II Consolidated with Emergent-Function Formalization OVST is a minimal conservative extension of canonical Loop Quantum Gravity (LQG), restoring the oriented volume sign degree of freedom arbitrarily discarded in standard kinematical formulations. This Version 4.0 manuscript supersedes all prior v3.x releases and introduces two major structural advances. Core Stage I Innovations (Retained & Refined):- Dual-branch direct-sum Hilbert space $\mathcal{H}_{\mathrm{OVST}} = \mathcal{H}_{\mathrm{LQG}}^{+} \oplus \mathcal{H}_{\mathrm{LQG}}^{-}$ preserving all standard LQG kinematics.- Weighted crossing-smooth rigged Hilbert space $\mathcal{H}_{\mathrm{cross}}$ with proven single-vertex polar-decomposition unitarity and full multi-vertex commutativity on physical gauge-invariant states.- Discrete Planck-brane crossing paradigm: continuous time and effective Hamiltonian dynamics emerge only as statistical coarse-grained limits. Version 4.0 Conceptual Corrections:- The negative-orientation branch $\mathcal{M}^{-}$ is strictly redefined as a quantum-geometric construct independent of classical areal radius. The orientation variable $\epsilon = \mathrm{sgn}(\det \tilde{E}) = \pm 1$ does not correspond to any negative-radius region in Schwarzschild–Kruskal geometry. The areal radius $\rho$ in the semiclassical effective description of both branches satisfies $\rho \geq 0$.- The nilpotent branch transition operator $\hat{\mathcal{T}}_{+\to-}$ (satisfying $\hat{\mathcal{T}}^{2}=0$) replaces the previously mislabeled "projection" terminology. Stage II Novel Framework:The $\lambda$ emergence function $\mathcal{E}_{3+1}(\lambda_{s}, \lambda_{t})$ is introduced as a two-parameter formalization of the semiclassical limit, separating spatial coarse-graining scale $\lambda_{s}$ from temporal crossing-statistical scale $\lambda_{t}$. The cross-product emergent algebra $\mathcal{A}_{\mathrm{emergent}}(\lambda_{s}, \lambda_{t}) = \mathcal{A}_{\mathrm{spatial}}(\lambda_{s}) \rtimes_{\sigma^{\lambda_{t}}} \mathbb{R}$ provides the rigorous bridge from discrete Planck-scale dynamics to continuous effective field theory, with Planck-suppressed $\mathcal{O}((\ell_{P}/L)^{2})$ residual corrections. Physical Applications (Conjectural Phenomenological Framework):- Fully background-independent Schwarzschild and Kerr singularity resolution via the Planck brane / Planck torus.- Globally unitary black hole information paradox resolution without firewall pathology.- Two-component cosmic dark matter interpretation from independent $\mathcal{M}^{-}$ pocket-universe baryons. All observational claims are explicitly demarcated as conjectural, with quantitative predictions designated as open challenges requiring completion of the $\lambda$ emergence function derivation. OVST 4.0 does not currently claim explanatory power for cosmological tensions (Hubble, $S_{8}$) or early-galaxy formation. Version History:This v4.0 supersedes v3.7 (DOI: 10.5281/zenodo.21396159) and all prior standalone Stage I preprints:- Emergent Time: 10.5281/zenodo.21321677- Crossing Operator & Unitarity: 10.5281/zenodo.21380298- Multi-Vertex Commutativity: 10.5281/zenodo.21391805- Dark Matter Semiclassical Interpretation v1.4: 10.5281/zenodo.21391190 Open Technical Challenges: Five core open problems are catalogued in Chapter XII, separating proven Stage I theorems from future Stage II formalization and observational prediction work. Research Discussion:The five open technical challenges in Chapter XII span operator algebra, ergodic theory, tensor network renormalization, numerical relativity, and observational cosmology. Technical correspondence is welcome on any aspect of OVST, from single-question clarifications to sustained joint work, at the correspondent's discretion. Contact: ybljc@qq.com. License: CC BY 4.0Author: Jinchuan Liu (Independent Researcher, Yibin, Sichuan, China)