The Stability of Matter as the Fundamental Axiom: Analytic Signal Framework for Unifying Quantum Mechanics and Cosmology

This paper advances the Unified Quantum Mechanics (UQM) framework by elevating the empirical stability of matter to the status of a cosmological first principle. While previous work established the operator formalism, this work posits that the foundational conflicts of 20th-century physics—Real vs. Complex, Deterministic vs. Probabilistic, and Continuous vs. Discrete—are resolved specifically by this stability axiom. The central innovation is the advancement of the Analyticity Mandate, which constrains all physical wavefunctions to be causal, positive-energy analytic signals ($E>0$). This mandate is mathematically implemented by identifying the imaginary unit $i$ with the physical, non-local temporal Hilbert transform operator ($i \equiv \mathcal{H}_t$), transforming the Schrödinger and Dirac equations into deterministic, real-valued field equations—the Unified Real Wave Equation.
     
     We uncover a fundamental symmetry at the heart of unification: the analytic signal constraint required to ontologically complete quantum theory simultaneously derives the geometric structure of General Relativity, demonstrating that both paradigms originate from the single axiom of matter stability. We establish that this analytic complexification constitutes a mathematically rigorous and physically necessary procedure, transforming the underlying real spacetime-energy field into a unified dynamical framework that naturally subsumes both quantum phenomena and gravitational evolution. This framework culminates in a unified Theory of Everything described by the fundamental action:
     $S_{\text{UQM}} = \int d^4x \sqrt{-g} \left(\frac{c^4}{16\pi G} (R - 2\Lambda) + \mathcal{L}_{\text{SM}}\right)$, where the Lagrangian density describes both geometry and matter as manifestations of a single real field $\psi_R$ under the Analyticity Mandate. The complete theory is expressed through the coupled system:
     $G_{\mu\nu} = \kappa T_{\mu\nu}[\psi_R] \quad \text{and} \quad \psi = \psi_R - i\mathcal{H}_t[\psi_R] \quad \text{with} \quad E > 0.$
     
     We present the following quantitative results and mechanisms: (1) We derive the Cosmic Cycle Period $T_{cycle} \approx 7.14 \times 10^{11}$ years for the cubic model (Spherical model: $\approx 7.00 \times 10^{11}$ years) and the maximum expansion factor $Z_{max} \approx 2.20 \times 10^{10}$ (Spherical: $\approx 1.36 \times 10^{10}$), by solving the cosmic equation of state governed by the electron's vacuum stability horizon $l_{crit} \approx 5.35$ cm (Spherical diameter: $D_{crit} \approx 6.64$ cm). (2) We identify the Fermionic Rigidity mandated by the Pauli Exclusion Principle as the specific geometrodynamic trigger for the non-singular Big Bounce, avoiding the entropy paradoxes of Conformal Cyclic Cosmology. (3) We restore Global Noether Conservation by reinterpreting cosmological redshift as mechanical work performed against the elastic tension of the vacuum plenum ($\rho_{\Lambda}$).
     (4) We formalize the \emph{Singh Stability Conjecture}, proposing a hierarchy of verification protocols including a Rapid-Dispersion interferometric experiment designed to test the specific cosmological parameter limits of the vacuum.
     
     The resulting ontology is a monistic Spacetime-Energy substance, where the Einstein Field Equation $G_{\mu\nu} = \kappa T_{\mu\nu}$ represents the phase equilibrium between the geometry of the vacuum and the density of matter. UQM thus completes Einstein's unification program by providing a deterministic, realist, and geometrically coherent foundation for all physical phenomena. We formally propose a community-wide verification program, defining definitive falsification tests through laboratory experiments, cosmological observations, and Gedankenexperiments. Within this program, we distinguish between parametric constraints and ontological validity, asserting that the ultimate falsification of the framework requires the empirical demonstration of stable negative-energy states.