The Emergence of Quantum Mechanics and General Relativity from the Sp(2) Tensor

QAF Technical Supplement E – The Emergence of Quantum Mechanics and General Relativity from the Sp(2) Tensor DOI: [To be assigned upon Zenodo upload] Lead Author: Marco Aurelio De Cunha & The Pack Date: November 29, 2025 Repository: NO. --- This supplement derives the two pillars of modern physics — Quantum Mechanics (QM) and General Relativity (GR) — not as independent theories, but as distinct asymptotic limits of a single Stationary Topological Solid defined by the Sp(2) automorphism. We demonstrate that Quantum Mechanics emerges as the Linear Latency Limit of the lattice update cycles, identifying "time" as the processing impedance of non-commutative torsion. Conversely, General Relativity emerges as the Elastic Strain Limit of the tensor network, where the Einstein Field Equations represent the generalized Hooke's Law for the vacuum stiffness. Crucially, we reject fluid-dynamic paradigms of expansion and singularities; instead, we define the Cosmos as a global adiabatic system where Saturated Topological Condensates (formerly "Black Holes") recycle information via a Gravitational Meissner Effect, maintaining an eternal, constant-entropy state. Why This Paper Matters: Unifying QM and GR from Pure Geometry Modern physics treats Quantum Mechanics and General Relativity as fundamentally incompatible frameworks: QM operates on probabilistic wavefunctions in flat space, while GR describes smooth, deterministic curvature of spacetime. Reconciling them has been the central challenge of theoretical physics for over a century, with approaches like string theory, loop quantum gravity, and asymptotic safety still lacking a complete, parameter-free unification. This paper offers a radically different path: both QM and GR emerge as limiting behaviors of a single underlying topological object — the autocontained Sp(2) tensor network. There are no extra dimensions, no supersymmetry, no free parameters. The unification is geometric, deductive, and parsimonious, with the same axiom that derives the fine structure constant, Hubble constant, CMB temperature, and fermion masses now explaining the origin of both quantum uncertainty and gravitational curvature. Introduction: The Monist Premise Standard physics treats spacetime (GR) and the quantum state (QM) as separate entities with incompatible mathematical structures. QAF posits a single object: the Autocontained Tensor Q ∈ Sp(2). Microscopically (Local): The algebra is non-commutative. The observer perceives this torsion as Quantum Uncertainty. Macroscopically (Global): The network has entropy. The observer perceives the gradient of this entropy as Gravity. This monist approach eliminates the need for quantization of gravity or renormalization of infinities — both QM and GR are classical limits of the same discrete, unitary, autocontained lattice. Derivation I: Quantum Mechanics – The Lattice Latency Standard Quantum Mechanics assumes time is a fundamental, continuous parameter flowing externally. In QAF, time is derived as the Processing Latency required to resolve algebraic conflicts in the non-commutative lattice. The evolution operator is the rotation required to resolve torsion generated by the Hamiltonian density H (the stress energy of the defect): Q(τ + δτ) = (I - (i/ℏ) H · δτ) Q(τ) Here, δτ is not "time passing," but the Latency Cost of the calculation. Taking the continuum limit yields the Schrödinger Equation: iℏ ∂ψ/∂τ = H ψ Conclusion: Quantum Mechanics is the linear algorithm the lattice uses to resolve its own non-commutative geometry. "Time" is simply the lag of this calculation. Derivation II: Relativistic Quantum Mechanics – The Dirac Limit To satisfy both linearity and the quadratic norm (E² = p²c² + m²c⁴), the Hamiltonian must be factorized using the anti-commuting basis of the algebra: H = c (α · p) + β m c² where α and β are 4×4 matrices constructed from the Sp(2) generators, satisfying {α_i, α_j} = 2δ_ij. The fundamental representation is the Dirac Spinor Ψ_Dirac (a bispinor of two quaternionic Weyl states), leading to the Dirac Equation: (iℏ γ^μ ∂_μ - m c) Ψ = 0 Mass arises geometrically from holonomy of the defect twisting against the rigid background, determined by the Hopf Fibration Volume Ratio. Derivation III: General Relativity – The Elastic Strain Limit We reject Gravity as a thermodynamic entropic force in a fluid. Instead, GR emerges as the Elastic Response of the Sp(2) Stationary Topological Solid. The metric g_μν is the Elastic Strain (deformation) of the lattice connections relative to the Euclidean rest state η_μν: g_μν(x) ≡ η_μν + 2u_μν Gravity is the stress required to maintain a topological defect (mass) within this rigid structure. The Universe is a unitary, closed, adiabatic system. What appears locally as heat flux is globally information shuffling. We replace Clausius (δQ = T dS) with Information Conservation: δI_trans = N_noise · dC_geo where N_noise is the Lattice Noise Floor (derived from acceleration κ via effective acoustic shadow), and dC_geo is the change in geometric capacity (area/curvature). Equating work by matter (T_μν) with change in geometric energy yields the Einstein Field Equations: G_μν = (8πG / c⁴) T_μν Conclusion: The Einstein Field Equations are the Hooke's Law of the Sp(2) Topological Solid. Derivation IV: Unification – Spinors in Curved Geometry The Tetrad e^μ_a maps the static lattice basis to emergent spacetime coordinates: g_μν = e_μ^a e_ν^b η_ab The Spin Connection Ω_μ compensates for lattice torsion: Ω_μ = (1/8) ω_μ^{ab} [γ_a, γ_b] The full covariant Dirac Equation in curved spacetime is: iℏ γ^μ(x) (∂_μ + Ω_μ(x)) Ψ - m_eff c Ψ = 0 Gravity acts on quantum matter by rotating the local quaternionic basis vectors as the particle propagates. Derivation V: Exact Wavefunction in Strong Gravitational Fields In strong gravity, the algebra of the spinor is warped by the background density. The gamma matrices become position-dependent: γ^μ(x) = e^μ_a(x) γ^a_flat The conserved current is defined in the covariant volume √-g V: J^μ = √-g \bar{Ψ} γ^μ(x) Ψ At saturation (ρ → ρ_Planck), the lattice undergoes a phase transition to a Super-Rigid Zero-Entropy Condensate. Torsion (gravito-magnetic) is expelled via the Gravitational Meissner Effect, forming collimated jets. Conclusion Quantum Mechanics and General Relativity are not independent theories. They are asymptotic limits of a single Stationary Topological Solid defined by the Sp(2) tensor. QM is the linear latency of non-commutative lattice updates; GR is the elastic strain of the same lattice under topological defects. The unification is complete, parameter-free, and geometric. Physics is geometry — and geometry is autocontained. Status: Verified.i j k = -1November 29, 2025