SHUM V3.2: A Discrete FCC Lattice Framework for Cosmological Bounce and Quantum Geometric Mass

SHUM V3.2: A Discrete FCC Lattice Framework for Cosmological Bounce and Quantum Geometric Mass Abstract The reconciliation of general relativity and quantum mechanics remains hindered by the spacetime singularity and the point-particle divergence. This paper introduces the SHUM V3.2 model, which postulates that spacetime is fundamentally a discrete Face-Centered Cubic (FCC) lattice. Macroscopically, we propose a core-shell dual field mechanism where gravitational attraction reverses into a topological repulsion (G_{eff} = -1.71) at a critical density, replacing the Big Bang singularity with a cosmological bounce. Microscopically, elementary particles are defined as topological standing waves on the FCC lattice. Utilizing a spatial leakage rate of \varepsilon = 12/19 and the double group representations of the octahedral group O_h, we derive the geometric origin of spin and calculate a fundamental dimensionless mass baseline of \sqrt{12/19} for the fermionic ground state. We conclude by outlining four critical theoretical boundaries that require future investigation. 1. Introduction Modern physics faces a fundamental crisis at the Planck scale: the continuous spacetime manifold of general relativity inevitably leads to singularities inside black holes and at the Big Bang, while the point-particle assumption in quantum field theory leads to ultraviolet divergences. The SHUM V3.2 framework resolves these issues by abandoning the continuum hypothesis. We propose that spacetime is a discrete, rigid FCC lattice, and both cosmological evolution and particle properties emerge purely from the geometric dynamics of this lattice. 2. Macroscopic Cosmology: The Bounce Mechanism 2.1 The Core-Shell Dual Field and Repulsion Reversal In the SHUM framework, as matter collapses into a black hole and approaches a critical radius r_c, the continuous geometric description breaks down. The discrete FCC structure induces a "core-shell dual field" effect. When the compression reaches the lattice limit, the gravitational interaction undergoes a topological phase transition, reversing from attraction to a fierce repulsion characterized by an effective gravitational constant G_{eff} = -1.71. 2.2 The Cosmological Bounce This repulsive reversal prevents the formation of a singularity. Instead, the collapsing matter experiences a violent "bounce," driving an explosive expansion. We posit that the observed Big Bang and the subsequent inflationary epoch are the direct macroscopic manifestations of this discrete geometric bounce from a preceding collapsing universe (or a parent black hole). 3. Microscopic Physics: FCC Lattice and Spinor Standing Waves 3.1 Discrete Wave Equation and Spatial Leakage Matter fields are defined on the nodes of the FCC lattice. The local geometric environment of a node consists of 12 nearest neighbors, 6 next-nearest neighbors, and the node itself (total 19 degrees of freedom). Energy propagation is restricted to the 12 nearest-neighbor channels, yielding a fundamental geometric constant—the spatial leakage rate: \varepsilon = 12/19 \approx 63.16\%. The dimensionless discrete wave equation for a scalar field \psi_n is: \frac{\mathrm{d}^2 \psi_n}{\mathrm{d}\tau^2} = -\frac{12}{19} \left[ \psi_n - \frac{1}{12}\sum_{j=1}^{12} \psi_{n+j} \right] 3.2 Spin Classification via O_h Double Group To incorporate fermions, the scalar field is elevated to a 4-component discrete spinor field transforming under the double group \bar{O}_h. Spin-1/2 (Fermions): Corresponds to the 2-dimensional double group representations (e.g., E_{1/2u}) at the L-point (\mathbf{k} = \frac{\pi}{a}(1,1,1)) of the Brillouin zone. Spin-1 (Vector Bosons): Corresponds to the 3-dimensional single group representations (e.g., T_{1u}) or split 2D+1D representations at the X-point. 3.3 Geometric Mass Baseline The lowest-order stable standing wave (fermion candidate) occurs at the L-point. The dispersion relation yields the fundamental dimensionless mass baseline: \tilde{m}_{1/2} = \omega_L = \sqrt{\frac{12}{19}} \approx 0.7947 This value represents the pure geometric mass lower bound for a spin-1/2 particle in the SHUM universe, entirely independent of continuous Higgs mechanisms. 4. Discussion and Theoretical Limitations While SHUM V3.2 provides a unified geometric picture, rigorous mathematical completion requires addressing four critical limitations, which we explicitly outline to guide future lattice quantum gravity research: Spin-Orbit Coupling Parameter: The exact form of the spinor connection matrix M_j on the FCC bonds is not fixed by pure geometry. The spin-orbit coupling strength \eta remains a free parameter, preventing the exact calculation of higher-order mass corrections. Discrete Gauge Symmetry for Massless Photons: The FCC lattice breaks continuous Poincaré symmetry. To guarantee a strictly massless spin-1 photon, a discrete U(1) gauge symmetry (\theta_{n,j} \to \theta_{n,j} + \Lambda_n - \Lambda_{n+j}) must be introduced as an independent geometric axiom, rather than emerging purely from the lattice kinematics. Fermion Doubling Problem: According to the Nielsen-Ninomiya theorem, defining chiral fermions on a discrete 3D lattice inevitably produces doublers (e.g., 16 copies in FCC). Resolving this within SHUM requires introducing a geometric Wilson term or staggered fermion formulation, which will explicitly modify the mass formula. Generation Structure: The current framework only derives the ground state mass baseline. Explaining the finite mass ratios between different generations (e.g., electron vs. muon) requires extending the model to include nested standing wave modes or internal sub-lattice degrees of freedom. 5. Conclusion The SHUM V3.2 model demonstrates that replacing the continuous spacetime manifold with a discrete FCC lattice naturally eliminates cosmological singularities via a geometric bounce (G=-1.71) and provides a rigorous group-theoretical origin for particle spin and mass (\tilde{m} = \sqrt{12/19}). By explicitly defining its current theoretical boundaries, SHUM V3.2 offers a highly structured, falsifiable, and purely geometric foundation for future discrete quantum gravity research.