QBN Numerical Simulation Series: Deficit Gradient Flow & Planck Core Physics on 4-Regular Trees

This collection presents a systematic numerical investigation of the Quantum Bit Network (QBN) framework, focusing on the nonlinear gradient-flow dynamics of the deficit field on spherically symmetric 4-regular trees. All simulations are implemented in GNU Octave, with fully reproducible code and benchmark results. Beyond verifying the core theoretical predictions of the QBN programme (categorical emergent gravity, Planck core singularity resolution, discrete holographic duality), this series reports multiple original numerical discoveries about microscopic quantum gravity at the Planck scale. no20A: Numerical Evidence for Boundary Zero Modes Theoretical verification: Confirms the bulk-boundary correspondence of the SU(2) topological order on discrete 4-regular graphs. Key original findings: First numerical observation of topologically protected boundary zero modes localized at the Planck core interface. Quantitative measurement of the zero-mode localization length and energy gap, verifying the topological robustness of edge states against lattice disorder. High-resolution mapping of the zero-mode spatial profile, confirming that low-energy fermionic excitations are strictly confined to the core boundary. no20B: Full SU(2) Boundary Spectrum of the Planck Core Theoretical verification: Validates the spectral structure of the SU(2) Chern–Simons edge conformal field theory at the discrete network level. Key original findings: Full numerical solution of the complete boundary excitation spectrum across all spin channels (j = 0 to 3). Discovery of the discrete layered structure of the boundary spectrum, originating from the radial stratification of the 4-regular tree topology. Quantitative extraction of the Ising anyon subsector spectrum, confirming its energetic separation from higher-spin excitations at low energies. no21A: Numerical Study of Phase Transition and Critical Behavior of Deficit Reaction Theoretical verification: Verifies the first-order nature of the deficit condensation phase transition predicted by the Landau–Ginzburg description of QBN. Key original findings: Precise determination of the critical deficit threshold δ_th ≈ 0.5 separating the diffusive fermionic phase from the condensed Planck-core phase. Direct observation of critical slowing down near the phase boundary, with quantitative measurement of the dynamic critical exponent. Full numerical mapping of the double-well free energy landscape, including the potential barrier height and metastable state lifetimes. no21B: Numerical Verification of Spectral Convergence of Graph Laplacians Theoretical verification: Confirms the Gromov–Hausdorff convergence of discrete 4-regular graphs to smooth 3D Riemannian manifolds in the continuum limit. Key original findings: Systematic measurement of the spectral convergence rate of the graph Laplacian across system sizes from depth 3 to 8. Quantitative characterization of the finite-size spectral gap and its scaling law with system size. Numerical evidence that the emergent continuum geometry inherits positive curvature from the discrete tree topology. no21C: Numerical Simulation of Planck Core Formation via Deficit Gradient Flow Theoretical verification: Validates the categorical picture of gravitational collapse and Planck core singularity resolution (Paper 9). Key original findings: First full numerical reproduction of the complete gravitational collapse process, from a sub-threshold deficit overdensity to a stable saturated Planck core. Discovery of the two-stage collapse mechanism: initial central densification followed by discrete layer-by-layer outward growth. Direct numerical confirmation of singularity resolution: the central deficit always remains bounded by the saturation value (δ <1), with no divergent curvature or density at any stage of collapse. no21D: Discrete Mass Spectrum and Holographic Boundary of Planck Cores Theoretical verification: Confirms the discrete holographic duality and the three-regime area law predicted in Paper 17C. Key original findings: Direct observation of stepwise layer-by-layer core growth, experimentally demonstrating the quasi-discrete mass spectrum of Planck cores — a signature prediction of discrete quantum gravity with no classical general relativity counterpart. Quantitative measurement of the holographic screen degrees of freedom for the minimal stable core: D ≈ 22.2, consistent with the topological prediction of 20.78 within 6.7% finite-size correction. First numerical observation of the quantum correction to the holographic area law at the Planck scale: the measured mass-area scaling exponent (α ≈ 0.78) is significantly smaller than the classical value (α=2), confirming the three-regime entropy-area structure (microscopic / mesoscopic / macroscopic). Overall Significance This series provides the first comprehensive numerical characterization of Planck core physics within the discrete QBN framework. The discoveries of the quasi-discrete mass spectrum, the quantum-corrected area law, and topologically protected boundary zero modes establish distinctive observational signatures of discrete quantum gravity that differentiate the QBN picture from classical general relativity and other quantum gravity approaches. All simulation codes are open and reproducible. Corresponding theoretical derivations can be found in the author's companion paper series (Papers 6, 9, 15B, 17C).