Relativistic + Gravitational Time Dilation (Phase 1)

Wave Propagation Speed in Flat Space (Phase 1)

Light Deflection During Transverse Motion (Phase 2)

Bound State Energy Levels in χ-Well (Phase 2)

Tunneling Rate Modulation by χ-Field (Phase 2)

Wavepacket Collapse vs χ-Gradient Interference (Phase 3)

EM Wave in χ-Gradient (Phase 1)

Dynamic coupling - time-varying χ (Phase 2)

Asymmetric coupling - directional energy transfer (Phase 2)

Nonlinear coupling - self-interaction via χ feedback (Phase 2)

Interference - coupled field mixing (Phase 2)

Saturation - coupling strength limit (Phase 2)

χ-field coupling to charge density ρ: ∇·E = ρ/ε₀

Moving charges create B-fields: ∇×B = μ₀J + μ₀ε₀∂E/∂t

Changing B-field induces E-field: ∇×E = -∂B/∂t

Complete Maxwell's equations with ∂E/∂t term

Verify c = 1/√(μ₀ε₀) emerges from lattice parameters

Energy density and flow: S = (1/μ₀)E×B

How χ(x,t) modifies local permittivity/permeability

EM field energy contributes to χ-field source

EM waves in χ-gradient (gravitational redshift for light)

Larmor formula emergence: P = (μ₀q²a²)/(6πc)

Frequency-dependent χ-field refraction creating rainbow effects

Dynamic χ evolution affects EM propagation

Cavity resonator modes in χ-field medium

Moving source/observer in lattice medium

EM wave scattering from χ-inhomogeneities

Charged particle in magnetic field radiates

Short EM pulse through χ-medium

Large-scale χ-evolution affects local EM properties

Electromagnetic potentials: gauge transformations leave physics unchanged

Charge conservation: ∂ρ/∂t + ∇·J = 0

Short EM pulse through time-varying χ(t) gate; measure differential delay

Global conservation — short

Global conservation — long

Wave integrity — mild curvature

Wave integrity — steep curvature

Hamiltonian partitioning — uniform χ (KE ↔ GE flow)

Hamiltonian partitioning — with mass term (KE ↔ GE ↔ PE flow)

Hamiltonian partitioning — χ-gradient field (energy flow in curved spacetime)

Dissipation — weak damping (exponential decay, γ=1e-3 per unit time)

Dissipation — strong damping (exponential decay, γ=1e-2 per unit time)

Thermalization — noise + damping reaches steady state

Momentum conservation — two-packet collision

Local frequency — linear χ-gradient (weak)

Local frequency — Gaussian well (strong curvature)

Local frequency — Gaussian well (broader potential)

Local frequency — Gaussian well (shallow potential)

Local frequency — linear χ-gradient (moderate)

Local frequency — Gaussian well (stable reference)

Time dilation — bound states in double-well potential (KNOWN: Packet becomes trapped, demonstrates bound state physics)

Time dilation — uniform χ diagnostic (isolate grid dispersion)

Time dilation — 2x refined grid (N=128, dx=0.5) [OPTIMIZED: matched baseline duration for fair convergence comparison]

Gravitational redshift — measure frequency shift in 1D potential well

Time delay — packet through χ slab (Shapiro-like)

Phase delay — continuous wave through χ slab (DEMONSTRATES: Klein-Gordon phase/group velocity mismatch - testable prediction!)

Local frequency — double well (ω∝χ verification)

Group delay — differential timing with vs without slab

3D radial energy dispersion visualizer — central excitation, volumetric snapshots for MP4

3D double-slit interference — quantum wave through slits showing χ-field localization

Gravitational redshift — frequency shift climbing out of χ-well

Gravitational redshift — linear gradient (Pound-Rebka analogue)

Gravitational redshift — radial χ-profile (Schwarzschild analogue)

Self-consistent chi from E-energy (Poisson) - verify omega~=chi at center (1D)

GR calibration - redshift to G_eff mapping (weak-field limit)

GR calibration - Shapiro delay correspondence (group velocity through slab)

Dynamic χ-field evolution — full wave equation □χ=-4πGρ with causal propagation (gravitational wave analogue)

Gravitational wave propagation — oscillating source radiates χ-waves, validate 1/r decay and propagation speed

Light bending — ray tracing through χ-gradient, measure deflection angle

Weak Equivalence Principle — mass-independent gravitational acceleration

ΔE Transfer — Low Energy

ΔE Transfer — High Energy

Spectral Linearity — Coarse Steps

Spectral Linearity — Fine Steps

Phase-Amplitude Coupling — Low Noise

Phase-Amplitude Coupling — High Noise

Nonlinear Wavefront Stability

High-Energy Lattice Blowout Test

Heisenberg uncertainty — Δx·Δk ≈ 1/2

Bound state quantization — discrete energy eigenvalues E_n emerge from boundary conditions

Zero-point energy — ground state E₀ = ½ℏω ≠ 0 (vacuum fluctuations)

Quantum tunneling — barrier penetration when E < V (classically forbidden)

Wave-particle duality — which-way information destroys interference

Non-thermalization — validates Klein-Gordon conserves energy (doesn't approach Planck)

Validates wave propagation isotropy on coarse grid

Isotropy — Coarse Grid

Isotropy — Fine Grid

Lorentz Boost — Low Velocity

Lorentz Boost — High Velocity

Causality — Pulse Propagation

Causality — Noise Perturbation

Phase Independence Test

Superposition Principle Test

3D Isotropy — Directional Equivalence

3D Isotropy — Spherical Symmetry

Dispersion Relation — Non-relativistic (χ/k≈10)

Dispersion Relation — Weakly Relativistic (χ/k≈1)

Dispersion Relation — Relativistic (χ/k≈0.5)

Dispersion Relation — Ultra-relativistic (χ/k≈0.1)

Causality — Space-like correlation test (light cone violation check)

Linear Momentum Conservation — Two-Packet Collision

Invariant Mass — Lorentz Invariance

Entropy Increase — Second Law of Thermodynamics

Irreversibility — Arrow of Time

Equipartition Theorem — Energy Distribution Across Modes

Thermalization Timescale — Relaxation to Equilibrium

Temperature Emergence — Boltzmann Distribution