================================================================================ TEST DESCRIPTION: 4D Wave Interference Engine ================================================================================ Author: Jonathan Shelton Date: 2026-03-20 Status: PRE-REGISTERED (test description written BEFORE coding) Protocol: Outcome-agnostic. Observe what the geometry produces. ================================================================================ 1. WHAT WE ARE TESTING ================================================================================ The 24-cell (Schläfli {3,4,3}, established geometry, 1852) is self-dual. The rotation mapping it to its dual is a π/4 (45°) isoclinic Clifford rotation in both invariant planes (COMPUTED from vertex coordinates). At 45°: cos²(45°) = sin²(45°) = 0.5. The geometry predicts an EQUAL division of energy between the two dual orientations. PREDICTION: A 4D wave interference simulation should show equal energy distribution between the dual orientations of the 24-cell geometry. OBSERVATION GOAL: How do the two equal halves interfere with each other? What patterns emerge from the interaction of two staggered-but-equal dual orientations in 4D space? ================================================================================ 2. WHAT WE EXPECT TO SEE (pre-registered) ================================================================================ Based on the established geometry of the 24-cell: A. EQUAL DIVISION: Energy should distribute equally between the two dual orientations (50/50). If we see significant asymmetry, the geometric prediction is wrong. B. INTERFERENCE PATTERN: The two staggered orientations should produce a characteristic interference pattern. We do NOT know what this pattern looks like — that is what we are observing. C. STABLE COEXISTENCE: The 45° offset should prevent complete destructive interference (annihilation) while allowing partial interference at intersection points. ================================================================================ 3. WHAT WOULD FALSIFY ================================================================================ - If energy distribution is NOT equal (significantly different from 50/50), the 45° stagger interpretation is wrong - If the two orientations fully annihilate despite the stagger, the coexistence prediction is wrong - If no coherent interference pattern forms (pure noise), the geometric structure is not physically meaningful ================================================================================ 4. WHAT WOULD CONFIRM ================================================================================ - Equal energy distribution between dual orientations (within numerical tolerance of the simulation) - A stable, non-trivial interference pattern at the intersection of the two orientations - Specific intensity nodes at the geometric vertices/edges of the 24-cell structure ================================================================================ 5. METHODOLOGY ================================================================================ ENGINE: 4D FDTD (finite-difference time-domain) wave equation ∂²ψ/∂t² = c² (∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z² + ∂²ψ/∂w²) + J(x,y,z,w,t) The wave equation extends naturally to 4D — no new physics is required, only an additional spatial dimension in the Laplacian. SOURCE CONFIGURATION: - Place sources at the 24 vertices of the 24-cell: permutations of (±1, ±1, 0, 0) - Apply the same f | t pulsed source used in 2D tests: J(r,t) = A(r) × sin(2πft) × W(t; r(r)) - The dual orientation (cell centers) provides the second set: (±1,0,0,0) ∪ (±½,±½,±½,±½) - The 45° Clifford rotation between them is built into the vertex placement, not imposed externally GRID: 4D array. Size constrained by memory: - 32⁴ = ~1M points (feasible on VPS, ~8MB per field) - 64⁴ = ~16.7M points (feasible with care, ~134MB per field) - 128⁴ = ~268M points (may exceed VPS memory) MEASUREMENTS: - Total energy in each dual orientation's Voronoi cell - Interference pattern at the midpoints between dual vertices - Symmetry order parameters (generalized Q_l for 4D) - Energy conservation check (must hold to numerical precision) VISUALIZATION: - 3D cross-sections of the 4D field at w = const slices - Energy density projections onto 3D subspaces - The 24-cell projects into 3D as rhombic dodecahedron — use this projection for visual inspection ================================================================================ 6. CONNECTION TO PRIOR TESTS ================================================================================ TLT-001: 1D isotropic source → radial rings (no lattice) ✓ TLT-002: 2D N=3 waves at 120° → hexagonal lattice ✓ (9/9 materials) TLT-003: 2D progressive compaction → cipher derivation ✓ THIS TEST (TLT-4D-001): 4D wave interference at 24-cell vertices → observe dual orientation interference → measure energy distribution between staggered halves → characterize the 4D interference pattern This follows the same output-agnostic methodology as TLT-001/002/003. The engine observes what the geometry produces. The geometry is established mathematics (Schläfli 1852). The wave equation is standard physics. The observation is what's new. ================================================================================ 7. FRAMERATE SWEEP: F4 vs H4 GOLDILOCKS TEST ================================================================================ The Fibonacci route predicts c_4D = 13/8 = 1.625c (H4 family, phi). The 24-cell's own geometry suggests 1 + sin(45°) = 1.707c (F4 family, √2). Steinberg measured 1.7 ± 0.2c (central value 1.7, range 1.5–1.9). SWEEP: Run the 4D simulation at multiple framerate values from 1.5 to 1.8 in steps of 0.025 (13 runs). At each value, measure: - Interference pattern coherence (does the pattern organize or fragment?) - Energy conservation quality (does the simulation stay stable?) - Symmetry order parameters (which framerate produces the cleanest geometry?) - Residual noise (which framerate minimizes disorder?) The framerate that produces the most organized, lowest-noise, highest- coherence interference pattern IS the natural framerate of the geometry. The simulation finds the Goldilocks zone — the sweet spot where the 4D wave equation and the 24-cell geometry are most compatible. CANDIDATE VALUES: 1.500 (lower bound of Steinberg error bar) 1.525 1.550 1.575 1.600 1.625 ← Fibonacci prediction (13/8, H4 family) 1.650 1.675 1.700 ← Steinberg central value 1.707 ← 1 + sin(45°), F4 family (√2) 1.725 1.750 1.775 1.800 (upper bound check) WHAT THIS RESOLVES: - If the Goldilocks zone centers on 1.625: Fibonacci route is confirmed, and the H4-F4 convergence needs explaining - If it centers on 1.707: the 24-cell geometry determines the framerate, and the Fibonacci route is an approximation - If it centers elsewhere: both routes are approximations of something deeper - If there IS no clear optimum: the framerate may not be determined by static geometry alone (could depend on dynamics) ================================================================================ 8. OPEN QUESTIONS THIS TEST MAY ADDRESS ================================================================================ - Does the 4D interference pattern produce shell-like structures at specific radii (connecting to Mackay magic numbers)? - Does the stagger produce a characteristic angular signature visible in 3D projections? - Does the interference at dual-vertex intersection points produce intensity nodes that map to known particle properties? - What is the natural framerate of the 24-cell geometry? - Is the F4/H4 distinction physically meaningful or are they close enough to be within the geometry's tolerance? ================================================================================