K4 · RESEARCH PROPOSAL
From a shared geometry to a tested control map.
K4 investigates how a regular tetrahedron’s coordinates, graph decomposition and symmetry organize physical control—and how measurement determines which possibilities survive contact with a real plant.
Purpose
The mathematical object comes first: four geometric corners and six edges. Balanced corner coordinates represent 3D space; oriented edge space splits into two orthogonal three-dimensional parts. The research asks how these structures help construct, identify and control a particular physical realization.
Electromagnetism is the worked example. Sensing, computation and coupled-fluid realizations remain separate plants with their own equations and evidence. Optional corner actuators add physical channels; geometric corners do not imply installed actuators.
Methods
Declare complete source paths, materials, dimensions and drive constraints. Compute an observation map in a stated regime. Measure inputs and outputs with background subtraction and a fixed sensor pose. Test held-out commands, inspect symmetry/model residuals, and use scaled SVD to identify authority.
Solve constrained inverse problems and test their commands. Close feedback loops with suitable sensing and stability checks. Local fields, gradients, volume objectives and mechanical behavior require different observations and acceptance tests.
What the V2 bench prototype contributes
V2 is the name of the physical six-edge bench prototype, distinct from the corner vectors v₀,…,v₃. The repository preserves a six-edge response catalogue, bounded multi-edge magnetic trajectories, and command-associated permanent-magnet motion. Its 31 August state record describes V2 as a characterized but uncalibrated prototype, retained after the W4 design pivot.
Local magnetic responseThe six-edge synthesis suite recorded positive and negative responses predominantly along each sensor axis, X, Y and Z. These are local command-to-field results at a fixed configuration; branch currents were not individually calibrated.
Placement-dependent motionObserved magnet motion establishes physical coupling in tested placements. It does not establish arbitrary translation, confinement or controlled spin.
Open records remain openSome later serial captures lack operator current, temperature, anomaly and motion closeout. They remain incomplete evidence.
V2 struts are recorded as 69.5 turns of 30-AWG wire with opposite-end terminals, approximately 386.5 μH mean inductance and 5.7–5.9 Ω resistance. These are historical readings, not parameters of the dimensionless simulator.
Source trail: repository CURRENT.md (31 August 2026), the EXP-04 response catalogue and EXP-05 experiment records under Build/experiments/. The archived apparatus page is dated context. No present wiring or operating state is inferred from it.
What the simulator contributes
Six prescribed edge-current signals are summed through a finite-filament field kernel. Three movable slices and a rotating 3D view show the same quasistatic snapshot at the playhead. Optional corner sources are point dipoles, distinct from the four geometric vertices.
Inspect superposition, compare local and spatial observations, and follow coordinated waveforms. Edge length is one normalized unit and colours are relative. The source geometry is chosen to make the composition visible. A defined apparatus supplies its own paths and material response; the frequency discussion explains how the dynamic model changes with the medium.
Open the field explorer →Expected results and open questions
Exact [A]: the coordinate transform, cycle/cut split and symmetry restrictions follow under stated assumptions.
Entailed, size untested: a validated map with adequate singular values and operating headroom permits an inverse command to test. Useful magnitude and bandwidth remain plant-dependent.
Open [C]: robust volume control, mechanical stabilization, MHD realizations and useful physical computation require dedicated models and measurements.