About K4
K4 is past the point where the next useful step is another abstract visualization. The inverse-solver mission is built and verified, a physical topology named v1build is bound into the solver, and the first build has a frozen specification. What follows is the mathematics that makes the instrument possible — and the discipline that keeps its claims honest.
Why K4 is structurally interesting
K4 is the complete graph on four vertices. It has six edges. Its cycle space has dimension three, and its cut space also has dimension three. That equal split is the first hinge: every six-component edge-current pattern decomposes into a cycle part and a cut part, exact graph-theoretic objects before geometry or electromagnetism enters.
i_edge = M·w + G·uSix physical edge currents as three cycle weights w and three cut weights u.
Mᵀ·G = 0 P_cycle + P_cut = I₆ P_cycle·P_cut = 0Cycle and cut bases are orthogonal; the two projectors are complete and non-overlapping. This is the Hodge decomposition of the edge-current space — two interwoven three-dimensional families with clean algebraic bookkeeping. It says nothing yet about the field; it says the current space itself has an exact loom.
The regular tetrahedron turns bookkeeping into selection rules
Place the four vertices at a regular tetrahedron and evaluate the magnetic field at its centroid. Under a T_d-symmetric current source, symmetry prevents cut-space edge drive from leaking into centroid B, while cycle-space drive remains full-rank and can address all three Cartesian B directions.
F₀·G = 0 rank(F₀·M) = 3 w = (F₀·M)⁻¹·B_targetAt the centroid, cut-to-B annihilation is exact for a symmetric source; cycle drive spans the three field directions; and a requested centroid field inverts directly into cycle weights. The qualifier is load-bearing: chiral helices break the required source symmetry, which is exactly why the build uses straight parallel bundles — they keep this null Schur-exact and multiply authority through parallel copper.
Three actuator families, not one
The complete architecture has thirteen hardware paths in three families. The edge family carries the cycle/cut decomposition. The four vertex coils form their own magnetic family — their opposing-pair combinations give Cartesian-aligned centroid-B authority, while the equal-drive [1,1,1,1] mode is a built-in symmetry diagnostic that should stay dark at the centroid. The corner electrodes form the electric family. Conflating them produces misleading degree-of-freedom counts and murky control language, so the project keeps them distinct.
Forward map, inverse solve, and provenance
The software packages the hardware into a forward map F. Given a target at a probe point, the solver returns a drive vector and a report carrying residuals, singular values, rank, null dimension, warnings, regime, model, claim class, and B-leak share. The null space stays exposed so a later layer can optimize for power or headroom.
x* = arg min‖F·x − y_target‖₂ (min-norm, null space exposed)Capacity, though, is an ellipsoid, not thirteen equal slots: field superposition is linear, but usable authority is shaped by singular values, current bounds, heating, headroom, and overlap between requested behaviors. Two requests can stay rank-compatible while making each other expensive. Rank is a skeleton; the live instrument prices the flesh.
Claim discipline
Every result carries an explicit class — algebraic, geometric, regime-limited, numerical, heuristic, or conjectural — and every promotion or demotion lands in an append-only ledger. This is not paperwork varnish: it stops a numerical coincidence, a symmetry theorem, and a bench heuristic from quietly blending into one confident sentence. When an audit found the headline E-side crossover frequency resting on a placeholder capacitance, it was demoted to heuristic on the spot — the culture working as intended.
Current scope and what comes after
V1 is the magnetic realization: edges plus vertex coils, in DC and the audio band. The build order is deliberate — calibrate single edges, assemble the full tetrahedron, then add the corner electrodes that bring the electric field into the instrument. The E-side is designed, not gated away; its control simply is not yet wired into the explorer or the automated operations while the exact math, cockpit, and visualization are worked out. Higher-frequency operation and multi-unit arrays sit further out, each behind a named kill-test.
Repo artifacts: build/V1_CANON_2026-06.md · build/EDGE_BOM_2026-06.md · build/K4_SYSTEM_SPEC_2026-06.md · web_cockpit/DESIGN_V1_2026-06.md · k4/k4_solver.py + test suite.