One tetrahedral graph. Two protected sectors. Now fiberized into a physical wire crystal.
The algebraic substrate is exact. The canonical EM realization is engine-verified. The physical wire-crystal build is now entering measurement.
Layer 1 — the K4 substrate
Six edges of a regular tetrahedron carry current. Graph theory splits the edge-current space exactly: three cycle modes and three cut modes — orthogonal by topology before geometry enters. When the source respects Td symmetry, Schur's Lemma forbids the two sectors from coupling at the centroid: cycle drives control centroid B; cut is centroid-silent and available for near-field structure, voltage patterns, and electric-field control. This null is not approximate. It is representation-theoretic, and it holds at all frequencies at the centroid.
Four vertex coils add a second magnetic family. Corner electrodes add the electric field. The full instrument has thirteen hardware paths and six effective centroid channels — three B, three E.
[A] cycle = T₁, cut = T₂ confirmed by explicit S₄ character computation. [G] F₀·G = 0 under T_d-symmetric source. Inverse solver: residuals ≤ 3×10⁻¹⁶ at the centroid. Every number carries its model and claim class.
Layer 2 — the K4 wire crystal
Each single-wire edge is replaced by a structured conductor fiber: a six-wire ring, a 6+6 staggered pair, or a 6/6/6 tri-shell lattice. Under common-mode drive, a six-wire fiberized edge reproduces a centerline conductor at centroid distance to 3.5×10⁻⁵ relative error — the finite-wire K4 solver is preserved.
From there the architecture opens: radial shell identity, angular modes, routed shell topology, optional helical chirality, and material-mediated damping. Chirality decomposes exactly as axial + azimuthal current components; the best three-shell azimuthal canceller is ++− (ρ²-weighted: 1+4−9 ≈ 0), not the naive +0−. Axial authority is preserved under any chirality pattern.
First physical build target: 36 wires (6 per edge × 6 edges). Bundle OD = 5.10 mm (7 × 0.729 mm insulated wire). Tests the load-bearing claim before adding tri-shell or chiral complexity. Full Wire Crystal →
The instrument — live, in this browser
Rotate and zoom the V1 build model below — grooved spars, parallel strand bundles, LW181 vertex coils at their canonical offsets.
For the full field instrument — drive the channels, solve to a target, watch field, load, and fidelity respond in real time — open the Explorer →
The ladder
K4 graph proof → EM engine → symmetry closure → fiberized edge → 36-wire build → Mode Foundry → 3D influence matrix. Each rung is verified before the next is claimed.
What exists now
An inverse solver with audit tests (residuals, singular values, regime, model, and claim class on every report) · a choreographer that compiles pattern sets into time-indexed drive trajectories · a calibrated physical topology (v1build: straight parallel bundles, Schur-exact cut null, loop-stack-matched vertex coils) · a frozen V1 build canon with parts list and acceptance numbers · a 3D influence matrix engine (B = G·I, 357×108, reproduces independent results to 5×10⁻¹⁶) · a Mode Foundry 2D fiber cross-section cockpit for geometry exploration · this field explorer.
Substrate math · Wire Crystal · Mode Foundry · Build & parts · Updates