K4 Wire Crystal
Executive summary · corrected 2026-06-13 · supersedes "+0−" chirality claim with "++" in all copies
Core thesis
The K4 project is a finite-wire electromagnetic architecture built on the six edges of a regular tetrahedron. In the canonical model each edge carries one current variable, giving a six-dimensional edge-current space that separates into two protected sectors:
ℝ⁶ = cycle ⊕ cutThe cycle sector provides magnetic-field authority at the tetrahedral centroid. The cut sector is centroid-silent in the canonical magnetic map but remains physically important for voltage structure, electric fields, near-field gradients, and higher-order modes.
The new architecture preserves this finite-wire foundation while replacing each single edge conductor with a structured conductor fiber. The first validated fiber is a six-wire ring: under common-mode drive it reproduces a single centerline conductor to high accuracy at centroid distance.
Essential bridge: a K4 edge can be fiberized without destroying the original finite-wire solver, provided the fiber is driven in common mode.
Claim ledger
Class A Exact graph-theoretic
V = 4, E = 6. Bare connected-K4 cycle rank β₁ = E − V + C = 3. ℝ⁶ = cycle ⊕ cut (exact). Cycle = T₁ (axial vector), cut = T₂ (polar vector) — confirmed by explicit S₄ character computation; orthonormal irreps.
Class B Symmetry-protected
Distinct irreps T₁ ≠ T₂ ⇒ Schur's Lemma forbids symmetry-respecting operators from mixing cycle and cut. Holds in ideal tetrahedral geometry. Chiral or helical sources break T_d → T and re-permit coupling (see Chirality).
Class C Engine-verified
Six wires at θ_k = k·60°, (ρ cos θ_k, ρ sin θ_k). Under common-mode drive I = [1,1,1,1,1,1] the fiberized edge reproduces a centerline conductor at centroid distance to relative error ≈ 3.5×10⁻⁵ (0.0035%). The 3D influence engine B = G·I is built and verified (G_edge 315×18, G_k4 357×108; reproduces independent results to 1.7×10⁻¹²).
Class D Frontier (roadmap, not yet claims)
Tri-shell routing as richer connected-current topology; conductive PLA as lossy damping boundary; helical grooves as controlled solenoidal behavior; chirality switching as a field-gradient handle; shell and routing defects as mode-selective features.
Fiber geometry
Four validated geometries, in order of complexity:
1. Six-wire ring — one shell of six conductors at 60°. Common-mode bridge verified to 3.5×10⁻⁵.
2. Offset 6+6 pair — two shells, second offset 30° (HCP packing; ρ₂ = √3·ρ₁).
3. Aligned 6/6/6 tri-shell — three shells sharing six angular columns; ρⱼ = j·d (d = insulated wire OD).
4. Staggered 6/6/6 tri-shell — three shells offset 0°/20°/40°.
For sixfold common-mode geometry the first geometric angular harmonic that survives C₆ is m = 6, not m = 3. (m = 3 is an imposed alternating-current mode, not a passive geometric harmonic.)
Insulated diameter rule
Pack by insulated OD, not bare copper. Aligned 6/6/6 radial lattice: ρⱼ = j·d, bundle OD = 2(ρ₃ + d/2) = 7d. With d ≈ 0.729 mm → bundle OD ≈ 5.10 mm.
Do not use the bare-copper figure (≈ 4.51 mm) — it omits enamel and underestimates groove sizing.
Tri-shell routing and DOF
Reduce the fiber to three shell channels per edge: I_e = [inner, mid, outer]. Across six edges = 18 branch currents — but NOT 18 independent controls. Once shells route through vertices, KCL applies. Controllable DOF = the routed graph's cycle rank:
| Routing | Loop DOF β₁ |
|---|---|
| Bare connected K4 | 3 |
| Identity tri-shell (9 edges each route its own shell) | 9 |
| Cyclic braided | 7 |
| Full shell mixing | 15 |
A K4 vertex joins three edges × three shells = 9 terminals, so a vertex needs a full 9-terminal junction model. A 3×3 matrix only describes a shell-to-shell motif.
Helical grooves and chirality
A helical shell current decomposes (uniform-helix, exact):
q_s = 2πρ_s N_s / L I_z = I_s / √(1 + q_s²) I_φ = h_s · q_s · I_s / √(1 + q_s²) h_s ∈ {+1, −1}A finite helical edge is a short solenoid, not a closed toroid: the azimuthal component is not perfectly confined and contributes a nonzero centroid magnetic field along the edge axis.
Three-shell chirality alphabet: +++ additive bias; ++− / +−+ / +0− shaping; +−− inner opposition.
Correction (Stage A.6, supersedes earlier "+0−" claim). The per-shell azimuthal weight at the centroid scales as ρ² → exactly 1 : 4 : 9 (inner : mid : outer), N-independent. The cleanest azimuthal canceller in the alphabet is therefore ++− (outer opposes inner+mid: 1 + 4 − 9 ≈ 0), NOT +0−. Measured |B_odd| at N=2:++− 3.5×10⁻⁸ < +−+ 5.2×10⁻⁸ < +0− 7.0×10⁻⁸.
Two further results: (1) centroid axial authority is preserved exactly under any chirality (B_even constant — a single-start helix threads each cross-section once at full current I). Chirality costs wire length √(1+q²), not centroid reach. (2) On all six edges with uniform handedness and common drive, per-edge azimuthal residuals partially add → net centroid residual ≈ 0.5% of baseline.
Physical build bridge
First build: 6 wires/edge × 6 edges = 36 wires. Tests the load-bearing fact: does a real six-wire fiberized edge preserve finite-wire K4 behavior under common-mode drive? Validates solver preservation, sixfold groove manufacturability, measured insulated OD, spacing, current sharing, heating, alignment tolerance, repeatability. Only after this floor holds: advance to tri-shell and chiral architectures.
Companion tool — Mode Foundry v0.3
2D fiber cross-section cockpit. 2D infinite-line Biot-Savart, axial-only; chirality as length-penalty / visual proxy. Use it to decide which local geometries deserve promotion into the 3D finite-edge influence-matrix solver.
Simulation roadmap
1. Full K4 common-mode check (six-wire bundles, all edges) — the 3D influence engine exists.
2. Non-common-mode detectability (alternating on one edge).
3. Tri-shell chirality scan, N = 0, 0.5, 1, 2, 3, 5 — DONE (stage_a6_chirality.py).
4. Routed shell-graph solver: explicit netlists, KCL, cyclic braiding.
5. Conductive-PLA proxy: damping vs heat.
Final claim
The K4 wire crystal is a Schur-protected finite-wire tetrahedral field architecture extended by structured conductor fibers. Proven bridge: a six-wire fiberized edge preserves the original finite-wire K4 solver under common-mode drive. Frontier: non-common fiber modes, tri-shell routing, helical chirality, and material boundaries add control handles whose true authority is set by graph topology, KCL, impedance, and measurement.
K4 graph + fiber lattice + shell routing + chirality + material damping = K4 wire crystalPreserve the finite-wire K4 solver first. Add complexity only when it earns its place.