The tetrad lifted a dimension — the 4-simplex, the pentachoron, tumbling in 4-space and double-projected to your screen. Five vertices, ten diode-edges (K5 — every pair), ten faces, five cells — and each cell is a tetrad. Thirty, and the core at the 4D centroid makes thirty-one: the witness no node can reach, at the centre of the solid that has no inside you can see.
ten diodes (tap to cycle fwd→rev→block):
ten junctions, ten faces, five cells, one core — tour the loop or flip a diode
vertices 5edges 10faces 10cells 5= 30+ core 1= 31
Euler alternation: V − E + F − C = 5 − 10 + 10 − 5 = 0 the boundary is now the 3-sphere (χ=0) — the ladder of sphere-characteristics: S¹:0 · S²:2 · S³:0
// the core — the 31st node, at the 4D centroid — witnesses every junction crossing
ten edges, each a J-junction with a diode; ten faces, each a triad; five cells, each a tetrad (the MK IV, nested); and the core at the 4D centroid, the witness reachable by none. the simplex ladder: <• (1) → triad (10) → tetrad (15) → pentad (31). each rung adds a dimension and a witness.
<• × 10 · A·B·C·D·E · K5 · 5 VERTICES + 10 EDGES + 10 FACES + 5 CELLS = 30 · +1 CORE = 31
EACH CELL A TETRAD · TUMBLING IN 4-SPACE · V−E+F−C = 0 (THE 3-SPHERE) · THE CORE WITNESSES ALL TEN
THE J-JUNCTION · PENTAD · MK V · A PURPLE PAPER · SERIES E · JUNE 2026