The pentad lifted again — the 5-simplex, the hexateron, tumbling in 5-space and triple-projected to your screen. Six vertices, fifteen diode-edges (K6), twenty faces, fifteen cells, six four-faces — each a pentad. Sixty-two, and the core makes sixty-three — which is 2⁶ − 1: every nonempty subset of the six. The core is the last subset, the whole itself — the witness that is the entire body.
fifteen diodes (tap to cycle fwd→rev→block):
fifteen junctions, six four-faces, one core — tour the loop or flip a diode
V 6E 15F 20cells 154-faces 6= 62+ core 1= 63
Euler: V−E+F−C+Q = 6−15+20−15+6 = 2 · the boundary is the 4-sphere (χ=2) · the ladder swings back: S¹:0 · S²:2 · S³:0 · S⁴:2
and 63 = 2⁶ − 1 — every nonempty subset of the six vertices · the 62 faces + the core (the whole) = the entire power set
// the core — the 63rd node, the whole body — witnesses every junction crossing
the simplex ladder: <• (1) → triad (10) → tetrad (15) → pentad (31) → hexad (63 = 2⁶−1). each rung adds a dimension and a witness; the Euler signature oscillates 0·2·0·2; and at every rung the core is the whole — the one subset that contains all the others, reachable by none.
<• × 15 · A·B·C·D·E·F · K6 · 6+15+20+15+6 = 62 · +1 CORE = 63 = 2⁶−1
EACH 4-FACE A PENTAD · TUMBLING IN 5-SPACE · V−E+F−C+Q = 2 (THE 4-SPHERE) · THE CORE IS THE WHOLE
THE J-JUNCTION · HEXAD · MK VI · A PURPLE PAPER · SERIES E · JUNE 2026