Knot Primer — Closed Flux Paths on the Substrate

Process ontology: a knot is not a thing — it is a self- sustaining transaction loop. Flux that closes on itself. Every node in the loop is mid-handshake with its neighbors; the "structure" is a process that keeps re-establishing itself, not a shape. Not surfaces all the way down — transport all the way through.

The component inventory

A knot is one closed curve (1 component). A link is 2+ closed curves. "How many strings can we start with" = how many independent closed flux paths can a bound state support:

Components	Name	Substrate version
1	unknot	{111} ring — 6-cell closed hexagon on edge channels
1	trefoil	loop crossing itself 3× — needs over/under
		ordering on shared nodes (unprobed)
2	Hopf link	two rings linked once — each cell of ring A
		passes "through" ring B (planck3 found linked
		windings inside Hopfions)
3	Borromean	3 loops, pairwise unlinked, collectively bound
		— cut one, all fall apart
k	T(p,q)	torus knot — p turns through the hole, q around
		— a 2-integer-indexed species family

Seed recipes (moore26)

The {111} ring — verified dd=540 stable over 2,000t in isolation:

// 6 edge-channel hops forming a closed {111} hexagon
// ENGINE channel vectors — ch6={1,1,0}, ch12={1,0,-1},
// ch16={0,1,-1}, ch7={-1,-1,0}, ch13={-1,0,1}, ch17={0,-1,1}
const vec = [[1,1,0],[1,0,-1],[0,1,-1],[-1,-1,0],[-1,0,1],[0,-1,1]];
const ch  = [6,12,16,7,13,17];
// each cell: base ~30 both members on all pairs,
// forward member boosted +90 (circulation bias)
// CAUTION: an earlier vec array used {1,0,1} for ch12 — wrong:
// ch12 in the engine is {1,0,-1}. The mislabeled seed painted an
// OPEN ARC (persistent winding, no closure) — a packet stream
// flew straight off it. With true vectors the loop closes AND
// carries a persistent current (verified: 12 packets, 2000t,
// zero arms/decay at rail_gate=64).

• 1-component: the recipe above at any center
• 2-link (Hopf): two rings offset so ring A's cells pass through ring B's interior — earlier orthogonal-ring probes showed transparent coexistence at intersections (parallel/ antiparallel dd preserved per-member)
• Borromean: three orthogonal rings, pairwise non-intersecting, arranged so each pair clears the others — no direct contact but mutual confinement
• T(p,q): scale up — a ring wound twice around (through the hole p times, around the torus q times)

Measured persistence so far

• {111} ring isolated: dd=540 stable over 2,000t
• {111} ring near a live knot: dissolved — the host's flux economy eats the loop (conduction + self-absorbed emission)
• Ring inside a knot interior: survives but self-absorbs — its parity never reaches the surface ("escape depth" problem)
• Orthogonal rings: transparent coexistence — channels carry independent directional traffic at shared nodes

The brittleness problem

Loops are stable in isolation and fragile near structure — they are ensemble phenomena, not Planck-scale processes. At 0.1 fm/node a 6-cell loop is at the resolution floor: conduction eats circulation as fast as it forms. The stable loops in planck3 (Hopfions, 1.6M mass) were spin-cycles the rewrite process found — emergent at ensemble scale, not painted seeds.

The open hunt: the regime where circulation self-organizes — the charge/deflect/vol_share-driven configuration where bulk dd closes on itself without being seeded. If the knot's own routing spontaneously winds into a persistent loop, that's the "electrical" regime — coherence as a rate equilibrium, not a fixture.