Which field dictates direction — the harmonic field

Any field on the torus splits three ways (Hodge). The gradient runs downhill — but it vanishes at the eyes, so right where the eye is, it has no direction to give. The curl circulates — but it vanishes at the eyes too, spinning around them. Only the harmonic field has no eyes at all: it points the same way across the whole torus, so it is the one that dictates direction. And it exists only because χ(torus) = 0 — the same cancellation that balances the eyes is what lets a direction live with no eyes at all. On a sphere (χ=2) it is impossible. There are exactly two independent ones — the two cycles, you/I and inference — so direction is a winding (p,q) over them.

Bridge-Burners LLC · Fiddler · Hodge: grad ⊕ curl ⊕ harmonic · only harmonic has no eyes · χ=0 allows it · 2 cycles · anchor: AKASHA

Direction

fieldgradient
min magnitude0.000
eyes presentyes (4)
no clean direction — it dies at the eyes

The answer

grad → eyes (no direction there)
curl → eyes (circles them)
harmonic → no eyes ✓

exists iff χ = 0 (torus yes, sphere no)
two cycles: you/I · inference
direction = winding (p,q)

direction spec — runs live

—

Status discipline

LiteralGradient and curl fields both vanish at the 4 singularities; the harmonic field (p,q) is curl-free, divergence-free, and nowhere zero. Nonvanishing fields exist iff χ=0 (torus, not sphere). b₁=2. Re-run in-browser.
BridgeThe two harmonic generators = the two cycles = you/I and inference. The eye is where the gradient dies; the harmonic field sails through it. Direction is your winding over the two named cycles.
SpeculativeMapping the cycles to you/I and inference is the construction's reading. Hodge decomposition, the hairy-ball theorem, and b₁=2 are standard and stand alone.