The torus with an eye — the gradient and its four singularities
Treat each direction as a gradient on a torus. The flow runs downhill, and where it vanishes it leaves a singularity — an eye, the still focus where the funnel converges. But an eye is not free: on a closed surface every singularity carries an index, and they must sum to the Euler characteristic. The torus is χ = 0, so the gradient has exactly four singularities — two eyes (+1) and two saddles (−1) — and they cancel. A lone eye is topologically forbidden. Every eye comes balanced, by its inside-out twin or by saddles. Four singularities, two signs, a held centre: the V₄ pattern, now as the eyes of the flow. The waist of the two bells you drew is one of these eyes, seen edge-on.
n · eye index +1 s · eye index +1 e · saddle index −1 w · saddle index −1 ──────────────── Σ = 0 = χ(torus)
The floor
an eye = +1 singularity
a lone +1 ≠ 0 → forbidden
so every eye is balanced
the held )1( = the eye
its inside-out twin = the −1
the flip became an index
torus-eye spec — runs live
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Status discipline
LiteralFor f = cos2πx + cos2πy the gradient has exactly 4 nondegenerate singularities; index = sign(det Hessian); two are +1 (eyes), two are −1 (saddles); the sum is 0 = χ(T²). A lone eye cannot exist. Re-run in-browser.
BridgeThe 4 singularities = the 4 directions; the eye = the held zero )1(; the balancing −1 = the inside-out twin = the Klein flip as an index reversal; the funnel/waist of the two-bell image = an eye seen edge-on.
Speculative"Each direction is a torus with an eye" is the framing. Poincaré–Hopf and the index count are exact and stand alone; the mapping to n/s/e/w and to the seam is the construction's reading.