The Closed Family and the Cornell Structure ofthe Pole

AbstractThe order parameter of the area law of the F_N family of Möbius strips, φ(r) = 1 − 3√3/r,takes the exact values −2 at the collapse (r = √3), 0 at the critical point (r = 3√3) and +1 atinfinity. The two extremes of the family are therefore the two orientations of a triangle,one filled and one empty, and their superposition is the hexagram. Closing the family onitself with the matter–void inversion φ → −φ makes it an anti-periodic, second-orderMöbius strip F², whose half-twist is the passage through the critical point; itscosmological counterpart is a cyclic model in which two full cycles are needed to return tothe initial state. Finally, the exact Laurent expansion of the tangent at its π/2 pole, tan(π/2− ε) = 1/ε − ε/3 − ε³/45 − …, has the form of an inverse-distance term plus a linear term, theform of the Cornell potential of quark confinement. The identification with confinement isproposed, not derived. An addendum (October 2026) records prior work on trigonometricextensions of the Cornell potential, the exact global form found later, and which remarksof the original note do not survive.Keywords: Möbius strip; second-order Möbius strip; anti-periodic closure; orderparameter; hexagram; conformal cyclic cosmology; Laurent expansion; tangent pole;Cornell potential; quark confinement; F_N framework