Octonions and the Cost of Integrability: Epit Realizations, First Jets, and Rational Curves

Octonion multiplication gives the round six-sphere a natural almost-complex structure JO, which is not integrable, while recent constructions provide integrable complex structures on the smooth six-sphere. We measure how far an integrable structure J must depart from JO to first order. With the round metric and its Levi-Civita connection fixed, we describe, for each tensor value, the linear space of first derivatives realizable by integrable germs, and project the octonionic derivative onto it. On the unit sphere the sharp local infimum is 12: every integrable germ satisfies ∥∇(J −JO)∥2 ≥ 12, and explicit germs approach this value. An exact rational certificate proves the unrestricted lower bound, giving the sharp local factor 1/2 on strict nearly Kähler six-manifolds. For metric-compatible competitors the sharp pointwise value is 16; on a closed strict nearly Kähler manifold it is attained globally exactly when a parallel complex line exists, which in the simply connected case singles out the standard nearly Kähler CP3 and flag manifold F(1, 2). A submanifold complex for both JO and an agreeing integrable structure has real dimension at most two. An explicit resolved-conifold neighborhood attains this dimension bound and agrees with JO on all six tangent directions along a great two-sphere. Such agreement forces a unique least firstorder correction of squared norm 40/a2 at radius a, whose integral 160π is conformally invariant. For an embedded rational curve C in a complex threefold X with normal bundle O(ℓ − 1) ⊕ O(−ℓ − 1), ℓ ≥ 0, the infimum of this integral over smooth matchings with the reference sphere is 160π + 16πℓ; it is attained exactly when the tangent–normal sequence of C splits holomorphically. The value 160π is attained exactly for the resolved-conifold neighborhood germ, so the optimized energy distinguishes holomorphic neighborhoods that are smoothly equivalent.