An Inside–Outside Symmetric Geometric Construction of the Calabi Conjecture and the Dimensional Boundary: Four-Dimensional Ricci Flatness, the Algebraic Equivalence of the Quartic, and the Open Problem of Curvature in Five Dimensions

This paper gives an inside–outside symmetric geometric construction of the Calabi conjecture in its simplest case (an elliptic curve coiled into a torus) and establishes a dimensional boundary for the validity of its concepts. The argument rests on a law of dimensional ascent: symmetry and rotation are one concept; dimension grows by +1, each ascent adding one global symmetry axis (a half-twist, J²=−1) that unifies the pair of opposites the previous dimension was forced to distinguish (front/back, inside/outside); n-dimensional space is divided by its n global ± symmetries into 2ⁿ subspaces, the fourth ± being the curvature inside–outside K⁺/K⁻; the "real 2n dimensions" of complex structures is a repackaging of the +1 ladder. Ricci flatness is identified as the unique non-metric value of curvature, designed exactly for the four-dimensional situation where distance has failed — a failure attested by three independent directions: the geometry of four-dimensional folding (two faces of one point meeting at infinity; sphere eversion impossible without breaking in 3D, smooth in 4D, with Smale's self-intersection theorem as the signature of the missing dimension), the mainstream topological turn (Donaldson/Freedman invariants, exotic R⁴), and the Galois structure of the quartic (the solvable chain's key layer V₄ being pure distance-free flips, against the rotation C₃ of the cubic). Mirror symmetry forced by the ζ-apex V=1/2 makes inside and outside curvatures cancel pointwise (K⁺+K⁻=0), yielding Ricci flatness as geometric necessity rather than as the solution of a Monge–Ampère equation, with the quartic's radical solvability as its exact algebraic equivalent. An operational criterion follows: the true dimension of a construction is the dimension of its operational rules, not its coordinate count; since the radical system is capped at degree four (Abel–Ruffini), constructions such as the E₈ and Leech lattices are deep recursions of four-dimensional rules over many coordinate copies — depth mislabeled as height — their computability itself locking the dimension (shape preservation across coordinate copies is the precondition of unity and hence of computation). The paper closes by posing, with two-sided honesty, the open problem of whether the Calabi question still holds and still has meaning in five dimensions and above, where the very concepts of curvature and balance may fail, the boundary between four and five being marked simultaneously by geometry and by the unsolvability of the quintic. Bilingual (English and Chinese PDFs).