Existence and Uniqueness of the Minimal Ternary Zero-Sum Structure: The 3 × 3 × 3 Queen-Line Geometry

Abstract

This paper studies the zero-sum display problem on the classical 3×3×3 queenline geometry over {0, 1, 2} 3 . The structure contains 27 points and 49 lines: 27 axis-parallel lines, 18 face-diagonals, and 4 space-diagonals. We characterize this structure by a corrected system of combinatorial axioms and prove: (1) The value algebra is forced to characteristic 3, and F3 is the unique minimal ternary alphabet; (2) Structure theorem: functions zero-sum on all 49 queen-lines are exactly affine functions, yielding a 4-dimensional solution space of 81 displays; (3) Duality theorem: 13 line directions and 13 projective classes of nonconstant displays form the projective plane PG(2, 3) via degeneracy incidence; (4) Coordinate reconstruction theorem: any structure satisfying the corrected axioms is isomorphic to the classical queen-line geometry, unique up to the full cube symmetry group of order 48. This version corrects errors in previous drafts regarding space-diagonal collinearity, degree sequence, automorphism group, and axiom strength, and removes conditions contradicting the classical structure.

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