Canonical-Height Gram of a Cubic Orbit

Abstract

Canonical height and Galois symmetry determine the constructed cubic-orbit Gram matrix.

Theorem 1.1 (Canonical-height block Gram matrix).

Lean statement: D5/S3/Factorization/Mordell/CubicOrbitCanonicalHeightGram.mordell_family_canonical_height_gram

Proof. Machine-checked in Lean as D5/S3/Factorization/Mordell/CubicOrbitCanonicalHeightGram.mordell_family_canonical_height_gram (✓ std3). ∎

Source. Repository-derived.

Commentary.

For non-torsion Mordell points with independent cubic rotations, the half-x-coordinate canonical height is positive at each point. Galois invariance and the horizontal three-rotation relation make distinct layers orthogonal and give each pair the block H times [[1,-1/2],[-1/2,1]], with positive determinant.

References

• Truth anchor: D5/S3/Factorization/Mordell/CubicOrbitCanonicalHeightGram.mordell_family_canonical_height_gram
• Dependency: D5/S1/Scale/GoldenCubicBlockCongruences
• Dependency: D5/S3/Factorization/Dedekind/GaloisScalarHeight
• Dependency: D5/S3/Factorization/Galois/GoldenCubicBlockPositiveRootTower
• Dependency: D5/S3/Factorization/Galois/GoldenCubicCommonInertiaAndSignature
• Dependency: D5/S3/Factorization/Mordell/CanonicalPointHeight
• Dependency: D5/S3/Factorization/Mordell/GoldenCubicBlockMordellTwists
• Dependency: D5/S3/Factorization/Mordell/PointVariableChange
• Dependency: D5/S3/Factorization/Mordell/SymmetricSquareAddition
• Dependency: D5/S3/Factorization/MordellTwoAdicNonTorsion
• Dependency: D5/S3/QuadraticForms/ParallelogramConstruction