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Diagonal Sign Inertia

Abstract

The inertia of a real diagonal Hermitian form is exactly its coordinate sign count.

Definition 1.1 (Real diagonal Hermitian form).

Lean statement: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.realDiagonal

Formalization. D5/S3/Weil/Pick/DiagonalSignNegativeIndex.realDiagonal (✓ std3).

Source. Repository-derived.

Commentary.

Real coefficients are embedded as the diagonal entries of a complex Hermitian matrix.

Definition 1.2 (Positive coordinate count).

Lean statement: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.positiveWeightCount

Formalization. D5/S3/Weil/Pick/DiagonalSignNegativeIndex.positiveWeightCount (✓ std3).

Source. Repository-derived.

Commentary.

This finite count records diagonal entries that are strictly positive.

Definition 1.3 (Negative coordinate count).

Lean statement: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.negativeWeightCount

Formalization. D5/S3/Weil/Pick/DiagonalSignNegativeIndex.negativeWeightCount (✓ std3).

Source. Repository-derived.

Commentary.

This finite count records diagonal entries that are strictly negative.

Theorem 1.4 (Diagonal inertia equals coordinate sign counts).

Lean statement: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.real_diagonal_inertia_eq_sign_counts

Proof. Machine-checked in Lean as D5/S3/Weil/Pick/DiagonalSignNegativeIndex.real_diagonal_inertia_eq_sign_counts (✓ std3). ∎

Source. Repository-derived.

Commentary.

Coordinate projectors give lower bounds for both signs, and the rank partition forces both bounds to be equalities.

Theorem 1.5 (Diagonal negative index counts negative weights).

Lean statement: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.real_diagonal_negIndex_eq_negative_count

Proof. Machine-checked in Lean as D5/S3/Weil/Pick/DiagonalSignNegativeIndex.real_diagonal_negIndex_eq_negative_count (✓ std3). ∎

Source. Repository-derived.

Commentary.

The negative component of the inertia package is exposed as a direct consumer theorem.

References

  • Truth anchor: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.negativeWeightCount
  • Truth anchor: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.positiveWeightCount
  • Truth anchor: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.realDiagonal
  • Truth anchor: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.real_diagonal_inertia_eq_sign_counts
  • Truth anchor: D5/S3/Weil/Pick/DiagonalSignNegativeIndex.real_diagonal_negIndex_eq_negative_count
  • Dependency: D5/S3/SpectralTopology/PointGapExactInertia