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Canonical zeta mirror fundamental symmetry

Abstract

The same-height zeta-zero mirror lifts to an involutive self-adjoint isometry with explicit negative odd directions.

Theorem 1.1 (The mirror is self-adjoint in inner-product form).

Lean statement: D5/S3/Midline/Cayley/CanonicalZetaMirrorFundamentalSymmetry.mirrorFundamentalSymmetry_inner_left

Proof. Machine-checked in Lean as D5/S3/Midline/Cayley/CanonicalZetaMirrorFundamentalSymmetry.mirrorFundamentalSymmetry_inner_left (✓ std3). ∎

Source. Repository-derived.

Commentary.

The multiplicity-preserving mirror permutation is represented by the repository’s ell-two reindexing linear isometry.

Theorem 1.2 (Every moved mirror coordinate gives a strict negative direction).

Lean statement: D5/S3/Midline/Cayley/CanonicalZetaMirrorFundamentalSymmetry.mirror_odd_vector_strictly_negative

Proof. Machine-checked in Lean as D5/S3/Midline/Cayley/CanonicalZetaMirrorFundamentalSymmetry.mirror_odd_vector_strictly_negative (✓ std3). ∎

Source. Repository-derived.

Commentary.

Antisymmetrizing a coordinate basis vector produces a nonzero minus-one eigenvector of the mirror.

References