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Part IIUnitary Torus Supply

Abstract

Actual part iiunitary torus supply supplies the parameterized matrix, finite-field or ordered-product identities consumed by the whole-SU K5 width and intrinsic prescribed-product proof paths. All original hypotheses and actual carriers are retained; this source provides no oracle for bare automorphism recognition.

Theorem 1.1 (actual steinberg iff hermitian).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTorusSupply.actual_steinberg_iff_hermitian

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTorusSupply.actual_steinberg_iff_hermitian (✓ std3). ∎

Source. Repository-derived.

Commentary.

The literal anti-diagonal Hermitian matrix equation is equivalent to Fix(field/inverse-transpose); neither presentation assumes unitary coverage. This connects the torus arithmetic to actual SU witnesses.

Theorem 1.2 (actual even unitary upper torus values).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTorusSupply.actual_even_unitary_upper_torus_values

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTorusSupply.actual_even_unitary_upper_torus_values (✓ std3). ∎

Source. Repository-derived.

Commentary.

One genuine SU torus before every upper SU target, even ranks>=4. The complete finite-field separation proof is consumed, not assumed.

Theorem 1.3 (actual odd unitary upper torus values).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTorusSupply.actual_odd_unitary_upper_torus_values

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTorusSupply.actual_odd_unitary_upper_torus_values (✓ std3). ∎

Source. Repository-derived.

Commentary.

The same actual full-upper-unitary kernel in every odd rank>=3. The central norm-one diagonal correction is the proved arithmetic one.

The results concern special-unitary matrix groups over finite fields. Their geometric setting is Nikolov and Segal, On finitely generated profinite groups, II: Products in quasisimple groups, Annals of Mathematics 165 (2007), 239-273, DOI 10.4007/annals.2007.165.239. Prescribed-product statements retain their field-size, rank and automorphism hypotheses.

References