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Part IIUnitary Upper Torus

Abstract

Actual part iiunitary upper torus 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.

Definition 1.1 (steinberg).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryUpperTorus.steinberg

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryUpperTorus.steinberg (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized steinberg statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Theorem 1.2 (steinberg entry).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryUpperTorus.steinberg_entry

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

Source. Repository-derived.

Commentary.

The true anti-diagonal unitary coordinate law.

Definition 1.3 (unitaryUpper).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryUpperTorus.unitaryUpper

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryUpperTorus.unitaryUpper (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized unitaryUpper statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Theorem 1.4 (actual unitary upper regular torus values).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryUpperTorus.actual_unitary_upper_regular_torus_values

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

Source. Repository-derived.

Commentary.

Actual upper-unitary VALUE coverage by one regular inner torus. The SAME determinant-one Steinberg-fixed h is chosen before ALL targets. Every witness remains upper-unitriangular and Steinberg fixed; the result is the exact unshifted noncommutative VALUE x^-1 h^s x h^-s. This is the concrete fixed-torus kernel, not arbitrary-auto uniformity.

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