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Part IIUnitary Odd P

Abstract

Actual part iiunitary odd p 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 (p).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.p

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.p (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized p 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 (actual p entry).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.actual_p_entry

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

Source. Repository-derived.

Commentary.

The parameterized actual p entry 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 1.3 (actual p multiply).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.actual_p_multiply

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

Source. Repository-derived.

Commentary.

The parameterized actual p multiply 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 p inverse).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.actual_p_inverse

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

Source. Repository-derived.

Commentary.

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

Definition 1.5 (Constraint).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.Constraint

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.Constraint (✓ std3).

Source. Repository-derived.

Commentary.

Actual central law for the vector quotient and true central layer.

Theorem 1.6 (actual p unitary iff).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.actual_p_unitary_iff

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

Source. Repository-derived.

Commentary.

The parameterized actual p unitary iff 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 1.7 (actual p upper).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.actual_p_upper

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

Source. Repository-derived.

Commentary.

The parameterized actual p upper 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 1.8 (actual p vector lift).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddP.actual_p_vector_lift

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

Source. Repository-derived.

Commentary.

Every vector in the actual P/P(2) has a genuine unitary lift. Diagonal central coordinates use relative trace; all off-diagonal coupling is retained, including characteristic two.

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