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Part IIUnitary Whole Prescribed Product

Abstract

For every positive q there are single constants N(q),C(q), chosen before all finite fields and ranks n>=6 above the field-cardinality cutoff. Arbitrary genuine compatible diagonal similitude and field-action tuples and positive divisors e(j)|q retain the original q/e(j). One actual SU correction tuple y is chosen before every actual whole-SU target b; x may depend on b. Each ordered value is x(j) inverse times ((alpha(j)*MulAut.conj(y(j) inverse)) to the q/e(j) power) applied to x(j). The genuinely proved K5 width closes whole-group coverage without a supplied decomposition premise.

Theorem 1.1 (actual uniform intrinsic whole q over e product).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryWholePrescribedProduct.actual_uniform_intrinsic_whole_q_over_e_product

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

Source. Repository-derived.

Commentary.

The separately proved absolute whole-SU width is genuinely consumed. There is no factorization, scalar-coverage or action-existence premise. This endpoint covers prescribed genuine D/Phi actions; recognition of arbitrary bare SU automorphisms remains a separate mathematical leaf.

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