Mináč–Nguyen–Tân Question 3.5

The primary source is Question 3.5:

Suppose that F has the n-sum property with respect to U. Is it true that F[G] also has the n-sum property with respect to U_{ε_G}?

The complete statement quantifies over a finite field F with more than two elements, every finite group G, a subgroup U of F's units containing minus one, and the original natural number n at least two. If every field element is a sum of exactly n elements of U, every element of the actual group algebra must be a sum of exactly n genuine units whose coefficient-sum augmentations lie in U. Repetition is allowed. Modular characteristic, noncommutative groups, characteristic two with more than two field elements, and the trivial group remain in scope.

Gardam's group-ring work provides the repository's existing native group algebra and genuine-unit context. This question concerns additive generation with a fixed number of terms and a restriction on each augmentation.

The paper proves several special cases and explicitly leaves arbitrary finite groups unresolved. The bounded literature and repository search associated with preregistration 15666 found no inspected exact settlement. The first-tier source classification and open-problem-resolution basis precede the Lean probe. No global priority claim follows from that bounded search.

The standard augmentation kills the Jacobson radical. The semisimple quotient is a product of finite matrix algebras. Its orthogonal factor idempotents select a simple factor on which augmentation is an injective unital ring map to F; that factor is identified with its scalar entries. The field hypothesis supplies exactly n prescribed scalar summands there. Rank normal form gives two-unit sums in every other factor, using a scalar distinct from zero and one; adding unit summands gives exactly n terms. Units lift through the nilpotent radical in the finite noncommutative ring. The lifting error can be added to one summand while preserving unit status and augmentation.

A mismatch in the native group-algebra or unit definitions, loss of a modular or noncommutative case, a changed number of terms, failure to preserve individual augmentations, or a nonstandard axiom would invalidate this settlement. A result restricted to coprime characteristic, abelian or solvable groups, or n times the group order would not settle the question.

D5/S3/ArithUnits/MinacNguyenTanQuestion35.result states precisely the source transfer using MonoidAlgebra, genuine units, and the standard augmentation. The result Describe carries OpenProblemResolutionClaim(Proved). The one result uses proof-local arguments; there are no companion theorems, private mathematical helpers, or wrapper results.

The source-specific unfinished-audit issue 15686 records the missing producer-accepted source binding and registration. The retained Reg/D5/S3/ArithUnits/MinacNguyenTanQuestion35.lean compiles the complete family, bridge, variation, sensitivity and dependence, including F = ULift (ZMod 3), G = ULift Unit, U = ⊤, n = 2 and a bottom-subgroup counterreadout. The attempted descriptor was rejected with producer exit 2 and contract.discovery:rigid_universes:Reg.D5.S3.ArithUnits.MinacNguyenTanQuestion35.registration_1. There is no declared_validated registration; the linked-issue route leaves the audit unfinished.

This is a first-tier externally named problem with admission basis open-problem-resolution. The mathematical unit is general and contains no bounded enumeration, checker, numeric reduction, or certified finite instance; its utility classification is none. Independent review, source-author audit completion or its linked-issue exception, canonical freeze, required checks, and integration are separate delivery obligations.

The author classifies the compiled proof as content, pending independent review. Preregistration 15666 anticipated bind-only; the original exact question was preregistered, not the content classification. The admission basis remains open-problem-resolution.

The literature search is limited to the versioned primary source and the bounded web and repository scope identified above. It does not establish worldwide absence of a prior proof. No resolution status is handwritten here; the canonical theorem and Scribe binding carry the mathematical result.