slug: sharma-primitive-polynomial-conjecture-4-2
bibkey: sharma2026primitivepolynomials
doi: null
url: https://arxiv.org/abs/2608.07262v2
triage: theorem
motivation_gids:
  - D5/S3/ConceptDynamics/InformationEscape/TheoremUnit

Sharma’s primitive-polynomial Conjecture 4.2

Problem

The target is the literal Conjecture 4.2 in Avnish K. Sharma’s Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures, arXiv:2608.07262v2. For every odd prime (p), every finite field (K) with characteristic (p) and (|K|=p^2), and every (\lambda\in K) satisfying the source multiplicative-generator condition (\operatorname{ord}(\lambda)=|K|-1), the source claim says that (X^p+X+\lambda) is the monic minimal polynomial of a multiplicative generator in an extension (L/K) of degree (p). “Primitive” here means multiplicative order; it does not mean Polynomial.IsPrimitive.

The primary statement, printed page 16, reads (mathematical typography transliterated):

Let p be an odd prime. Then, for every primitive element λ∈F_(p²), the polynomial x^p+x+λ is primitive over F_(p²).

The negated statement proved here is [ \neg\bigl[\forall p\text{ odd prime},\ \forall K\text{ finite field of characteristic }p\text{ with }|K|=p^2,\ \forall\lambda\in K, \ \operatorname{ord}(\lambda)=|K|-1\ \Longrightarrow \exists L/K,\alpha\in L:\ [L:K]=p\ \land \operatorname{minpoly}_K(\alpha)=X^p+X+\lambda\ \land \operatorname{ord}(\alpha)=|L|-1\bigr]. ] The extension is a finite field extension. The minimal polynomial is monic. The existential Conjecture 4.1 is not the target.

The Lean definition fullClaim preserves all of these quantifiers and the source conclusion. The candidate coefficient is the source leading coefficient, so the public result is exactly Not fullClaim after unfolding the coefficient parameter.

Motivation

The source strengthens its existential Conjecture 4.1 to every primitive lambda when the base field has cardinality p squared. Its reported tests cover primes 3 through 37. A primitive lambda at p=41 for which no primitive extension generator has the stated minimal polynomial negates precisely that stronger universal claim.

Gap

The bounded source, library and ownership audit in #8851 found no inspected resolution of Conjecture 4.2 in its checked scope. The finite tests in Remark 4.2 do not establish the unbounded universal claim. The required formal gap is a primitive-lambda counterexample with a certificate applying to every hypothetical root in the source extension, not a sampled root.

Route

The refutation instantiates (p=41) with (K=\operatorname{QuadraticAlgebra}(\mathbb Z/41\mathbb Z,3,0)) and (\lambda=5+u), where (u^2=3). Eight kernel-checked coefficient-list product identities evaluate the certificate polynomial at any root α of (X^{41}+X+\lambda) and prove (z^{83}=\alpha). The same source proves (|K|=1681) and orderOf λ = 1680 for the source polynomial (X^{41}+X+\lambda).

Assuming the conjecture for this λ would produce a degree-41 extension (L), so (|L|=1681^{41}) and a primitive α of order (|L|-1). Finite-field exponentiation applied to (z), together with (z^{83}=\alpha), forces α to have order dividing ((1681^{41}-1)/83). This is a strict positive divisor bound below the claimed order, giving the contradiction and the full claim negation.

Evidence

The source theorem is D5/S3/ArithUnits/SharmaPrimitivePolynomialRefutation.result. Its private certificate_evaluation theorem supplies the universal root certificate at SourceLeadingCoefficient. The proof of result combines that certificate with the primitive-lambda witness and the hypothetical degree-41 extension to negate the full source claim. Information registration belongs to the separate Reg package under #5214; this D5 module has no inline registration or judge imports.

The utility header requests the typed closed-negation relation from D5/S3/ArithUnits/SharmaPrimitivePolynomialRefutation.fullClaim to D5/S3/ArithUnits/SharmaPrimitivePolynomialRefutation.result under certified-instance / refutes. The native producer owns its source-bound evidence. Following canonical Freeze, the existing result Describe binds this problem slug to that theorem through OpenProblemResolutionClaim(Refuted); the generated Blueprint projects the typed binding.

Falsifier

A defect in the source quantifiers, the order/minimal-polynomial definition, the degree-41 certificate, the eight-product Horner identities, or the finite extension cardinality argument would invalidate the route. A finite sample of primes or lambdas would not address the universal claim.

Triage

theorem, targeting the published Conjecture 4.2 under open-problem-resolution, subject to final review. The computational content is a certified-instance with the utility basis refutes: the conclusion is the negation of the full universal conjecture. The current dossier’s resolution is derived from that typed result binding. Final review and delivery remain pending; no merge or KPI claim is made.

ASSUMED-UNVERIFIED

The numerical discovery preceded preregistration #8851. The source hashes and bounded literature/library/ownership search results are in the Library note. No inspected resolution was found in the checked arXiv, OpenAlex and Crossref scope; Semantic Scholar was rate-limited. The repository ownership scan covered issue/PR bodies but not every comment. These limits do not establish worldwide absence of prior resolution, and no priority claim is made. Final independent review, required CI and ordinary merge remain caller-owned obligations.