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This paper aims to explain the Finite Field Kakeya Conjecture using foundational axioms within Prenary Mathematics.

July 27, 2026Open Access

V5-有限域挂谷猜想的偏元解释:从减法不可清零到多项式方法的必然性(V5-A Prenary Explanation of the Finite Field Kakeya Conjecture in the Prenary Mathematics Framework: From the Non-Zero Subtraction Axiom to the Necessity of the Polynomial Method)

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This paper aims to explain the Finite Field Kakeya Conjecture using foundational axioms within Prenary Mathematics.
Translated Dvir's polynomial method into Prenary axioms in discrete space.
Analyzed the relationship between polynomial degree, normalization constraints, and finite fields.
Provided three falsifiable predictions for independent verification.
The optimal polynomial degree d is identified as q^{n-1}, which serves as a natural upper bound related to normalization constraints.
The optimality of the Schwartz-Zippel Lemma is linked to the existence of minimal non-zero elements from the Non-Zero Subtraction axiom.
A specific dimension-dependent sequence for the size constant C_n is characterized, with C_2 = 1/2, C_3 ≈ 0.618, and C_4 ≈ 0.786.

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Provides a foundational axiomatic explanation for the Kakeya conjecture in finite fields, suggesting implications for primal mathematics.

Song Chen (2026) studied this question.

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