What does this research mean for the field?

Key mathematical results concerning minimal logarithmic energy on the 2-sphere, which are central to Smale's 7th Problem, can be comprehensively synthesized and formally verified using the Lean 4 interactive theorem prover. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To review continuous potential theory and derive discrete logarithmic energy minima for spherical point configurations addressing Smale's 7th problem.

August 30, 2026Open Access

Minimal Logarithmic Energy on the 2-Sphere and Smale's 7th Problem

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CNCharles EDOU NZE

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Implication

Mathematical monograph demonstrates minimal logarithmic energy bounds on the 2-sphere, highlighting complete machine-checked formal verification in Lean 4.

Key Points

To review continuous potential theory and derive discrete logarithmic energy minima for spherical point configurations addressing Smale's 7th problem.
Analyzed exact algebraic pairwise distances and interaction energies for Platonic solid configurations (N = 2, 3, 4, 6, 8, 12).
Applied continuous potential theory, the Brauchart-Hardin-Saff asymptotic expansion, and Riesz s-energy transitions on the 2-sphere.
Formally certified spherical metric identities, combinatorial pair counts, and energy integral bounds using the Lean 4 interactive theorem prover with zero axioms and zero sorry placeholders.
Demonstrated that the normalized Lebesgue surface measure is the unique equilibrium measure with continuous energy integral I(σ) = 1/2 − ln 2.
Established exact discrete energies for regular polyhedral arrangements, including E = −3 ln 2 for the regular tetrahedron (N = 4).
Achieved 100% machine-checked verification of geometric distance identities and continuous energy bounds without warnings or unproven steps.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a93f0f56c1a8fb52e79d797 https://doi.org/10.5281/zenodo.22148779

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Implication

Mathematical monograph demonstrates minimal logarithmic energy bounds on the 2-sphere, highlighting complete machine-checked formal verification in Lean 4.

Minimal Logarithmic Energy on the 2-Sphere and Smale's 7th Problem

Discussion

Member takes

Implication

Key Points

Cite This Study

Minimal Logarithmic Energy on the 2-Sphere and Smale's 7th Problem

Discussion

Member takes

Implication

Key Points

Cite This Study

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