preprint · Zenodo (CERN European Organization for Nuclear Research)
A unified mathematical formula termed the congruential potential connects four traditionally distinct areas of number theory: prime pairs, Goldbach representations, quadratic forms, and dyadic dynamics. This formulation, paired with an associated singular series, provides empirical predictions that closely align with numerical measurements. Testing against Sophie Germain primes shows measured-to-predicted ratios approaching 0.9993 at one hundred million, while predictions for Goldbach representations closely match external charge fluctuations. In dyadic systems related to the Collatz process, the two-adic grammar matches observations to four decimal places. The framework also identifies specific logarithmic signatures in sequence slopes and a uniform 3.3 percent logarithmic bias within Goldbach representations. The findings constitute verified phenomenology based on numerical observation rather than formal mathematical proof.
Long-standing challenges in number theory, such as Goldbach representations and the Collatz process, are usually treated in isolation. By demonstrating that a single underlying mathematical structure accurately describes numerical behaviour across all four distinct domains, this work offers mathematicians a cohesive empirical baseline to explore hidden connections and guide future theoretical investigations.
The abstract does not indicate an application pathway or commercial use case for this theoretical research.
AI-generated from the published abstract. Always read the original work before citing.
Résumé Nous définissons le potentiel congruentiel S = ∑_q [k log(1 - 1/q) - log(1 - ν_q/q)] et sa série singulière 𝔖 = e^{-S}. Quatre secteurs apparemment disjoints — paires de premiers, représentations de Goldbach, formes quadratiques, dynamique dyadique — dérivent de cette seule forme. Validations : ratios mesure/prédiction 0.9917, 0.9981, 0.9993 pour Sophie Germain à 10^6, 10^7, 10^8 ; fluctuation de charge externe 1.9985 contre 1.9998 prédit (Goldbach) ; grammaire 2-adique exacte au dix-millième (Collatz). Trois remarques nouvelles : signature ln 2 de la pente de n → 2n+1 ; signature dyadique ℓ ln 2 des pentes de Cunningham ; biais logarithmique uniforme de 3.3% (Goldbach). Positionnement : phénoménologie vérifiée, pas preuve. Abstract We define the congruential potential S = ∑_q [k log(1 - 1/q) - log(1 - ν_q/q)] and its singular series 𝔖 = e^{-S}. Four apparently disjoint sectors — prime pairs, Goldbach representations, quadratic forms, dyadic dynamics — derive from this single form. Validations: measured/predicted ratios 0.9917, 0.9981, 0.9993 for Sophie Germain at 10^6, 10^7, 10^8; external charge fluctuation 1.9985 against 1.9998 predicted (Goldbach); 2-adic grammar exact to four decimals (Collatz). Three new remarks: the ln 2 slope signature of n → 2n+1; the dyadic signature ℓ ln 2 of Cunningham slopes; a uniform 3.3% logarithmic bias (Goldbach). Positioning: verified phenomenology, not proof.
This page summarises published work. The authoritative version sits with the publisher.
DOI: 10.5281/zenodo.22190260
Is something wrong with this record? Report it or request removal.
Discussion
Have you built on this work, tried to replicate it, or seen it applied in practice? Share what you know. Verified researchers and MARATTO™ domain experts can open a discussion, and any member can reply. Contributions are reviewed before they appear.
No discussion yet. Open the first thread.
New to MARATTO™? Create a free account.
