preprint · Zenodo (CERN European Organization for Nuclear Research)
A mathematical formulation known as the unified congruential potential connects four previously separate areas: prime pairs, Goldbach representations, quadratic forms, and dyadic dynamics. Defined through a specific logarithmic summation over primes and an associated singular series, this expression yields quantitative predictions across these distinct mathematical fields. Computational tests demonstrate strong consistency between theoretical predictions and empirical measurements. For Sophie Germain prime pairs, accuracy ratios between measurement and prediction range from 0.9917 to 0.9993 across scales up to one hundred million. In Goldbach representations, external charge fluctuations reach 1.9985 compared to a predicted 1.9998, alongside an identified uniform 3.3 percent logarithmic bias. In dyadic systems, calculations align with two-adic grammar to four decimal places, accompanied by distinct logarithmic slope signatures. The framework is positioned as computationally verified phenomenology rather than formal mathematical proof.
Unifying disparate problems in number theory allows researchers to recognise shared structural behaviour across prime distributions, additive questions, and arithmetic dynamical systems. By testing theoretical formulas against large-scale computations, this approach demonstrates that seemingly isolated mathematical phenomena follow common quantitative regularities, providing a computational basis for future pure mathematics research.
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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.
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DOI: 10.5281/zenodo.22200635
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