preprint · Zenodo (CERN European Organization for Nuclear Research)
A mathematical formulation termed the congruential potential and its related singular series unites four seemingly disconnected areas: prime pairs, Goldbach representations, quadratic forms, and dyadic dynamics. Numerical tests demonstrate strong agreement between empirical measurements and theoretical predictions. These tests include measured-to-predicted ratios of 0.9917, 0.9981, and 0.9993 for Sophie Germain primes up to ten to the power of eight. In addition, an external charge fluctuation of 1.9985 aligns closely with a predicted value of 1.9998 for Goldbach representations, and a two-adic grammar for Collatz dynamics achieves accuracy to four decimal places. New observations identified within this framework include a specific logarithmic slope signature for the mapping of n to 2n plus 1, a dyadic slope signature for Cunningham chains, and a uniform 3.3 percent logarithmic bias in Goldbach representations. The results represent verified phenomenology rather than formal mathematical proof.
Uncovering common structures across distinct problems in number theory helps researchers better understand the distribution of prime numbers and related dynamical systems. By linking Sophie Germain primes, Goldbach representations, and dyadic maps under one phenomenological framework, this work provides unified empirical patterns that can guide future theoretical investigations into classical, long-standing mathematical problems.
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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.22204698
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