preprint · Zenodo (CERN European Organization for Nuclear Research)
A mathematical framework termed the congruential potential and its associated singular series provides a unified foundation across four domains: prime pairs, Goldbach representations, quadratic forms, and dyadic dynamics. Numerical tests demonstrate close alignment between theoretical predictions and empirical measurements. For Sophie Germain prime counts up to one hundred million, measured-to-predicted ratios reach 0.9993. In Goldbach representations, the external charge fluctuation measures 1.9985 compared to a predicted value of 1.9998, alongside a uniform 3.3 per cent logarithmic bias. In dyadic dynamics linked to the Collatz problem, two-adic grammatical predictions match observations to four decimal places. The formulation reveals distinct slope signatures involving the natural logarithm of two across specific integer sequences and Cunningham chains. The results establish a verified phenomenological model rather than formal mathematical proof across these number-theoretic systems.
Unifying disparate problems in number theory under a single mathematical formula helps researchers understand hidden connections between prime distribution, additive number theory, and dynamical systems. While the work presents verified numerical phenomenology rather than rigorous mathematical proofs, highly accurate predictive models can guide future formal inquiries and reveal systematic structures within longstanding unsolved 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.22190259
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