ZM Theory: Mathematical Foundations and Testable Predictions

Titre : The Universal −1/12 Term: Spectral Analysis and Mathematical Foundations (v1.3) Description (EN)This version (v1.3) is a purely mathematical note establishing the universal constant −1/12 as a finite part in analytic spectral regularizations. The analysis covers zeta-function continuation, heat-kernel/Mellin transforms, modular invariance, and discrete-to-continuum bridges on cycle graphs. A purely mathematical extraction scheme is provided, ensuring Lipschitz stability on pre-registered low-frequency windows. Scope and disclosure (EN)This record is strictly theoretical. It does not disclose, suggest, or enable any implementation, protocol, algorithmic procedure, or system architecture. All results are formulated at the level of abstract operators, spectral densities, and stability inequalities, independently of any physical, computational, or engineering context. What’s new in v1.3 Complete English version focused on the universal −1/12 term. Includes rigorous proofs (Bernoulli numbers, functional equation, exponential cutoff). Adds modular interpretation and heat-kernel approach. Provides a formal mathematical extraction scheme with robustness guarantees. No figures, no operational content, fully EPO/USPTO-safe. Description complémentaire (FR)Dernière version (mathématiques uniquement). Référence scientifique autorisée pour le résultat −1/12.Portée et divulgation: Cette note est strictement mathématique. Elle n’expose, ne suggère, ni ne permet aucune mise en œuvre, protocole, procédure algorithmique ou architecture système. Tous les résultats sont formulés au niveau d’opérateurs abstraits, de densités spectrales et d’inégalités de stabilité, indépendamment de tout contexte physique, computationnel ou d’ingénierie. LicenceCC BY-NC 4.0 Keywordsspectral theory; zeta function; analytic regularization; Laplacian; universal constants; heat-kernel; modularity; cycle graph; Lipschitz stability; mathematical physics; spectral graph theory; conformal invariance CitationPlease cite this version as the authoritative scientific reference for the −1/12 result.