The Arithmetic Origin of Black Holes and the Emergence of Time — with the Hourglass Framework, 14-Operator Calculator, and Field Taxonomy

This work presents the Arithmetic Black Hole Model (ABHM), a structural framework mapping the multiplicative depth function Ω (n) to a gravitational shell geometry, with primes at the horizon, composites in the interior, and the singularity at n=0. It develops analogies to Hawking temperature, time dilation, and black hole entropy within the arithmetic lattice. Part XIX introduces the Hourglass Framework: a two-family operator structure unifying the power mean family (Pythagorean, Harmonic, Tropical) and the shift/Diamond family under one master formula Ω₂, ₊ (a, b). The distinguished additive node at (c=0, k=1) is identified exactly. The integer n=2 is proved to be the unique positive algebraic marker satisfying n+n = n×n. The interpretive synthesis INTEGER + GEOMETRY = TIME is presented with explicit epistemic labeling separating exact results from geometric interpretation. This release includes the full paper (ABHMFinalᵥ2. docx), a 12-slide presentation of the 14-operator framework (operatorsₚresentationᵥ2. pptx), an open-source browser-based calculator implementing all 14 operators with inverses (diamond-calculator-v2. html), and a README inviting the community to build upon this work. Companion papers: Classification Law (10. 5281/zenodo. 18756471) and Sieve Firewall / RH Analysis (10. 5281/zenodo. 18854321). The Arithmetic Black Hole Model: A Structural Analogy on the Multiplicative Semigroup of Integers https: //zenodo. org/records/19078891 Open source. Public prior art established March 19, 2026. Soli Deo Gloria.