The Breathing Universe: 802_Mathematical Compendium and Operational Manual

This publication serves as the formal mathematical backbone and operational reference of the Breathing Universe Model (BUM). While earlier papers introduced the conceptual, physical, and phenomenological structure of the model, the present compendium consolidates all core definitions, relations, conservation laws, and limiting cases into a single, internally consistent mathematical framework. The model is based on a single physical substrate: a continuous vacuum capable of storing, redistributing, and coherently organizing tension. All physical phenomena are expressed through the dynamic vacuum-tension field H(t), which generalizes the relativistic constant c squared and governs the exchange between localized inertial mass and distributed field energy via the universal identity E = m · H(t). Conservation is enforced explicitly through the relation d/dt [ m(t) · H(t) ] = 0, ensuring that energy is neither created nor destroyed but continuously redistributed between matter and vacuum tension. The compendium defines the full hierarchy of BUM quantities, including vacuum tension, zero-line balance, field regimes (light, color, shadow, zero), operators, meta-operators, snapping thresholds, oscillatory coherence, and time-crystal phases. All symbols are standardized, dimensional consistency is enforced, and limiting cases are provided that recover Special Relativity, General Relativity, quantum correspondence regimes, and the Lambda-CDM cosmological limit. In addition to serving as a reference, the document functions as an operational manual. It specifies how the equations are to be used in analytical derivations, numerical simulations, and observational testing, including parameter constraints, reduction rules, and admissible approximations. No new physical mechanisms are introduced; the role of this work is to ensure mathematical closure, interpretive consistency, and reproducibility across the entire Breathing Universe research program.