The Carlo Unified Engine Mathematical Specification

This document presents the complete mathematical specification of the Carlo Unified Engine, aunified conceptual framework that formalises the internal structure and behaviour of the CarloEngines. Version 1.0 defines the shared architecture that underlies the ReGenesis Engine (RE), theRebirth Engine (RB), the Continuum Engine (CE), the Blackout Equation (BE), the Carlo VisualLanguage (CVL), and the Universal Carlo Pseudocode (UCP). The unified engine is expressed as a tuple U = (S, O, U_rule, C_rule, R_rule, A, V), where S is thestate space, O is the operator set, U_rule defines update behaviour, C_rule defines collapseconditions, R_rule defines regeneration behaviour, A defines attractor structures, and V defines thegraph‑based visual representation. Each engine is a specialisation of this general form. The specification includes: • A formal definition of the state space S, including configurations, trajectories, and structuralelements. • A complete description of the operator set O, where each operator is a function o: S → S capableof modifying values, labels, structure, or lifecycle state. • The ReGenesis Engine (RE), defined through neighbourhood functions, evaluation functionals,selection rules, regeneration dynamics, and attractor behaviour. • The Rebirth Engine (RB), defined through lifecycle labels, transition rules, absorbing archivedstates, and structured lifecycle evolution. • The Continuum Engine (CE), defined through incremental operators, continuity constraints,discrete time evolution, and convergence to attractors. • The Blackout Equation (BE), defined through a coherence functional C(c), a threshold θ, and areset map governing collapse and recovery. • The Carlo Visual Language (CVL), which maps states to diagrams and defines operator‑inducedgraph transformations, attractor diagrams, and collapse as graph contraction. • The Universal Carlo Pseudocode (UCP), which provides the unified operational loop that selectsengines, applies operators, performs updates, triggers collapse, executes regeneration, andproduces visual representations. Together, these components form a single, internally consistent mathematical system. All behaviourdescribed in the document follows strictly from the definitions provided. No physical, cognitive, ormetaphysical claims are made; the framework is purely conceptual and self‑contained. This specification serves as the foundational reference for all Carlo Engine implementations,simulators, visualisers, and theoretical extensions. unified engine, mathematical specification, conceptual framework, state space, operators, update rules, collapse rules, regeneration rules, attractor structures, visual representation, ReGenesis Engine, regeneration dynamics, neighbourhood functions, evaluation functional, selection rules, attractor sets, Rebirth Engine, lifecycle transitions, lifecycle labels, absorbing states, Continuum Engine, incremental evolution, continuity constraints, discrete time dynamics, Blackout Equation, coherence functional, threshold collapse, reset map, Carlo Visual Language, graph representation, diagram transforms, graph contraction, attractor diagrams, Universal Carlo Pseudocode, unified operational loop, engine selection, operator application, collapse trigger, regeneration trigger, trajectory evolution, composite system, structured change, configuration dynamics, transformation processes, coherence thresholds, regeneration operators, convergence behaviour, abstract engines, internal logic systems, formal definitions, non‑physical models, conceptual mathematics, Carlo Unified Engine