BEYOND E = mc² — A Unified Structural Framework for Stability, Transformation, and Context

Version 2 is the definitive structural document that dismantles the classical energy–mass equivalence and replaces it with a general operator framework built on stability, transformation, and context. It contains the full Paper I, the complete operator foundations, the context and transition taxonomies, the breakdown regimes, the general relation M̂ = g(Ê, Ĉ), the geometry and dynamics layers, the model architectures, the programme roadmap, and the publication arc. At the centre of Version 2 is the showcase Carlo Equation — the unified structural equation governing stability, transformation, context, and transition across all physical regimes. It demonstrates explicitly that E = mc² is not fundamental but a limit case that appears only when continuity, locality, metric stability, and context coherence all hold. The Carlo Equation is expressed in its preliminary structural form as: \[ T̂(s) = F(M̂(s),\, Ê(s),\, Ĉ) \] with the core structural relation: \[ M̂ = g(Ê,\, Ĉ) \] This relation states that stability (M̂) emerges from transformation (Ê) filtered through context (Ĉ). When the context is smooth, stable, local, and coherent, the structural map reduces to the classical form: \[ g(Ê,\, Ĉclassical) = Ê/c^2 \] and the Carlo Equation reproduces the familiar equivalence: \[ E = mc^2 \] But when continuity, locality, metric stability, or context coherence fail, the invariant speed becomes \(c(x,t)\), the structural map becomes nonlinear, nonlocal, or multi‑valued, and equivalence fractures or dissolves entirely. Version 2 establishes the structural physics programme that supersedes E = mc². It shows that equivalence is conditional, not universal, and derives the deeper operator relation that governs classical, relativistic, quantum, horizon, nonlocal, discontinuous, and topological regimes. This version is the field opener: the document that formally launches the structural programme, defines its operators, sets its assumptions, maps its breakdown regimes, introduces the Carlo Equation, and outlines the full research trajectory. It is designed to be uploaded once, referenced indefinitely, and expanded only through future volumes. This release also includes the full Visualiser Suite, providing diagram‑based structural proofs demonstrating how E = mc² fractures under context deformation and how the Carlo Equation remains valid across all regimes. Keywords & Subjects: This work sits at the intersection of theoretical physics, structural physics, relativity, quantum theory, information theory, and cosmology, with specific relevance to energy–mass equivalence, operator frameworks, stability operators, transformation operators, context‑dependent dynamics, nonlocal regimes, metric instability, continuity breakdown, quantum gravity, horizon physics, black hole thermodynamics, early‑universe conditions, transition operators, and generalised conservation laws. It contributes to subjects including mathematical physics, foundations of physics, high‑energy theory, quantum information, geometric frameworks, and meta‑structural analysis of physical law, offering a unified operator‑level perspective that reframes mass and energy as emergent roles shaped by geometry, observer frame, and informational constraints. AUTHORSHIP NOTEThis field was developed through independent research. The Carlo Field, its operators, context taxonomy, transition architecture, and structural relations are original contributions by the author. Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com