The Circular Foundations of Energy–Mass Equivalence: Mapping Assumptions, Exposing Circularity, and Reframing E = mc²

This deposit contains the full text of The Circular Foundations of Energy–Mass Equivalence: Why E = mc² Was Used to Prove Itself, along with supporting diagrams and structural materials. The paper examines the historical derivations of energy–mass equivalence and demonstrates that they rely on four implicit assumptions — continuity, locality, metric stability, and context coherence — which already encode the equivalence they claim to prove. When any assumption breaks, the identity fractures. The work shows that E = mc² is not a universal primitive but a limit case of a more general structural relation: M̂ = g(Ê, Ĉ) Using the Carlo operator system (M̂ for stability, Ê for transformation, Ĉ for context, T̂ for transition), the paper maps how equivalence holds only within specific regimes and collapses in quantum, horizon, nonlocal, and discontinuous contexts. The circularity is illustrated through intuitive metaphors — such as a security camera trying to survey itself or a fridge trying to refrigerate a fridge — to make the self‑referential nature of the classical derivations clear. The deposit also includes a visual compendium of diagrams and structural maps that help readers approach the paper’s logic, including: operator diagrams transition engine schematics regime breakdown charts assumption fracture maps circularity loops context taxonomy diagrams These materials are provided to make the paper accessible to readers from physics, philosophy of science, and structural systems research. What the paper proves That historical derivations of E = mc² are structurally circular. That equivalence depends on four hidden assumptions. That breaking any assumption fractures the identity. That E = mc² is a regime‑specific limit case, not a universal law. That a general relation M̂ = g(Ê, Ĉ) resolves the circularity. How to approach this deposit Read the main paper first — it establishes the circularity argument and the operator system. Use the compendium diagrams — they make the structure intuitive and help visualise the assumptions and fractures. Refer to the operator definitions — they clarify how stability, transformation, context, and transition interact. Use the regime charts — they show exactly where equivalence holds and where it collapses. This deposit is intended as a complete, self‑contained introduction to the circularity problem and its structural resolution. This release includes a custom 3D Carlo Field visualiser that renders the collapse of the classical E = mc² loop and the emergence of the operator lattice, showing the transition from assumption‑bound equivalence to the full structural framework. 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