The Circular Foundations of Energy–Mass Equivalence: The Carlo Field Compendium and Interactive Operator-Space Visualiser

E = mc² is treated as a self-evident universal law, but its historical derivations quietly assume the thing they claim to prove. This deposit compiles the full Carlo Field framework — the paper that makes this argument, three supporting compendia, and an interactive 3D tool for exploring the structure directly — into a single 31-page edition plus a standalone browser-based visualiser. The argument The paper, The Circular Foundations of Energy–Mass Equivalence: Why E = mc² Was Used to Prove Itself, shows that both classical and relativistic derivations of mass–energy equivalence depend on four unstated structural assumptions: continuity (A1), metric stability (A2), locality (A3), and context coherence (A4). Because classical energy definitions already embed mass, and relativistic geometry already guarantees continuity and locality, every standard derivation encodes the conclusion into its premises before the argument begins — "mass and energy are equivalent because E = mc²; E = mc² is universal because mass and energy are equivalent." The paper illustrates this with two structural metaphors: a security camera that can only survey the part of itself already in frame, and a fridge that cannot refrigerate itself without an external gradient. To resolve the circularity, the paper introduces the Carlo Field operator system — M̂ (stability), Ê (transformation), Ĉ (context), and T̂ (transition) — and reframes equivalence as the limit case of a general relation, M̂ = g(Ê, Ĉ), which reduces to E = mc² only when all four assumptions hold. Break any one of them and the identity fractures: continuity breaks under quantum discontinuity, metric stability breaks near horizons, locality breaks in nonlocal regimes, and context coherence breaks under discontinuous or topological conditions. What's inside the compendium Part I — the foundational paper. The full argument above, with the abstract, all eight sections, and the regime-dependence table showing exactly where equivalence holds (classical, relativistic) versus fractures (quantum, horizon, nonlocal, discontinuous). Part II — Compendium One: structural diagrams, equations, maps & metaphors. Redraws every argument from the paper as a visual structure: the core operator triangle, the transition engine T̂(s) = F(M̂, Ê, Ĉ), the derivation of E = mc² as a classical limit case, the circular-derivation loop diagram, the security-camera and fridge metaphors as illustrated cards, the context taxonomy, and the full fracture map tying each broken assumption to its regime. Part III — the Advanced Expansion Compendium. Formalizes the framework as an operator algebra: commutation relations ([M̂,Ê] = iĈ, [Ê,Ĉ] = iT̂, [Ĉ,M̂] = iÊ) forming a closed algebraic loop, operator norms, a discrete numerical update model, regime-dependent forms of g(Ê,Ĉ) (including the quantum, horizon, and nonlocal variants), and six applications — regime classification, collapse detection, stability analysis, context engineering, transition mapping, and Carlo-Field simulators. Part IV — Simulator Architecture, Engine Spec & DSL. A complete computational specification: the eight-component simulator architecture (core engine, model layer, regime manager, collapse monitor, recorder, visualiser, config loader, CLI), the engine's method interfaces, and a full domain-specific language (.carlo files) for declaring operators, regimes, parameters, and collapse thresholds — with a worked example model. The interactive visualiser carlo_field_visualiser.html is a self-contained, dependency-free 3D scene (Three.js, opens in any modern browser, nothing to install) that turns the operator algebra into something you can manipulate directly. The four operators sit at the vertices of a tetrahedron, connected by all six relational edges, with particles flowing continuously along the M̂–Ê–Ĉ triangle to represent transition as the framework's structural engine rather than a side effect. Selecting a regime from the control panel doesn't just change a label — it visibly fractures the specific edge tied to the assumption that regime breaks, updates the live g(Ê,Ĉ) formula for that regime, and flips the central E = mc² readout to E ≠ mc² with a glitch animation. A collapse trigger simulates ‖T̂‖ → ∞. Clicking any node opens its full definition; drag to orbit, scroll or pinch to zoom, and keyboard shortcuts (1–6 for regimes, Esc, R to reset) are supported throughout. It degrades gracefully to a static reference page if WebGL is unavailable. Files included Carlo_Field_Compendium.pdf — the full 31-page compiled edition carlo_field_visualiser.html — the interactive companion (open directly in a browser) Author Matthew Carlo, Cognitive Engine Architect, Holyhead, Wales, United Kingdom. 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. Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com