What question did this study set out to answer?

This framework aims to describe how physical systems are represented differently across various scales and constraints.

April 30, 2026Open Access

The Mechanics of All (TMOA): A Unified Framework for Scale-Dependent Physical Representation

Key Points

This framework aims to describe how physical systems are represented differently across various scales and constraints.
Introduces a projection operator for mapping underlying substrates into representations based on scale and cost constraints.
Defines a base field equation relating physical properties to scale and interaction effects.
Presents three empirical cases to support the framework's predictions.
Demonstrates hydrodynamic electron flow in graphene as a transition from particle to fluid representation.
Shows Gaussian boson sampling exhibits exponential representation cost divergence.
Confirms single-photon qutrit behavior, illustrating non-binary basis cardinality.

Abstract

The Mechanics of All (TMOA) is a unified structural framework describing how physical systems are represented across scale, constraints, and interaction hierarchies. The framework introduces the projection operator Πbasis (σ), which maps an underlying substrate ψ into representations Rb selected as a function of scale (σ) and cost constraints. TMOA formalizes that physical descriptions are not intrinsic properties of systems but regime-dependent projections. The base field equation is defined as: ∂²ψ/∂t² = c²∇²ψ + λ∂²ψ/∂σ² + N (ψ) where σ represents logarithmic scale and N (ψ) captures nonlinear interactions. This publication presents three independent empirical cases supporting the central prediction of TMOA: - Hydrodynamic electron flow in graphene (basis transition: particle → fluid) - Gaussian boson sampling (Jiuzhang 3. 0) demonstrating exponential representation cost divergence- Single-photon qutrit confirming non-binary basis cardinality These cases converge on a single invariant: the substrate remains unchanged while the representation varies with scale, constraints, and observer capability. Authorship and AI Notice: This framework, its concepts, predictions, and structure are entirely authored by Nuno Lopes (XCYT). Artificial intelligence tools were used strictly to assist in mathematical formalization, symbolic derivations, and consistency verification. AI did not originate the framework. Usage and Fairness: This work is intended to be freely used for research, education, and non-commercial purposes with proper attribution. If this framework is used in commercial products, systems, or services that generate revenue, users are encouraged to contact the author to arrange a fair contribution or licensing agreement. Contact: info@disrupt. it. com

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