Foundations of a relational description of the proton. QCD, topological quantisation and tests across scales.

BKT-107-A02Foundations of a relational description of the proton. QCD, topological quantisation and tests across scales BKT-107-A02-EN-20261108-v1.6 Introduction The persistence of a physical structure is a property of its dynamics, conservation laws and admissible responses to interactions. Mass, radius and a selected form factor describe different aspects of the same configuration, but a single observable does not determine its entire organising mechanism. This problem is particularly relevant to the proton, whose properties simultaneously include baryon number, spin, mass, electromagnetic and axial currents, and distributions of energy and stress. A consistent model connects these quantities through a common action and specified quantisation rules. The PJM-USC-GTWSSF-GTCW programme considers the hypothesis of a common relational and informational organisation of physical processes. In this work, the programme's concepts receive operational definitions. Closure means that a configuration belongs to a class specified by explicit consistency constraints. Non-closure is a quantitatively defined violation of those constraints. Relational contact describes a specified coupling or response between channels, while a channel denotes a map connecting a state to selected operators and measurable quantities. Relational information denotes the distinguishability of configurations through observables, correlators and response functions. The distinction between the consistency of a configuration and its persistence can be expressed through two conditions. The consistency defect vanishes if and only if the configuration belongs to the defined class of states. Persistence of that class means that the evolution of every state belonging to it remains within the same class for the times under consideration, within the interval of existence of the solutions. The first condition defines configuration consistency, while the second specifies its preservation under the dynamics. The passage from the first condition to the second is the content of a theorem about a particular generator. This formulation leads to calculable questions concerning energy bounds, persistent circulation, admissible perturbations and the robustness of observable properties. BKT-107 introduces the framework of relational closure, non-closure and contact, together with its associated estimators. BKT-107-A01 develops dynamical realisations, variable couplings, reduction with memory and the description of stresses. The BKT-107-A02 supplement connects this framework to the definition of an effective action, the operator description of QCD, topological quantisation and conditional relations across scales. Its central objective is to establish a sequence from a defined class of states, through dynamics and operators, to jointly constrained predictions. The ring of six oscillators provides an explicit mathematical model in which circulation protection can be proved for an entire class of initial states. The sharp bound and its rational certificate distinguish a theorem about a family of trajectories from the observation of a single simulation. Controlled symmetry breaking leads to quantitative protection conditions over finite horizons. Changes in coupling are accompanied by an account of both the modified dynamics and the modified current operator. The complementary reduction framework with memory determines how eliminated degrees of freedom, initial preparation and truncation affect the error of an effective description. In the proton sector, the construction is based on QCD operators and chiral effective actions. External sources define currents and the energy-momentum tensor, giving masses, moments, radii and stresses explicitly specified meanings. Topological baryon number, classical configuration persistence and fermionic quantisation constitute distinct components of the construction. Soliton calculations connect energy, inertia and current responses at fixed action coefficients. This exposes limitations of the minimal current model even when the masses used for calibration are reproduced. Comparing a contact interaction with a finite-range kernel determines the effect of the local approximation on the classical energy balance. The variational principle specifies the direction of the change relative to the infimum of the nonlocal functional. An additional testing domain comprises energy-momentum tensor form factors and baryon transport. Positivity criteria and complete uncertainty information constrain admissible stress reconstructions. Distinguishing radii associated with different operators enables the comparison of observables with different physical meanings. Baryon transport, considered also in relation to STAR measurements, introduces a confrontation channel distinct from the nucleon's static properties. The cosmological extension examines how quantities from different epochs can be connected through the parameters of one specified field model. In the canonical realisation with a quartic interaction, the scale of an ideal Thomas-Fermi core is related to the amplitude of an earlier radiation-like phase. This relation has explicit conditions concerning the solution branch, gradients, state preparation and thermal history. The analysis of a common mediator supplements it with constraints on response-matrix rank, pole structure and the transport of coefficients across scales. The contribution of this work is to connect explicit realisations, rigorous persistence conditions, controlled approximations and identifiable relations between observables. Separating calibration data from independent tests determines which agreements follow from parameter selection and which test the mechanism itself. In hadron physics, the framework assesses the joint consistency of masses, currents and stresses. In effective theory, it controls reduction and locality, while in cosmology it connects structural constraints to the early energy budget. The relational hypothesis thereby takes the form of a set of theorems and relations whose scope, accuracy and possible physical implications are specified by equations.