The proton as a dynamic relational structure. BKT-107-A01 / DSR

The Proton as a Dynamic Relational Structure. Self-Sustaining Organization, Internal Twist, a Metafield Node, and Physical Identification BKT-107-A01 / DSR v2.9 Abstract This work develops a mathematical and physical description of persistent structural organisation under autonomous energy exchange, taking the proton as a physical reference system. Within the PJM USC GTWSSF GTCW research programme, the concepts of relational closure, non-closure and contact are realised in explicitly specified models of variable couplings, geometric channels and fields. Energy conservation, retention, local tolerance, protection of the direction of circulation and orbital stability are characterised by distinct mathematical conditions. The main result concerns the full symmetric ring of six oscillators. For normalised energy H ≤ 6.1, modal angular momentum L₁ ≥ 4.2 and |η(0)| < 1, it is proved that |η(t)| < 1 and J_full(t) > 353√3/1350 ≈ 0.452899 for every t ≥ 0, where η is the geometric coordinate and J_full is the spatially averaged energy current. The certificate covers a family of states with additional modal excitations under a fixed symmetric coupling law, with all quantities expressed in dimensionless units. A numerical example in which one coupling is increased by 5% exhibits reversal of the mean current despite conservation of total energy. The geometry of the channel axis and its independent internal twist are connected to reduction error control through the equation residual and the operator graph norm. For total, stationary, spherically symmetric and conserved energy–momentum tensors of short range, in the Breit-frame representation and under specified regularity and moment-convergence conditions, the bounds r_D²/r_mech² ≥ 3/5 for non-negative normal stress and ≥ 5/7 with additionally non-negative shear are obtained. The radius r_D is defined by the slope of the D form factor, while r_mech is the mechanical radius. The metafield formalisation includes a common action, energy relative to a background and memory arising from the elimination of variables. Under the hypotheses 𝒢_M = 4π and R_cl = r_E, with the input radius r_E = 0.84075 ± 0.00064 fm, the conditional relation E_cl = 4ℏc/r_E gives 938.8141 ± 0.7146 MeV. Here 𝒢_M denotes the mass coefficient, while R_cl = r_E identifies the structural radius with the charge radius; the stated uncertainty includes only propagation of the uncertainty in that radius. The results provide a framework connecting geometry, dynamics, tolerance control and observation maps. The theorems apply to specified models, while identifying them with the proton entails additional conditions on quantisation, charge realisation and consistent predictions of observables using shared parameters. Keywords: proton; relational structure; Law of One Mechanism; metafield; autonomous energy circulation; variable couplings; relational closure; internal twist; orbital stability; reduction error control; energy–momentum tensor; form factors; identifiability. Introduction The persistence of the proton presents a particularly demanding problem in the study of dynamic physical structures. Its recognisable identity coexists with internal interactions, correlations and variation in its local description. In quantum chromodynamics, the proton is a dynamic bound state of quark and gluon fields. The relational perspective focuses on the organisation of couplings and flows, and on the conditions under which a specified class of organisation can preserve its properties while its local realisation evolves. The aim of this work is to formulate this question in computable terms, encompassing equations of motion, invariants, tolerance bounds and maps to measurable quantities. The study forms part of the PJM USC GTWSSF GTCW research programme, developed around the Law of One Mechanism. The programme seeks common principles governing the formation, reorganisation and stabilisation of physical structures. In the approach considered here, structural identity is associated with the preservation of a class of admissible relations, while amplitudes, geometry, twist and rates of energy exchange may change. Information has an operational meaning: it denotes specified correlations, dynamical constraints and the way a state is read out. A common mechanism is examined through an evolution generator, specified reduction maps and relations between parameters across sectors of the description. Annex BKT-107-A01 develops the definitions of BKT-107/CND-EST in mechanical and field models. Closure denotes conformity with a specified physical class, non-closure is measured by a defect of that conformity, and the dynamical form of relational contact corresponds to explicit coupling between channels. Distinguishing these concepts allows energy balance, retention, local tolerance, current direction and orbital stability to be studied separately. Self-sustaining organisation here means the preservation of organisation at zero external power and with non-zero structural quantities. Energy can be exchanged between sectors of the system while the total energy remains conserved. Autonomous ring models with variable couplings provide constructive realisations. For a specified family of states of the full symmetric ring, preservation of the admissible range of the geometric coordinate and positivity of the mean energy current are proved for all time. This protection also includes additional modal excitations satisfying the stated energy conditions. A separate result concerns the orbital stability of the circulating orbit that minimises energy at fixed non-zero modal angular momentum. The analysis of non-uniform coupling also supplies a concrete example in which protection of the circulation direction is lost while energy balance is preserved. The theorems and numerical examples thus establish conditions for persistence and mechanisms by which they can be violated. The transition from geometry to dynamics involves distinguishing the curvature of a channel’s axis from its internal twist. The field operator, projection onto collective variables and equation residual determine a controlled reduced description. Under explicit assumptions on regularity, coercivity and feedback coupling, estimates of the local twist error and conditions for tolerance preservation are obtained. The metafield formalisation develops the hypothesis of a shared substrate for fields and couplings through a common action, an energy–momentum tensor and the energy of a configuration relative to its background. Eliminating some variables introduces memory into the dynamics of the observed subsystem. A separate φ⁴ field model in 1+1 dimensions provides an example of nonlinear perturbation control with translational freedom taken into account. Connecting the construction to the proton involves geometry, quantum numbers and responses to physical probes. Analysis of the energy–momentum tensor yields conditional constraints on stress profiles, mechanical radii and derivatives of the D form factor. The mass–radius relation is examined under explicit hypotheses concerning normalisation and the identification of radii. The geometry of knots and braids serves a heuristic role, while its connection to the quark and gluon description remains a hypothesis requiring quantitative identification. Certificates for preservation of a model class and predictions of the physical proton’s lifetime are distinct questions. The physical content of the proposed description is specified by a common map from parameters to mass, charge, spin, and electromagnetic and gravitational form factors. Parameters determined in calibration retain the same values in predictions for independent data, with correlations and reduction error taken into account. This criterion allows the construction to be assessed as a representation of the same physics, an effective approximation or a source of distinct predictions. The contribution of this work is to connect the specification of a relational class, the law of motion, dynamical certificates, reduction control and observation maps within a single framework for studying structural persistence.