BKT–37Y07 Operator Identifiability, Renormalized Wilson Junctions, and Hierarchical Closure of the Proton

Abstract

Operator Identifiability, Renormalized Wilson Junctions, and Hierarchical Closure of the Proton The BKT–37Y07 publication constitutes the next stage of the multipart proton research programme developed from the principal BKT–37 article within the PJM–GTWSSF–USC–GTCW framework. The main BKT–37 paper introduced the proton as a stable and dynamic relational-structural node physically realized by QCD. The subsequent studies were not independent variants of the same claim. Instead, they developed distinct layers of the problem: the physical configuration space, the Dirac operator, the energy–momentum tensor, GPDs and GFFs, orthogonal-residual geometry, resonance transitions, identifiability conditions, and a programme of falsification. Within this genealogy, BKT–37Y01 developed the description of the proton as a dynamic coupling node and introduced the configuration space, Dirac-operator spectra, the EMT/GFF formalism, and the foundations of orthogonal-residual projection. BKT–37Y02 organized the matching between QCD and the effective description, with particular attention to total energy–momentum conservation and the prevention of double counting. BKT–37Y03 formalized the gauge quotient, admissibility domain, path classes, return operator, and the monodromy problem. BKT–37Y04 extended the model to a resonance-relational description and separated physical states, operators, and their observable projections. The BKT–37B and BKT–37C1 appendices developed the USC state space, closure tensors, effective action, metrology, multichannel residual geometry, covariance structure, and out-of-sample validation. BKT–37E audited the mass–radius branch and the 4ℏc/rE relation, whereas BKT–37F established a form-factor-based sphericity-defect protocol. BKT–37Y05 integrated the multilayer model, the proton–Roper transition, the programme of three Wilson channels, and its extension to light nuclei. Finally, BKT–37Y06–SYN consolidated these modules into an integral proton model, separated Standard-Model consistency from a possible residual beyond QCD, and established a registry of frozen predictions and negative controls. BKT–37Y07 does not mechanically accumulate the claims of the preceding publications. It critically reconstructs them in the language of quantum field theory, operator theory, statistical geometry, and full multisector validation. Concepts that previously had a pictorial or interpretive role are replaced with operational objects. A “layer” denotes the response sector of a specified operator rather than a material concentric shell. A “channel” denotes a class of gauge-invariant transport and coupling rather than a classical conduit. “Information” denotes the structure of admissible relations and constraints rather than an additional substance or energy carrier. The main BKT–37 article therefore remains the ontological and model-level point of departure, whereas BKT–37Y07 translates its principal postulates into operator, statistical, and experimental conditions. The publication presents an integrated theoretical and methodological study of the operator identifiability of proton structure relative to a comprehensive reference model incorporating QCD, QED, electroweak interactions, GPDs, GFFs, PDFs, continuum lattice QCD, chiral effective field theory, bound-state QED, coupled-channel analyses, and reaction, detector, and reconstruction models. Its central objective is to determine whether, after accounting for the complete known variability of the reference model, a common low-dimensional response direction remains that cannot be reduced to a refit of QCD parameters, PDF variations, experimental systematics, data selection, reaction modelling, or reconstruction effects. The formal starting point is a regularized Euclidean QCD measure deformed by renormalized operator functionals DkR. The exact response identity ∂λk⟨Oi⟩λ=−Covλ(Oi,DkR) connects a microscopic operator source with an observable response direction of the system. For N observables and r independent sources, the local response Jacobian satisfies rankK≤min(N,r). This bound gives precise mathematical content to the earlier hypothesis of a common mode and low-dimensional closure. It does not, however, constitute independent evidence for new dynamics. The physical distinctness of an operator requires its non-triviality to be demonstrated modulo BRST-exact classes, equations of motion, total derivatives, field redefinitions, and directions equivalent to an ordinary refit of QCD parameters. A major development relative to the preceding publications is the construction of a gauge-invariant candidate for a three-arm baryonic Wilson junction. Quark fields or curvature tensors are transported by Wilson lines to a common junction point and subsequently contracted into an SU(3)c singlet. At the same time, the article corrects overly strong earlier topological interpretations. An individual open Wilson transporter is not a physical observable; the product of three open arms does not form a gauge-invariant monodromy; and the unit modulus of the eigenvalues of an SU(3) holonomy is not a criterion of proton stability. A resonance width is associated with a complex pole of the scattering amplitude or with an effective Feshbach operator obtained after eliminating open channels, rather than with a loss of holonomy unitarity. The empirical and numerical part includes an audit of MMGPDs analyses, continuum lattice QCD, gravitational form factors, operator-dependent radii, PDF uncertainties, low-Q2 form factors, Rosenbluth sensitivity, detector models, and the proton–Roper transition sector. The radii obtained from MMGPDs remain consistent with continuum lattice QCD at below the 1σ level. The electric–magnetic radius difference, ΔrEM=0.0290 fm, does not exceed 0.729σ over the full physically admissible correlation range. Approximately 94.7% of the reported variance of the proton electric radius originates from the PDF block. The supplied simulation tests yield pboot=0.799, an out-of-sample prediction of −0.0106±0.059, and subspace similarity below the adopted threshold of 0.90. These results remain consistent with the null hypothesis and do not currently demonstrate a stable orthogonal residual irreducible to the complete QCD/QED reference model. BKT–37Y07 extends the earlier description of local proton closure to a hierarchy of composite structures. The surface channel includes the gauge-invariant electromagnetic response transferred to atoms and molecules. The deep channel includes short-range nucleon interactions and weak p↔n currents. Hydrogen, the H2 molecule, the deuteron, and deuterium are analysed as successive closure classes without replacing QCD, QED, chiral EFT, or established binding mechanisms with relational terminology. The local stability of a proton is necessary but insufficient for the stability of a higher-order system; global closure also depends on interfaces, antisymmetry, conservation laws, energy balance, and the accessibility of decay channels. The principal result of the publication is the transition from the interpretive model developed across the BKT–37 series to a falsifiable C0–C8 validation protocol. The protocol covers operator admissibility, completeness of the reference model, full covariance treatment, identifiability after profiling, rank testing, stability across an ensemble of null models, out-of-sample prediction, transfer without amplitude refitting, and independent replication. The current status of the programme is formally and methodologically strong but dynamically unresolved. A distinct empirical content of the PJM–GTWSSF–USC hypothesis would arise only after demonstrating a renormalized, non-redundant, and transferable response direction that is not absorbed by the full QCD–EFT–QED hierarchy and that predicts new data in at least two independent sectors.