introduction Annex BK–Y17A is the technical and mathematical extension of the main BK–Y17 article. Its purpose is to formulate a projection apparatus for testing whether scalar quantities used in physics — such as mass, energy, resonance width, chiral condensate, Chern number, or topological susceptibility — are informationally complete, or whether they are effective projections of a fuller relational-informational structure. The central hypothesis of the annex is: X_scalar = P_X(S_full) where X_scalar denotes a scalar observable, P_X the projection operator, and S_full a fuller relational structure including geometry, topology, coupling relations, dynamics, information, and observational or computational channels. The annex does not reject scalar quantities as physical tools. It defines a procedure for testing when a scalar description is sufficient and when it must be supplemented by relational descriptors. This publication is Annex A to the main BK–Y17 article and provides the formal foundation for the deconstruction of the scalar paradigm within the PJM–GTWSSF–USC–GTCW framework. In the main BK–Y17 article, the problem is formulated at the methodological level: a scalar quantity may be a valid effective observable, but it does not necessarily exhaust the information contained in the physical structure of a system. Annex BK–Y17A addresses how this problem can be written mathematically and prepared for numerical testing. The document defines a projection apparatus based on the full relational structure S_full and the projection operator P_X. Within this apparatus, a scalar is treated as the result of reduction, marginalization, partial trace, tensor-network contraction, or an observational procedure leading from a fuller structure to a single measurable number. Such a number remains physically meaningful, but its informational completeness becomes a separate research question. The annex introduces a set of mathematical and numerical tools: the full relational representation space, the projection operator, mutual information I(A:B), informational difference Delta I_X, the relational estimator delta_eff, the complexity weight Omega_chi, and MPS, PEPS, and GGPEPS representations. The formalism is directed toward sectors in which reduction to a scalar may hide correlations, topology, gauge structure, or channel dependence: QCD-like sectors, QED, QCD topology, Chern insulators, hadronic resonances, and systems modelled by tensor networks. A central feature of the annex is the distinction between formal definitions, model predictions, and empirical results. Numerical values reported in the annex should be treated as model predictions or numerical hypotheses for independent validation, not as completed empirical confirmation of the theory. The Roper-resonance sector is one example: the annex develops the relation between effective mass and mutual information and indicates model values to be tested through lattice QCD, MPS, PEPS, GGPEPS, VQE, coupled-channel analyses, and high-energy data. An important model equation is: m_eff = m_0 + kappa_I I(A:B) This equation is not presented as a new fundamental law of mass and does not replace the Higgs mechanism, QCD, or standard definitions of mass in field theory. It has the status of a first-order calibration hypothesis: it tests whether effective mass differences may be linked to differences in relational mutual information within well-defined sectors. The parameter kappa_I is a calibration parameter with units of energy per bit, not a fundamental constant. The annex also establishes criteria for positive, negative, and unresolved outcomes. The projection hypothesis is strengthened sectorally only if a stable relational-informational excess over the scalar description remains after controlling for null models, uncertainties, ansatz choice, factorization, operator basis, and standard physical mechanisms. If scalar-matched null models reproduce all relational diagnostics within uncertainty, the hypothesis is constrained or falsified in that sector. BK–Y17A should be read together with the main BK–Y17 article and with Annexes BK–Y17B and BK–Y17C. The main article defines the scalarity problem and the physical context; BK–Y17A develops the formal apparatus and estimators; BK–Y17B develops the numerical-calibration layer; BK–Y17C develops the operator-geometric reconstruction of degrees of freedom lost in scalar reduction. EN — keywords BK–Y17A; PJM; GTWSSF; USC; GTCW; scalar paradigm; projection apparatus; projection operator; effective quantities; mass; mutual information; delta_eff; Delta I_X; Omega_chi; MPS; PEPS; GGPEPS; lattice QCD; VQE; QCD-like sectors; QED; Schwinger model; Chern number; QCD topology; Roper resonance; tensor networks; information geometry; sectoral falsification
introduction Annex BK–Y17A is the technical and mathematical extension of the main BK–Y17 article. Its purpose is to formulate a projection apparatus for testing whether scalar quantities used in physics — such as mass, energy, resonance width, chiral condensate, Chern number, or topological susceptibility — are informationally complete, or whether they are effective projections of a fuller relational-informational structure. The central hypothesis of the annex is: X_scalar = P_X(S_full) where X_scalar denotes a scalar observable, P_X the projection operator, and S_full a fuller relational structure including geometry, topology, coupling relations, dynamics, information, and observational or computational channels. The annex does not reject scalar quantities as physical tools. It defines a procedure for testing when a scalar description is sufficient and when it must be supplemented by relational descriptors. This publication is Annex A to the main BK–Y17 article and provides the formal foundation for the deconstruction of the scalar paradigm within the PJM–GTWSSF–USC–GTCW framework. In the main BK–Y17 article, the problem is formulated at the methodological level: a scalar quantity may be a valid effective observable, but it does not necessarily exhaust the information contained in the physical structure of a system. Annex BK–Y17A addresses how this problem can be written mathematically and prepared for numerical testing. The document defines a projection apparatus based on the full relational structure S_full and the projection operator P_X. Within this apparatus, a scalar is treated as the result of reduction, marginalization, partial trace, tensor-network contraction, or an observational procedure leading from a fuller structure to a single measurable number. Such a number remains physically meaningful, but its informational completeness becomes a separate research question. The annex introduces a set of mathematical and numerical tools: the full relational representation space, the projection operator, mutual information I(A:B), informational difference Delta I_X, the relational estimator delta_eff, the complexity weight Omega_chi, and MPS, PEPS, and GGPEPS representations. The formalism is directed toward sectors in which reduction to a scalar may hide correlations, topology, gauge structure, or channel dependence: QCD-like sectors, QED, QCD topology, Chern insulators, hadronic resonances, and systems modelled by tensor networks. A central feature of the annex is the distinction between formal definitions, model predictions, and empirical results. Numerical values reported in the annex should be treated as model predictions or numerical hypotheses for independent validation, not as completed empirical confirmation of the theory. The Roper-resonance sector is one example: the annex develops the relation between effective mass and mutual information and indicates model values to be tested through lattice QCD, MPS, PEPS, GGPEPS, VQE, coupled-channel analyses, and high-energy data. An important model equation is: m_eff = m_0 + kappa_I I(A:B) This equation is not presented as a new fundamental law of mass and does not replace the Higgs mechanism, QCD, or standard definitions of mass in field theory. It has the status of a first-order calibration hypothesis: it tests whether effective mass differences may be linked to differences in relational mutual information within well-defined sectors. The parameter kappa_I is a calibration parameter with units of energy per bit, not a fundamental constant. The annex also establishes criteria for positive, negative, and unresolved outcomes. The projection hypothesis is strengthened sectorally only if a stable relational-informational excess over the scalar description remains after controlling for null models, uncertainties, ansatz choice, factorization, operator basis, and standard physical mechanisms. If scalar-matched null models reproduce all relational diagnostics within uncertainty, the hypothesis is constrained or falsified in that sector. BK–Y17A should be read together with the main BK–Y17 article and with Annexes BK–Y17B and BK–Y17C. The main article defines the scalarity problem and the physical context; BK–Y17A develops the formal apparatus and estimators; BK–Y17B develops the numerical-calibration layer; BK–Y17C develops the operator-geometric reconstruction of degrees of freedom lost in scalar reduction. EN — keywords BK–Y17A; PJM; GTWSSF; USC; GTCW; scalar paradigm; projection apparatus; projection operator; effective quantities; mass; mutual information; delta_eff; Delta I_X; Omega_chi; MPS; PEPS; GGPEPS; lattice QCD; VQE; QCD-like sectors; QED; Schwinger model; Chern number; QCD topology; Roper resonance; tensor networks; information geometry; sectoral falsification
short introduction The BK–Y17 article is the main publication of a research package devoted to the limitations of the scalar paradigm in physics. The work develops the projection hypothesis, according to which a single scalar quantity such as mass, energy, resonance width, cross section, Hall coefficient, or a Planck quantity may be a correct effective observable while not exhausting the physical structure of the system. The aim is to formulate a testable research programme in which relational, topological, channel-dependent, and correlational observables are examined as carriers of information not reducible to a single scalar. introduction This publication presents a formal and methodological treatment of the reduction of complex physical structures to scalar quantities. The starting point is the distinction between a scalar as a correct measured or effective parameter and the fuller physical structure from which this parameter is reconstructed, projected, or derived in an observational procedure. The central element of the work is the projection hypothesis: X_scalar = P_X(G, T, R, D, I, C) where X_scalar denotes an observable scalar quantity, P_X the projection operator to a selected observable, G geometry, T topology, R coupling relations, D dynamics, I structural information understood as compatibility and admissibility conditions of configurations, and C observational, decay, detection, and reconstruction channels. This equation does not replace the Standard Model, general relativity, QCD, nuclear physics, condensed matter physics, or measurement theory. It provides a formal expression of the thesis that a scalar can be a projection of a fuller structural object rather than its complete description. In the article, the projection operator is interpreted as a class of known reduction procedures: expectation value of an observable, partial trace over hidden degrees of freedom, marginalization of relational variables, renormalization coarse-graining, or an effective observational operator including process dynamics, final-state channel, detector response, reconstruction, and statistical fitting. The argument covers sectors and examples such as the Planck scale, mass, fundamental constants, the t tbar threshold, the Roper resonance, the Higgs sector, chirality, neutrinos, Hall phenomena, and response tomography. As the leading numerical closure, the article introduces the dimensionless relational-width indicator of the Roper resonance, C_R = Gamma_R / M_R. For the reference values used in the demonstration, M_R = 1440 ± 30 MeV and Gamma_R = 350 ± 100 MeV, one obtains C_R = 0.243 ± 0.070. Relative to the model threshold C_crit = 0.10, this gives Pi_R = C_R / C_crit = 2.43 ± 0.70. This result is not treated as evidence for the USC hypothesis, but as a falsifiable example showing that the central-mass scalar can be measurement-correct and structurally incomplete at the same time, because it does not explicitly encode width, decay channels, and amplitude dynamics. The article serves as the main document of the BK–Y17 series. Detailed developments of the projection apparatus, numerical predictions, calibrations, the Planck-scale map, and the operator-geometric reconstruction of lost degrees of freedom are presented in three technical-mathematical annexes: BK–Y17A, BK–Y17B, and BK–Y17C. These annexes should be treated as formal and computational complements to the main work. The aim of the publication is not to replace existing physical theories, but to establish a test programme: if, after controlling for known standard parameters, relational, topological, channel-dependent, or correlational observables provide independent information relative to a single scalar, the projection hypothesis is strengthened sectorally; if they do not provide such information, it is weakened or falsified sectorally. keywords scalar paradigm; effective quantities; mass; energy; projection operator; Universal Structural Code; PJM; GTWSSF; USC; GTCW; response tomography; Roper resonance; Higgs sector; chirality; neutrinos; Hall phenomena; Planck scale; information geometry; sectoral falsification
This record provides the bilingual publication package of document BKT-43A/USC-RA1/ZENODO-20260707-v13, entitled “Formalization of Relational Closure and Non-Closure Estimators for Quantum-State Stability”, with the subtitle “A New Class of Mathematical-Physical Tools for Studying and Diagnosing the Stability of Quantum Systems.” The work presents a family of relational closure and non-closure estimators as a coherent mathematical-physical apparatus for diagnosing the stability of quantum states. The core formalism includes a sectoral closure estimator, an attractor-topological-spectral extension, a logarithmic multichannel aggregator, a covariance-aware variant, a geometric estimator based on the Bures distance, and a differential-dynamical layer. The article includes a canonical tabular compendium of the formulas, parameters, formal meaning, and physical interpretation of the estimators. The model-computational verification is based on Bell-state dephasing, the Bures channel, quantum Fisher information, Lindblad dynamics, a finite Kitaev-chain test, and the analysis of quantum-error correlations. The results have formal and computational status: they confirm the internal consistency of the definitions, agreement with analytic results of the reference model, and correct treatment of channel redundancy by the covariance-aware estimator. They do not yet constitute external experimental validation. This record continues earlier models and published formal materials. In particular, it develops the sectoral (\delta_{\rm eff}) and (\delta_{\rm eff,ATS}) model from BKT-Y16-20260622-v10.2, extends the three-level relational non-closure estimator apparatus from BKT-43 USC-RA1 20260619 v1.3, and organizes earlier Zenodo records related to the (K^{\log}_{QM}) estimator, relational geometry, and response tomography: 10.5281/zenodo.19893953, 10.5281/zenodo.20798037, 10.5281/zenodo.20797198, and 10.5281/zenodo.19824700. The response-tomography record is used as methodological context, not as validation evidence for the quantum calculations presented in the present article. The package contains the Polish version, the English mirror version, LaTeX source files, figures, MasterData tables, auxiliary scripts, bibliography, and editorial-technical audit reports. The purpose of the record is to provide a complete, reproducible, and citable formalization package for the estimator family before further editorial assessment and possible experimental validation.
Publication title The Relational Non-Closure Estimator and Its Derivatives as a Tool for Information Physics within the LOM–GTSFC–USC–GTCW Framework This publication presents a flagship formal-computational result developed within the LOM–GTSFC–USC–GTCW framework: the relational non-closure estimator and its derivatives as a measurable diagnostic tool for stability, decoherence, entanglement, topological protection, and structural reconfiguration in physical systems. The main objective of the work is to move from the descriptive idea of relational compatibility to a mathematical object that can be computed, differentiated, compared with null models, and evaluated through PASS/FAIL criteria. In this formulation, the stability of a physical system is not reduced to a single observable such as energy, mass, purity, fidelity, entropy, coherence time, or a topological invariant. It is represented as the result of compatibility among multiple physical channels: phase, entanglement, decoherence, spin-chiral, topological, and geometric-informational channels. The central object of the work is the diagnostic package \left(K_{\log}^{(s)}(\boldsymbol{\lambda}),\nabla_{\boldsymbol{\lambda}}K_{\log}^{(s)}(\boldsymbol{\lambda}),H^{(K,s)}(\boldsymbol{\lambda})\right),] where (K_{\log}^{(s)}) denotes the sectoral relational non-closure estimator, (\nabla_{\boldsymbol{\lambda}}K_{\log}^{(s)}) describes the direction in which non-closure increases or decreases with respect to control parameters, and (H^{(K,s)}) is the Hessian describing the curvature of the relational cost surface. The value of the estimator indicates the degree of channel mismatch, the first derivative indicates the direction of stability evolution, and the second derivative enables the diagnosis of thresholds, critical points, and accelerated loss of coherence. The publication includes the full formal apparatus, descriptions of equation components, physical interpretation, a table of derivative formulae, numerical calculations, MasterData, plots of (K(\lambda)), (\dot{K}), (K''), gradient maps, and tests against null models. The computational section uses demonstrative model systems, including Bell-state decoherence, a two-parameter gradient map of non-closure, and a minimal topological test inspired by the Kitaev chain. The epistemic status of the results is explicitly separated. The work reports a model-computational PASS for the proposed diagnostic apparatus, because the estimator is computable, differentiable, monotonic in controlled demonstrative models, and comparable with null models. At the same time, its empirical status remains INCONCLUSIVE until the apparatus is applied to independent experimental data, full covariance matrices, external test systems, and preregistered validation procedures. The significance of the work lies in proposing a formal bridge between the relational-informational language of USC and standard tools of mathematical physics, including cost functions, sectoral estimators, gradients, Hessians, covariance variants, quantum information geometry, the Bures metric, and the quantum Fisher information matrix. The estimator does not replace quantum mechanics, field theory, decoherence theory, or topological models of matter. It acts as a diagnostic layer that synthesizes multiple observables into a single axis of relational stability and enables quantitative analysis of whether a physical system maintains coupling compatibility, loses it, reconfigures, or approaches a critical threshold. Keywords LOM; GTSFC; USC; GTCW; Universal Structural Code; relational non-closure estimator; information physics; decoherence; quantum entanglement; quantum information geometry; Bures metric; quantum Fisher information matrix; topological protection; Kitaev chain; MasterData; PASS/FAIL; null models.
Introduction This article presents the quantum module of the USC formalism, namely the Universal Structural Code, developed as a classificatory layer within the PJM/LOM programme and the GTSFC–USC–GTCW framework. The work does not modify quantum mechanics, the Schrödinger equation, the density matrix formalism, Lindblad dynamics, or the standard theory of decoherence. It presents an additional operational description of quantum state stability, based on a relational and topological analysis of phase coherence, entanglement, purity, decoherence, spin chiral structure, and topological protection. The central element of the article is the estimator KlogQMK^{\mathrm{QM}}_{\log}KlogQM, which aggregates selected observables of quantum mechanics into a single scale of relational non closure. In this formulation, a quantum state is classified not only through energy, entropy, purity, or a single entanglement measure, but through the multichannel consistency of its relational structure. Quantum stability is therefore described as the simultaneous preservation of phase coherence, topological robustness, entanglement structure, spin correlations, and a low level of decoherence. The significance of the work for established physics lies in the ordering of standard quantum mechanical observables within a new diagnostic language. The estimator KlogQMK^{\mathrm{QM}}_{\log}KlogQM can be treated as a cost function or classification map acting on density matrices ρ∈D(H)\rho \in \mathcal{D}(\mathcal{H})ρ∈D(H). In its developed form, the formalism incorporates the geometry of quantum states, including fidelity, the Bures distance, the Fubini Study metric in the pure state limit, and the covariance aware form of channel mismatch. In this way, the module is embedded in the known tools of quantum information physics and does not introduce a competing dynamical apparatus. The significance of the work for quantum computer architecture is related to the problem of maintaining coherence, entanglement, noise robustness, and logical stability of quantum states over time. In practical quantum architectures, the mere preparation of a qubit or an entangled state is not sufficient for useful information processing. The essential issue remains the ability of a system to preserve the relational structure of the state under noise, dephasing, environmental coupling, and local errors. The estimator KlogQMK^{\mathrm{QM}}_{\log}KlogQM is presented here as a candidate multichannel indicator of quantum state degradation, applicable to the comparison of Bell states, decohered mixtures, topologically protected systems, the Kitaev chain, the toric code, and future fault tolerant architectures. Two levels of demonstration are presented. The first concerns a Bell state subjected to dephasing, where the estimator increases monotonically with the loss of phase coherence, entanglement, and purity. The second concerns a minimal topological test based on the Kitaev chain, where the topologically protected phase obtains a lower value of KlogQMK^{\mathrm{QM}}_{\log}KlogQM than the trivial phase. These results have the status of an operational demonstration, not a full ontological validation of USC. Their significance lies in showing that the quantum module of USC can be formulated in a language consistent with established quantum mechanics, state geometry, and quantum information theory. The main contribution of the article is the indication that quantum state stability can be analysed not only through individual measures, such as purity, von Neumann entropy, concurrence, or fidelity, but through their relational structure. In this sense, the article proposes a transition from the classification of individual properties of a state to the classification of its multichannel coherence. The question of whether a state is pure, entangled, or topological is supplemented by the question of whether the set of its properties forms a stable, relationally closed quantum configuration. The scientific status of the work is defined as QM-PROGRAM. This means that the presented formalism has the character of a mathematically coherent and testable classification programme, consistent with established physics, while still requiring further numerical validation. The next stage is defined as a technical annex containing full mathematical and physical derivations, simulations of open quantum systems, Lindblad tests, analysis of the noisy Kitaev chain, the toric code, null models, ROC/AUC curves, and comparison of the KlogQMK^{\mathrm{QM}}_{\log}KlogQM estimator with individual standard measures. In a broader sense, the article shows that USC can be formulated as a tool for classifying and testing the stability of quantum structures. This form of the formalism creates a bridge between the geometry of quantum states, quantum information theory, decoherence, topological protection of states, and practical quantum computer engineering. If further numerical tests confirm the advantage of the estimator over null models, the KlogQMK^{\mathrm{QM}}_{\log}KlogQM module may acquire the status of a useful diagnostic tool for analysing the stability of quantum states and quantum computing architectures. Keywords Universal Structural Code, USC, PJM, LOM, GTSFC, quantum mechanics, geometry of quantum states, decoherence, entanglement, fidelity, Bures distance, density matrix, Kitaev chain, topological protection of states, quantum computers, KlogQMK^{\mathrm{QM}}_{\log}KlogQM, relational closure, quantum information, open quantum systems.
Conditional relational tomography of a nucleus Coupling geometry, memory and coherence through reconfiguration Robert Kupski BKT–96–INT v2.1 Abstract Conditional relational tomography of a nucleus is formulated within the Law of One Mechanism (LOM) and the GTSFC–USC–GTCW framework.The study examines the connection between coupling organisation, stable spatial response, temporal memory and coherence read out after reconfiguration. A model of two coupled fields specifies a positive energy functional, dynamics, preparation and observation maps. Eliminating the environment determines static softening, the spatial response kernel and memory in the reduced description. With an additional stationary Gaussian preparation of the uniform mode, the same stiffness matrix determines the contrasts of two directional phase probes. The principal result is an exact, conditional space–memory–coherence relation obtained by eliminating the shared coupling from three types of readout. At dimensionless coupling g = 1, length scale L₀ = 1 fm, dimensionless fluctuation scale η = 0.2 and relative environmental mobility μ_env = 1, the common combination equals 1/4. The spatial response kernel has moments m₂ = 1 fm², m₄ = 16/3 fm⁴, m₆ = 72 fm⁶ and m₈ = 1792 fm⁸, while the ratio of logarithmic contrasts for the soft- and hard-direction probes equals 3. These are consequences of a specified realisation, without independent fitting of each characteristic. The amplitude description is supplemented by positive semidefinite Gram matrices and invariant loop phases, which contain information not determined by pairwise compatibility magnitudes alone. Operator bounds on the transfer error of selected observables under a transformation are derived, together with conditions for identifiability and propagation of joint covariance. Inhomogeneity, inertia and finite preparation time delimit the transfer of relations between readouts. ATLAS, STAR and ALICE results provide complementary research anchors with distinct selections, kinematic domains and measurement operators. A limited calibration to reconstructed ATLAS data reproduces the response trend but establishes neither predictive superiority over the nNNPDF reference nor memory and phase-preparation parameters. The contribution is a construction of testable constraints linking different manifestations of one coupling organisation. Model results remain distinct from empirical identification of the metafield and from establishing dynamics irreducible to quantum chromodynamics. Introduction Investigating nuclear structure involves both the spatial distribution of its response and the mechanisms governing the persistence of that response, its dependence on history and the correlations revealed in interaction products. A radius, a cross section, a relaxation time and an angular-distribution harmonic are different projections of the system under study. Their joint description becomes more restrictive than a collection of independent parametrisations when a specified coupling organisation connects their values through definite relations. The central problem of this work is precisely the possibility of connecting spatial response, memory and coherence in this way. Within LOM, the formation, admissibility and stabilisation of physical structures are hypothesised to arise from shared rules of relational–informational organisation. Structural information here denotes constraints, coupling directions, deformation costs and rules of admissible transformations, rather than an additional substance or merely a measure of experimental uncertainty. GTSFC describes shared coupling structures and their sectoral realisations; the Universal Structural Code (USC) organises conditions of compatibility, composition and tolerance. The metafield denotes a postulated layer of this organisation. In this hypothesis, particles and fields are considered its physical realisations, while valid results of quantum mechanics and field theory are retained. The cosmological scope of GTCW remains separate from the nuclear observables analysed here. The conceptual basis is the development of persistent relational-node descriptions in the BKT–37 programme and the distinction between reconfiguration, sector preservation and recovery of the full state in BKT–37–REK. BKT–74–QREC distinguishes retention of organisation from state recurrence and dynamical attraction, whereas BKT–94 connects a shared generator with cross-channel identifiability. BKT–96 applies these distinctions to nuclear response and interferometry, where geometric shape, relaxation history and phase relations have their own explicitly specified readout maps. The observational anchors come from studies of ultraperipheral nuclear interactions. ATLAS measurements constrain partonic-response contrast, geometric selection through neutron topologies and components of jet photoproduction. STAR and ALICE studies provide complementary amplitude, polarisation and diffraction constraints. In two-source interference, distinguishability of products such as positive and negative pions remains distinct from distinguishability of their production alternatives. The common object of description is therefore the complete process leading to a specified measurement outcome. “Conditional tomography” denotes reconstruction dependent on source, reaction, selection and detector models, rather than direct recovery of an arbitrary nuclear profile. The mathematical realisation is based on a positive functional of two coupled fields. Its second variation determines stiffness, while elimination of the environmental sector yields the spatial response operator. Once mobility is specified, the same elimination determines a memory kernel in the reduced dynamics. Positive spatial kernels admit a spectral representation connecting shape to a hierarchy of moments. With a shared Gaussian preparation of the uniform mode, stiffness also determines fluctuations read out by phase probes. Energy, the evolution law, preparation and probe sensitivity remain distinct but consistently specified components of this realisation. For the family with dimensionless local coefficients a = h = 2, coupling 0 ≤ g < 2 and gradient costs c = L₀²/4 and d = L₀²/2, the dimensionless quantity r = m₂/L₀² is defined by the second moment m₂ of the spatial response kernel and the fixed length scale L₀. The quantity ℛ = ln(γ_s)/ln(γ_h) is the ratio of logarithmic contrasts of unit soft- and hard-direction probes. M₀ and M₁ denote the zeroth and first memory-kernel moments in dimensionless time with unit mobility of the visible field; μ_env > 0 is the relative environmental mobility. Under stationary Gaussian preparation with a positive fluctuation scale, the derived relation is: (2r − 1)/[2(r + 1)] = [(ℛ − 1)/(ℛ + 1)]² = M₀/2 = μ_env M₁ = g²/4. This condition restricts the independent freedom of three types of readout. It identifies neither a spatial moment with memory nor phase contrast with an entanglement measure. At g = 1 and μ_env = 1, it gives r = 1, ℛ = 3, M₀ = 1/2 and M₁ = 1/4. Finite preparation time modifies contrasts according to the dynamics approaching the stationary distribution, while extensions with inertia can preserve statics despite a different response history. The three-way relation is therefore a result for a specified model class, not a universal identity for arbitrary nuclear processes. The general description of amplitude organisation uses a complex Gram matrix. Positive semidefiniteness ensures nonnegative probabilities for admissible amplitude combinations, while loop phases specify relations not determined by pairwise overlap magnitudes alone. This construction is broader than the particular Gaussian preparation used in the three-way relation. An operator description of a transformation, in turn, specifies preservation of selected observables and bounds on their transfer error without identifying such preservation with recovery of the entire quantum state. The research value of a shared model lies in predicting readouts not used to determine its parameters. The analysis therefore addresses not only existence of a solution but also parameter indistinguishability, uniqueness of predictions on calibration-equivalent sets and cross-channel covariances. ATLAS, STAR and ALICE results constrain different components of this problem; they do not automatically establish a single joint fit or identical parameters for different nuclei. The achievement is a computable realisation of the organisational hypothesis and a set of conditions for its independent testing. The predicted interdependence of responses has physical content even when its microscopic realisation remains consistent with quantum chromodynamics. Empirical identification of the metafield and conclusions about additional dynamics constitute separate levels of inference.
Description QSTH M.1B is the operational continuation of QSTH M.1A, Genealogy of Alpha-I-Dim and Equations E1–E10. M.1A established the algebraic genealogy of the reference Alpha-I-Dim construction and identified effective entropy S_eff and the structural parameter κ(Γ) as imported quantities whose physical meaning could not be supplied by algebra alone. M.1B addresses that open requirement by constructing a first explicit finite operational framework for structural configurations, equivalence, representation, statistical measure, dynamics, entropy and independent classification. The study defines typed relational configurations with a decidable admissibility predicate and equivalence under relabelling; constructs a faithful incidence representation; distinguishes representational equivalence, physical equivalence and indistinguishability under a limited readout; and states the conditions under which deterministic and probabilistic dynamics descend consistently to configuration classes. A finite operational statistical entropy is then constructed from a prespecified state space, measure and readout. Independently, a dynamical classifier based on closed recurrent sectors is introduced, together with an audit of exact charge protection versus finite escape time. The central integration result is deliberately restrictive. The independently defined sector count is not a universal entropic coefficient for Equation E10. The same entropy may coexist with different sector structures, while genuine internal degrees of freedom may increase entropy without creating an additional recurrent sector. The independent test is therefore allowed to reject the proposed identification rather than being normalized to reproduce it. Accordingly, the formal mathematical and statistical stage B1-B7 is complete within its declared domain, while physical horizon identification, M-independence and the continuation to E11+ remain open. This publication does not claim a new universal constant, a microscopic theory of horizon entropy, or empirical confirmation of QSTH. Its contribution is an auditable operational framework, explicit counterexamples, reproducible finite tests, and a precise boundary between mathematical construction and physical identification. A separate companion study, QSTH M.1B-SR1, examines whether a richer spectral response can provide a conditional bridge between an independently defined observable and the thermal entropy of specified interface carriers.