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ASFM as a Real-Space Field Foundation for Quantum Correlation Structure: A Noether-Ring Derivation of the Tsirelson Bound from Finer-Grid Finite Residual-Field Resonance Nodes

Andreas Pernt

Published Jul 4, 2026• 1 citation
Publisher

Abstract

The Asymmetric Source Field Model (ASFM) is a general real-space residual-field theory that addresses multiple open questions in quantum and nuclear physics. The present work applies this framework specifically to quantum correlation structure. Observable structures are modeled as spatially bounded residual-field interference organizations with resonance structure arising from harmonically scaled spherical source-field superposition. Constructive and conjugate-modulating sectors generate spatially bounded, nonzero, spatially persistent coherence nodes characterized by phase coherence, amplitude ordering, eigenmode structure, resonance capability, and node-to-node coupling response. In the Bell/CHSH reconstruction, the two measurement sites are not treated as independent carriers of pre-existing local values. They are local eigenmode readouts of one shared, spatially bounded residual-field interference topology with resonance structure. ASFM is therefore not a Bell-local hidden-variable model; the Bell assumption not satisfied by ASFM is local separability, equivalently local factorization. Bell-type correlations arise from non-separable residual-field topology, not from detector unfairness, sample rejection, or post-selection. The present work formulates the corresponding norm-preserving CHSH ring structure. The four CHSH configurations are interpreted as linked readout states of a closed resonance ring in the joint residual-field eigenmode space. A physically admissible closed ring may redistribute phase, frequency, amplitude, and eigenmode orientation, but it may not create residual-field readout norm. The associated normalized ring coherence represents the closure fraction of the conserved ASFM ring-norm class. The Bell-local boundary corresponds to the absence of an additional nonclassical ring-closure contribution, whereas complete norm-preserving closure corresponds to the Tsirelson limit. The paper distinguishes effective and direct Noether-ring coherence. Effective coherence is extracted from the simulated CHSH value. Direct coherence is predicted independently from phase closure, frequency compatibility, amplitude balance, and residual-node eigenmode coherence. Agreement between these quantities and their stable positive correlation constitute the principal non-circular validation criterion. The numerical reconstruction reports stable interference-resonance windows in which fixed-ring bounded readouts are Bell-violating and Tsirelson-safe. The leading confirmed window at a grid resolution of 65 cubed and harmonic range K6-K24 reaches a mean absolute CHSH value of 2.8118, with no Class-III seed in high-statistics confirmation. Across all confirmed interference-resonance windows, fixed-ring component tests show that phase, frequency compatibility, amplitude readout, eigenmode transport, coupling, and CHSH angular geometry are necessary. Controlled degradation of each component monotonically reduces the absolute CHSH value and produces no super-Tsirelson outcomes. The strongest near-Tsirelson windows form a more specific structural subclass characterized by compact pair topology, stable bounded readout, favorable amplitude response, and stable two-axis eigenmode structure. Direct structural coherence is retained as a global classifier. The principal non-circular refinement is an independently computed CHSH-mode anisotropy factor that incorporates the local orientation, transport, and angular dependence of the eigenmodes before CHSH evaluation. Within this application of the broader ASFM theory, quantum entanglement is interpreted as the effective readout of coupled, spatially bounded residual-field interference nodes with resonance structure embedded in a shared field topology, rather than as a connection between isolated particles. The Tsirelson bound is reconstructed operationally as the fixed, norm-preserving closure limit of an admissible CHSH residual-field readout ring.

DOI
10.5281/zenodo.21198210

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The Asymmetric Source Field Model (ASFM) is a general real-space residual-field theory that addresses multiple open questions in quantum and nuclear physics. The present work applies this framework specifically to quantum correlation structure. Observable structures are modeled as spatially bounded residual-field interference organizations with resonance structure arising from harmonically scaled spherical source-field superposition. Constructive and conjugate-modulating sectors generate spatially bounded, nonzero, spatially persistent coherence nodes characterized by phase coherence, amplitude ordering, eigenmode structure, resonance capability, and node-to-node coupling response. In the Bell/CHSH reconstruction, the two measurement sites are not treated as independent carriers of pre-existing local values. They are local eigenmode readouts of one shared, spatially bounded residual-field interference topology with resonance structure. ASFM is therefore not a Bell-local hidden-variable model; the Bell assumption not satisfied by ASFM is local separability, equivalently local factorization. Bell-type correlations arise from non-separable residual-field topology, not from detector unfairness, sample rejection, or post-selection. The present work formulates the corresponding norm-preserving CHSH ring structure. The four CHSH configurations are interpreted as linked readout states of a closed resonance ring in the joint residual-field eigenmode space. A physically admissible closed ring may redistribute phase, frequency, amplitude, and eigenmode orientation, but it may not create residual-field readout norm. The associated normalized ring coherence represents the closure fraction of the conserved ASFM ring-norm class. The Bell-local boundary corresponds to the absence of an additional nonclassical ring-closure contribution, whereas complete norm-preserving closure corresponds to the Tsirelson limit. The paper distinguishes effective and direct Noether-ring coherence. Effective coherence is extracted from the simulated CHSH value. Direct coherence is predicted independently from phase closure, frequency compatibility, amplitude balance, and residual-node eigenmode coherence. Agreement between these quantities and their stable positive correlation constitute the principal non-circular validation criterion. The numerical reconstruction reports stable interference-resonance windows in which fixed-ring bounded readouts are Bell-violating and Tsirelson-safe. The leading confirmed window at a grid resolution of 65 cubed and harmonic range K6-K24 reaches a mean absolute CHSH value of 2.8118, with no Class-III seed in high-statistics confirmation. Across all confirmed interference-resonance windows, fixed-ring component tests show that phase, frequency compatibility, amplitude readout, eigenmode transport, coupling, and CHSH angular geometry are necessary. Controlled degradation of each component monotonically reduces the absolute CHSH value and produces no super-Tsirelson outcomes. The strongest near-Tsirelson windows form a more specific structural subclass characterized by compact pair topology, stable bounded readout, favorable amplitude response, and stable two-axis eigenmode structure. Direct structural coherence is retained as a global classifier. The principal non-circular refinement is an independently computed CHSH-mode anisotropy factor that incorporates the local orientation, transport, and angular dependence of the eigenmodes before CHSH evaluation. Within this application of the broader ASFM theory, quantum entanglement is interpreted as the effective readout of coupled, spatially bounded residual-field interference nodes with resonance structure embedded in a shared field topology, rather than as a connection between isolated particles. The Tsirelson bound is reconstructed operationally as the fixed, norm-preserving closure limit of an admissible CHSH residual-field readout ring.
In the Asymmetric Source Field Model (ASFM), light is described as a field of interference nodes, which in physics would correspond to light quanta. This work examines whether the correlations of polarization-entangled light pairs arise because both nodes of a pair lie on the field of the same short-lived null-field sources. For point-symmetric source arrangements the orientations of both nodes agree at all times, and mixing two perpendicular source planes sets the shared orientation continuously to any angle while the partner node follows exactly. Under the principle that the sources and their carrier wave respond to a measurement as a whole, the setting at one node is fully transferred to the other. A readout from the independent fine structures of the two polarizer outputs yields Malus's law. In the limit of many contributions the Tsirelson bound 2√2 is the upper limit of the model for every transfer strength. The seven data points of Aspect, Grangier and Roger (1982) are reproduced with χ² = 2.1 without freely fitted parameters. The signed outcomes at each node remain balanced. The field power in the detector cone of the other node depends by a few per cent on whether a measurement takes place, but not on the setting, which is a testable prediction beyond quantum mechanics. The English and German manuscripts are included.
This preprint formulates Schrödinger’s cat as a boundary case of macroscopic residual-cluster stabilization within the Asymmetric Source Field Model (ASFM). Instead of treating the cat as a fundamental object placed into competing macroscopic states, the manuscript describes macroscopic reality as stabilized wave-field topology.The ASFM framework starts from the residual field relationwhere constructive source-field components and destructive compensation-field components generate finite, nonzero residual interference structures. Residual nodes are treated as real-space coherence structures, while the quotient structure provides the algebraic stabilization framework.The manuscript distinguishes residual nodes, cluster classes, quotient mappings, causal-retarded field action, topological correlation, and non-claimed simultaneous marginal control. Measurement is described as residual-cluster stabilization rather than as the collapse of a fundamental macroscopic object. The work also states explicit failure conditions and open validation points for future ASFM measurement dynamics.
Version 1.0 (March 2026) We demonstrate that CHSH Bell inequality violations emerge naturally in the Lattice Field Medium (LFM) framework when measurement is modeled as a non-separable interaction with a shared wave state. The CHSH parameter S = 2.829 matches the Tsirelson bound (2√2 = 2.828) to 0.04% — without injecting quantum mechanics, nonlocal signaling, hidden variables, or superdeterminism. Four measurement models compared (1,000,000 trials each, seed=42): Model Description S Value Interpretation A Local deterministic phase readout 2.000 Classical bound (exact) B Local probabilistic 1.416 Sub-classical C Non-separable joint sampling (LFM) 2.829 Bell violation D Analytical quantum mechanics 2.828 Tsirelson bound The key insight: In LFM, entangled particles share a common wave field governed by GOV-01 and GOV-02. Measurement is an irreducibly joint interaction with this globally coherent state — NOT independent local readout. The correlation function P(a,b) = cos²(θ_ab/2) is not postulated but emerges from the overlap integral of the measurement-perturbed field. Physical mechanism:The LFM wave field Ψ(x,t) satisfies GOV-01 (Klein-Gordon lattice equation). Two spatially entangled solitons share a single non-factorizable wave function. When a measurement interaction is modeled as a joint projection onto the full bipartite state, the resulting CHSH correlations acquire the cos²(θ/2) dependence characteristic of the Tsirelson bound. Local models A and B cannot access the cross-term — they are limited to S ≤ 2 by construction. Significance: This result closes the last remaining gap (IT-05: Entanglement / Bell inequality) in the LFM Physics Equation Emergence Catalog, achieving 100% effective coverage of all 208 cataloged physics equations across 16 domains — zero verified gaps remaining. The catalog entry IT-05 has been upgraded from GAP/CONDITIONAL to DERIVED/NOVEL. Contents: manuscript.pdf — Main paper (LaTeX compiled) experiments/bell_chsh_experiment.py — Reproducible experiment (numpy only, no dependencies) experiments/summary.json — Machine-readable results figures/fig_bell_chsh_comparison.png — S-value comparison across four models figures/fig_bell_correlations.png — Correlation functions by model zenodo_v1.0/ — Complete distribution package Reproducibility: python experiments/bell_chsh_experiment.py (requires numpy only). Results are fully deterministic. Related work: LFM-PAPER-045 (full framework reference, v26.0), LFM-PAPER-065 (electromagnetism from phase), LFM-PAPER-047 (spin-1/2 emergence from Lorentz symmetry). Falsifiable: The non-separable measurement model makes a specific prediction — local models always yield S ≤ 2 regardless of field amplitude; only joint-projection models reach S ≈ 2√2. This is experimentally distinguishable from standard QM only at the level of the measurement interaction model, not the S-value itself.
Relational Formation and the Bell Bound investigates an alternative structural reading of Bell-type correlations. Rather than assuming that alternative measurement settings reveal values attributable to a single pre-existing complete state, the paper introduces a minimal Relational Formation Algebra (RFA) in which a definite relational state is formed under the realized context. The framework is developed through four stages, R0–R3, describing a pre-relational possibility space, contextual relational formation, intrinsic compatibility through composition, and subsequent observational projection. A further condition, R4 (local substructure consistency), requires globally context-dependent relational states to retain locally consistent marginal structure. Two reproducible blind numerical experiments explore the resulting correlation domain. An unrestricted R0–R3 ensemble produces Bell violations but also permits CHSH values above the Tsirelson bound. In a structured bipartite realization satisfying R4 to numerical precision, Bell violations remain common while none of 30,000 randomized trials exceeds 2\sqrt{2}. The paper then examines the corresponding binary, cross-compatible operator algebra and the elementary origin of the 2\sqrt{2} norm bound. The proposal is not presented as a refutation of Bell’s theorem or CHSH, nor as an independent derivation of quantum mechanics. Instead, it asks whether the assumption of a pre-existing jointly attributable order can be replaced by a framework in which complete order is formed relationally while local consistency is retained. The conceptual distinction is summarized by the proposition: “Relation does not select an order. Relation forms the order.” The work includes the numerical parameters and fixed random seeds required to reproduce the reported simulations and references the original EPR (1935), Bell (1964), and CHSH (1969) papers. AI assistance: ChatGPT (OpenAI) was used as a collaborative tool for mathematical exploration, numerical simulation, structural analysis, and preparation of the manuscript. The conceptual direction, interpretation, selection of hypotheses, and responsibility for the published content remain with the human author.
Bell correlations represent one of the deepest structural features of quantum mechanics. Experimental violations of Bell inequalities demonstrate that observed two-state correlations cannot be reproduced by classical local hidden-variable probability spaces satisfying Kolmogorov factorization constraints. Standard quantum mechanics accounts for these violations through Hilbert-space geometry, projector-valued measurements, and noncommuting observables. However, the geometric origin of this structure is normally postulated rather than derived. In this paper we investigate whether the measurement geometry underlying Bell correlations can emerge naturally from the spectral closure structure of Time–Scalar Field Theory (TSFT). The goal is not to replace quantum mechanics, nor to construct a classical hidden-variable model reproducing Bell violations. Instead, the objective is narrower and mathematically sharper: to determine the conditions under which admissible scalar-time closure sectors become equivalent to quantum two-state measurement geometry. Starting from the scalar-time field Θ(x^μ) and its associated fluctuation operator, we construct a self-adjoint spectral problem governing admissible coherence modes. Under finite-energy, normalizability, and phase-return conditions, the admissible spectrum becomes discrete. We then examine degenerate two-dimensional closure eigenspaces and show that closure-preserving basis transformations naturally induce unitary rotational structure. After quotienting by physically irrelevant global phase, the admissible state geometry reduces to an emergent SU(2)-covariant two-state manifold. Within this framework, binary measurements arise as closure-compatible projection operations on the degenerate sector. The associated correlation function acquires the form E(a, b) = −a · b, matching the standard quantum singlet-state correlation law. Maximization of the corresponding Clauser–Horne–Shimony–Holt (CHSH) functional then yields the Tsirelson bound S_max = 2√2, as a geometric consequence of closure-preserving scalar-time measurement structure. The derivation presented here does not assume Hilbert-space quantum mechanics as an initial axiom. Rather, the central claim is that under sufficiently constrained spectral closure conditions, the admissible geometry of scalar-time coherence sectors becomes mathematically equivalent to the two-state measurement geometry responsible for Bell correlations.