Paper T2A: Bell Bounds, Nonfactorizable Relational Coherence, and No-Signalling in Holosphere Theory An Exact Real-Valued SU(2) Representation and Bell-Assumption Audit
Paper T2A develops a real-valued representation of Bell correlations within Holosphere Theory and identifies the assumptions responsible for the classical and quantum Bell limits. The analysis first derives the classical Clauser–Horne–Shimony–Holt bound from measurement independence, ordinary probability, and Bell-local factorization. It then represents the two-party singlet state as a normalized, nonseparable real bipartite state, with local analyzer settings described through real forms of spinor rotations and binary measurement projectors. Applying the Holosphere compatible-path or path-volume probability rule, expressed as the squared norm of the projected state, produces the standard singlet joint probabilities and the cosine correlation between analyzer directions. The optimal four-setting configuration reaches the Tsirelson value while the general real-operator analysis prevents stronger correlations within the declared binary-observable framework. Summing over either remote outcome produces locally unbiased probabilities that are independent of the remote setting, preserving operational no-signalling and preventing faster-than-light communication. The canonical model retains measurement independence and local measurement operations but does not retain outcome independence or Bell-local factorization. The paper also distinguishes exact realification from a microscopic physical derivation. It does not yet derive the singlet state, probability frequencies, detector response, decoherence, or relational persistence from fundamental Holosphere lattice dynamics. Paper T2A therefore establishes a mathematically consistent and reproducible Holosphere representation of ordinary quantum Bell correlations while leaving the specifically Holosphere physical mechanism and independent empirical confirmation as open research requirements.
Paper T2A develops a real-valued representation of Bell correlations within Holosphere Theory and identifies the assumptions responsible for the classical and quantum Bell limits. The analysis first derives the classical Clauser–Horne–Shimony–Holt bound from measurement independence, ordinary probability, and Bell-local factorization. It then represents the two-party singlet state as a normalized, nonseparable real bipartite state, with local analyzer settings described through real forms of spinor rotations and binary measurement projectors. Applying the Holosphere compatible-path or path-volume probability rule, expressed as the squared norm of the projected state, produces the standard singlet joint probabilities and the cosine correlation between analyzer directions. The optimal four-setting configuration reaches the Tsirelson value while the general real-operator analysis prevents stronger correlations within the declared binary-observable framework. Summing over either remote outcome produces locally unbiased probabilities that are independent of the remote setting, preserving operational no-signalling and preventing faster-than-light communication. The canonical model retains measurement independence and local measurement operations but does not retain outcome independence or Bell-local factorization. The paper also distinguishes exact realification from a microscopic physical derivation. It does not yet derive the singlet state, probability frequencies, detector response, decoherence, or relational persistence from fundamental Holosphere lattice dynamics. Paper T2A therefore establishes a mathematically consistent and reproducible Holosphere representation of ordinary quantum Bell correlations while leaving the specifically Holosphere physical mechanism and independent empirical confirmation as open research requirements.
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.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
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.
This paper proves a finite-dimensional structural recovery-and-obstruction theorem for Bell-local factorization inside projective phase structure. The primitive object is a completion-locked projective probability datum: a bipartite state, fixed local projective contexts, the Born–Haar reduction rule, and fixed normalization. A projective measurement context carries a compact relative phase torus, and Haar averaging over that torus gives the nonselective Born/Lüders projection channel. The Born probability table is therefore determined before any hidden-variable or Bell-local representation is tested. Bell-local factorization is determined as a derived classical recovery representation of the locked Born table. The paper proves canonical recovery regimes: pure product rays are exactly Segre-factorized projective phase classes, mixed separability yields Bell-local recovery, and one-sided commutative local readout pairs yield Bell-local recovery through a joint-outcome hidden variable. Conversely, a Clauser–Horne–Shimony–Holt (CHSH) value larger than 2 proves obstruction: no Bell-local representation exists for the chosen datum. No-signaling follows from the trace identities of the projective Born law, while Tsirelson’s bound 2√2 follows from the operator geometry of admissible binary projective observables. The theorem chain therefore determines the hierarchy: projective phase carrier → Born–Haar reduction → Segre/separable recovery or Bell-factorization obstruction → no-signaling and Tsirelson admissibility. This hierarchy also determines the intermediate cases: Bell–CHSH violation is a property of the complete projective measurement datum, not of entanglement alone. Product states, separable mixtures, commutative readouts, and nonoptimal settings can remain below the classical threshold even inside quantum theory. The constructive spin instantiation is derived explicitly. For the diagonal SU(2) action on ℂ² ⊗ ℂ², rotational invariance selects the unique spin-zero singlet line. The singlet is entangled by failure of Segre factorization, its Pauli spin correlations satisfy E(a,b) = −a·b, and optimal admissible spin readouts attain the exact Tsirelson value 2√2. In this derivation, 2√2 is not introduced by experiment-fitting; it is the Hilbert-space admissibility ceiling of the projective phase datum, realized exactly by the spin-zero SU(2) sector. License note: Distributed under CC BY-NC-ND 4.0.
Abstract Bell’s theorem renders quantum correlations distinct from those of any local-realistic model. Although being stronger than classical correlations, quantum correlations are limited by the Tsirelson bound. This bound, however, applies for Hermitian, commutative operators corresponding to non-signaling observables in Alice’s and Bob’s spacelike-separated labs. As an attempt to explore theories beyond quantum mechanics and analyze the uniqueness of the latter, we examine in this work the extent of non-local correlations when relaxing these fundamental assumptions, which allows for theories with non-local signaling. We prove that, somewhat surprisingly, the Tsirelson bound in the Bell–Clauser–Horne–Shimony–Holt scenario, and similarly other related bounds on non-local correlations, remain effective as long as we maintain the Hilbert space structure of the theory. Furthermore, in the case of Hermitian observables we find novel relations between non-locality, local correlations, and signaling. We demonstrate that such non-local signaling theories are naturally simulated by quantum systems of parafermionic zero modes. We numerically study the derived bounds in parafermionic systems, confirming the bounds’ validity yet finding a drastic difference between correlations of ‘signaling’ and ‘non-signaling’ sets of observables. We also propose an experimental procedure for measuring the relevant correlations.