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Spin Set as Coherence Disclosure Coupled Spin and Bell Correlations Sections 166 through 169

Philip Lilien

Published Oct 22, 2026• 0 citations
Publisher

Abstract

We now extend the single-spin construction to two spin subsystems. The same rotation representation connects intrinsic spin with singlet and triplet sectors once we adopt quantum composition. With the joint quantum measurement rule, we recover the singlet correlations used in Bell tests. This supplies a precise relation within our taxonomy, while exposing a further requirement: a geometric expression for a correlation must be accompanied by a consistent joint probability law.

DOI
10.5281/zenodo.23574348

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We now extend the single-spin construction to two spin subsystems. The same rotation representation connects intrinsic spin with singlet and triplet sectors once we adopt quantum composition. With the joint quantum measurement rule, we recover the singlet correlations used in Bell tests. This supplies a precise relation within our taxonomy, while exposing a further requirement: a geometric expression for a correlation must be accompanied by a consistent joint probability law.
A unifying taxonomy must explain both shared structure and physical differences. We test a common disclosure requirement: rotating a preparation and its analyzer together should preserve the probabilities. The spinor and vector actions of one Clifford rotor connect spin projection, helicity, and photon polarization while producing different rotation weights. With specified quantum measurements, we recover the angular laws and prove a precise obstruction to identifying the representations. The physical choice of representation remains an input.
When two particles are created together in a single quantum event, they can become entangled: their properties remain correlated no matter how far apart they travel. Measure one, and the outcome of measuring the other is no longer independent. This correlation is the central puzzle of quantum mechanics. Classical physics sets a hard limit on how strong such correlations can be. The CHSH inequality (Clauser, Horne, Shimony, and Holt, 1969) formalises that limit: for any theory in which each particle carries its own independent set of predetermined answers, the correlation measure S cannot exceed 2. Bell-type experiments consistently violate this bound, with the maximum ideal value predicted by quantum mechanics being S = 2√2. Something does not fit. Several explanations have been proposed. Some invoke extra dimensions. Others appeal to wormhole geometries connecting the particles. The mainstream position is that the outcomes "collapse and become correlated," which states the observation without explaining the mechanism. All of these either add dramatic ontological furniture or repackage the mystery as an answer. This paper proposes a simpler reading. The value 2√2 is not a mystery to be explained by exotic physics. It is the geometric signature of a jointly structured bipartite state whose correlation was fixed at the source, at the moment of entanglement, not at measurement time. The two particles are not two independent objects that later communicate. They are two readout surfaces of a single joint preparation. The cosine correlation law, the Tsirelson bound, and the full Bell violation follow directly from the geometry of the two-qubit state space and the relative orientation of the measurement settings. No signal, no extra dimension, no new physics is required. Three companion papers in this series explored the opposite hypothesis: The Speed of Light and Its Limits [1] examined whether entanglement could circumvent relativistic constraints; Quantum Entanglement as a Communication Channel [2] proposed a synchronized-clock experiment to detect timing signatures; and Parametrizing Correlation Timing [3] developed a Lorentz-structured ansatz for concurrence-dependent delays. If those experimental proposals return null, the present reading is reinforced: not by proof, but by elimination of the alternatives that require more. A final section develops geometric and regime-based frameworks consistent with the source-structural reading, including the measurement torus as the natural geometry of Bell experiments, a jointness index based on Concurrence, and connections to emergent spacetime programmes. These are offered as directions for future development.
A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum S=(√3 ℏ)⁄2; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.
This paper shows how spin-1/2 structure and Bell inequality violations are reproduced within the USSFT multiplicity framework. We show that if physical predictions depend only on Omega = |Z|^2 and Z is defined up to an overall phase, then rotational invariance of Omega is compatible with projective representations of SO(3), equivalently linear representations of its double cover SU(2). This compatibility with the double-cover group explains the 4pi periodicity characteristic of spinors. For entangled pairs, the Complex Multiplicity Amplitude Z encodes correlations as a global functional over the joint constraint surface, yielding the quantum mechanical cos^2(theta/2) correlation and violating Bell inequalities up to the Tsirelson bound 2sqrt(2). Marginal probabilities remain invariant under distant measurement choices (no-signaling), preserving compatibility with relativity. Assuming measurement independence (free choice of settings), the account does not require superdeterminism or retrocausality; correlations arise from the global structure of Omega over the entangled system's joint constraint surface. The microscopic mechanism by which the U-field encodes entanglement correlations remains an open problem. This is the ninth paper in the 18-paper USSFT technical series, completing the Quantum Mechanics block (Papers VII-IX); for the U-field Lagrangian foundations see Papers I-VI, for the quantum operator formalism see Paper VII (DOI: 10.5281/zenodo.19675374), and for the Born rule derivation see Paper VIII (DOI: 10.5281/zenodo.19675825).