Quantum Correlations and the Tsirelson Bound in a Physical–Observation Dual-Axis Structure—From the Physobser Axiom and U(1) Phase Transport to CP1, the Born Pairing, and Bell Correlations
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 \(S^3\), and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber \(S^3/U(1)=CP^1≃S^2\). The universal bilinear composite of two independent systems is \(C^4\). 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 \(p(P∣x)=Tr(ρ_xP)\), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form \(E_ρ(a,b)=a^TT_ρb\). Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix \(T_ρ=−I_3\), producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to \(2\sqrt{2}\). Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus \(CP^1\), 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.
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 (S^3), and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber (S^3/U(1)=CP^1≃S^2). The universal bilinear composite of two independent systems is (C^4). 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(ρ_xP)), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form (E_ρ(a,b)=a^TT_ρb). Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix (T_ρ=−I_3), producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to (2\sqrt{2}). Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus (CP^1), 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.
When the conditions under which a record is formed remain fixed, a physical theory may represent that record completely by a unary law of the state observed. This paper asks what structure is forced when the same independently certified physical identity gives different complete records under different formation conditions. The initial datum distinguishes two typed empirical roles, while the state identity of the formation side is derived from its complete behavior. Failure of factorization through the first projection proves that the formation-side role cannot be deleted. The complete row and column behaviors of the primitive law then determine two kernel pairs and their coequalizers. Unique descent retains every observable distinction while removing every redundant label, and the resulting realization is terminal among all exact role-preserving realizations. The product pairs the reduced roles; a pullback extracts the combinations on which records can actually be formed. Only after these steps does exponential transposition identify each surviving formation class with one complete response map. The resulting bijection supplies the state criterion and establishes the observation-state object. Compatibility with the antecedent theory preserves the independently given physical state space and yields the physical–observation law in which a fact is generated jointly by the state observed and the state of observation. The same construction gives exact criteria for conditional, label-free, and joint identifiability and selects the same-source pullback on which transport may subsequently be defined. A nonzero coherent response on that pullback yields the phase-transport fundamental equation, the first fundamental equation written directly on observation space. Quantum theory subsequently enriches this canonical interface. Coherent response and full circular phase select the complex field; reversible binary mixing yields CP1 ∼= S2 ; independent composition creates the four-dimensional event domain; response-event identity glues probabilities across complete contexts; and Gleason representation with marginal consistency gives the Born trace pairing. The two-qubit correlation tensor then yields the singlet cosine law, the CHSH inequality, the Tsirelson bound, and the Horodecki spectral criterion. Bell violation is thereby identified with the absence of a positive global answer section across incompatible observation states. The quantum formalism is thereby closed into one architecture: complex state geometry describes coherent physical possibility, complete observation states specify the ways in which facts may form, the Born interface joins them, and quantum instruments carry a realized fact into subsequent dynamics. Fixed observation sections recover classical unary laws, while smooth equivariant enrichment lifts relativistic frame geometry and internal gauge geometry to the same joint domain. The central result is ontological. Physobser Theory admits observation into objective physical reality not as a mind, an external readout, or a perturbation, but as an irreducible state-bearing axis; a fact is generated at its lawful interface with physical state. Category theory is the proof instrument. The discovered physical content is the dual-axis constitution of facts, which supplies a common foundation for quantum, classical, relativistic, and gauge physics.
Physics ordinarily writes an observation as y = h(x), locating change in the state x while leaving the conditions that form the fact in the background. We prove that, whenever those conditions vary at fixed certified first-state identity, the complete law possesses a natural and irreducible dual-axis structure. The primitive datum is an operationally role-faithful table µ0 : P × A → Y⊥, with ⊥ denoting certified nonimplementability and with no probabilistic, linear, or quantum structure assumed. A fixed-first-role witness blocks every exact factorization through the first projection. Coequalizing identical complete rows and columns then produces behavioral state objects X and S, a unique law Fe : X ×S → Y⊥, and its feasible joint domain. This realization is terminal among active typed exact realizations: every competitor maps uniquely onto it, and the fibers of that map are exactly the competitor’s behavioral redundancies. Moreover, S is canonically bijective with the realized family of complete observation maps, admits no further lossless factorwise quotient, and is invariant under role-preserving empirical covers. When antecedent physics supplies a compatible physical-state map, the construction lifts canonically to Fephys : Xphys × S → Y⊥, yielding the Physobser Axiom. Classical laws are its fixed-observation slices. Quantum theory realizes the two axes as preparation state and observation event, with the Born pairing as their interface; contextuality and Bell nonclassicality then express failures of a context-free one-axis reduction. Because the value object is arbitrary, the theorem applies equally to distributions, channels, trajectories, correlations, and structured records. The result establishes a general foundation for Physobser Theory: a fact is generated by the state of what is observed together with the objective state through which it becomes a fact. Category theory is the instrument by which minimality, invariance, and essential uniqueness are proved; it is not the physical content of the theory. The physical content is the objective co-determination of facts by the two irreducible state axes.
Enlarging observation space extends the available event families, but its effects on correlation bounds and dynamical spectra depend on different assumptions. Within the quantum representation of the physical--observation dual-axis structure (PODA), the real Gram geometry of the Born pairing yields the dimension-independent CHSH bound 22 and a two-dimensional saturating configuration. For a fixed d-dimensional maximally entangled state, arbitrary-rank binary projections give an even--odd dependence, whereas rank-one events yield 2+(42−4)/d. These bounds admit an experimental comparison at fixed preparation. We then introduce an independent joint field with a quadratic action and positive gradient energy along observation space. On the full CP1 with its standard projective metric and Hopf connection, we derive the covariant eigenvalues and degeneracies for integer phase weight. In the minimal nonzero sector ∣q∣=1, the mass levels obey Mn2=mX2+[2n(n+2)+1]mS2, where mX is an independent local mass parameter and mS is determined by the lowest observation eigenvalue and the gradient coefficient. The two lowest distinct mass levels uniquely reconstruct both scales, giving mX2/mS2=(7−r)/(r−1) with r=M12/M02∈(1,7]. Higher levels satisfy relations such as 3M22=8M12−5M02 without further parameters. Exactly equal mass spacings occur if and only if mX=mS. These spectral results depend on the specified action, full observation geometry, and phase representation. The scale ratio remains a dynamical parameter, while the rest-energy relation retains the form E0,n=Mnc2.
A standard Bell observable is a scalar correlation associated with a selected pair of local measurement directions. We formulate it as a projection of a complete two-particle correlation tensor and distinguish two angular sectors. The central result is a reversal of the sign multiplying the cosine law: for coincident calibrated settings, the Bell observable is positive for photons and negative for a fermionic singlet. This distinction provides an operational criterion for experimental identification. For photons, the positive sign follows from averaging two projection amplitudes over the complete non-inertial phase interval; the fermionic sign follows from the negative exchange holonomy of the phase--momentum sector. Mapping phase directions to linear-polarizer axes produces the corresponding double-angle dependence. The photon Stokes tensor has positive linear-polarization components and a negative circular-polarization component, whereas the fermionic singlet has an isotropic negative tensor. We introduce a motion-dependent extended tensor and a frequency-dependent non-inertial susceptibility. A phase-synchronous two-arm experiment combines optical modulation, rotation, and seeded multiaxial piezoelectric vibration. Independent motion measurements separate common and differential components and allow controlled variation from correlated to independent and oppositely driven motion; a single rigid platform is the simpler common-frame limit. Bell-setting and state-correlation vectors express the sign reversal as a scalar projection. The construction yields binary joint probabilities and recovers the Tsirelson bound with oppositely signed optimal CHSH combinations.
Union Dipole Theory postulates two distinct channels of transfer: the particle moves through the permeable medium at a bounded speed, while a disturbance within that medium is stated to propagate at an unbounded one (Al-Mayahi, 2014). This paper asks what that structure implies for correlated pairs, and obtains the singlet joint distribution and the Tsirelson value from a single principle. The principle is single-valuedness of the internal phase — the same condition that produced the phase-closure result of the companion paper. Applied to a pair, it does two things at once. It indexes admissible transitions by integers, so that a shared pair quantum resolves into one channel and not partly into both. And, applied to the two pair channels as degrees of freedom of the medium, it fixes their action integrals, which for a linear medium are proportional to the stored energy. The channel weights are therefore not an added rule about rates: they are what closure gives. One condition supplies both the counting and the quadratic weighting. The consequences follow without further assumption. Writing the pair on its shared internal phase, the two channel amplitudes are 2cos(θ/2)cos φ and −2sin(θ/2)sin φ. Their closed action integrals stand in the ratio cos²(θ/2) to sin²(θ/2); a single shared quantum normalises them; and the correlation is E = −a·b with a CHSH maximum of exactly 2√2. An inversion symmetry then fixes the four joint outcome probabilities uniquely and makes the single-station marginals exactly one half at every angle, so no-signalling holds for the construction and not only for the architecture. Three negative controls are computed rather than asserted, and each is derived in closed form. Extracting outcomes by any local rule gives −(1 − 2θ/π) and |S| = 2 exactly. Sending the indivisible transition to the channel of larger instantaneous drive gives the same linear correlation and the same value 2, and no threshold weight repairs it. Weighting the channels by amplitude rather than by action gives S = 1.657 at the standard settings and, on optimisation over the four settings, exactly 2. All three controls therefore terminate at the Bell bound, and the distance between 2 and 2√2 is attributable to the closure measure and to nothing else in the construction. Two structural results complete the account. The correlation is independent of distance because it is a ratio: any common factor multiplying both channel weights — the inverse-square attenuation of the medium included — cancels identically between numerator and denominator, so the disturbance law is obeyed in full while the correlation does not register it. And a residual anisotropy of the medium leaves every prediction unchanged: perfect anticorrelation at zero angle survives at every order, and the CHSH maximum remains exactly 2√2 on the circular sections of the response ellipsoid. Finally, generalising the channel weighting to an arbitrary exponent shows that the correlation is an ordinary cosine for exactly one value, n = 2, and that the maximal CHSH value is strictly increasing in the exponent and passes through 2√2 at exactly that value. The position of the quantum ceiling is thereby tied to a constitutive property — the quadratic, that is linearly elastic, response of the medium — which is itself what the closure condition selects. The framework is not a local hidden-variable theory, does not evade Bell’s theorem, and lies outside the class the theorem constrains; this is stated as a classification and not as an achievement. What is read rather than derived is named, and what remains open is named: in particular, no dynamical account is given of how the shared medium delivers the single transition to one channel across two separated stations. Two appendices follow. The first compares the predicted angular form with Bell correlations reported on three physically unrelated platforms, spanning CHSH values from 2.07 to 2.71, all of which report the same angular form: strength varies, shape does not. The second sets out a two-coordinate hardware diagnostic which follows from the result and which is the subject of United Kingdom patent application GB 2619873.9, filed 26 August 2026. Companion papers: Al-Mayahi, A. (2026a). Internal Time as a Counted Quantity. https://doi.org/10.5281/zenodo.22110704 — Al-Mayahi, A. (2026b). Union-Dipole Continuum Electrodynamics. https://doi.org/10.5281/zenodo.22110528
Classical theories typically model a complete record as a unary function of a physical state, treating observation conditions as external parameters. However, quantum mechanics exhibits a fundamental tension: the same independently certified physical input can yield different complete records when observation conditions vary, giving rise to the century-old measurement problem and the EPR paradox. In this work, we derive the necessary state architecture to resolve this without conflating the physical input with the conditions through which its record is formed. By applying behavioral equivalence and redundancy elimination within the category of sets, we prove that the formation-side difference cannot be mathematically removed. This forces the “observation state” to emerge as an irreducible, objective, state-bearing axis, establishing a strict physical–observation dual-axis (PODA) architecture. We emphasize that this dual-axis structure represents two non-interchangeable projection responsibilities, rather than orthogonal coordinates or dynamically decoupled systems. Under stated coherence and composition conditions, this dual-axis geometry naturally yields the complex field, binary projective geometry, and the Born trace pairing. Crucially, it further derives the Schrödinger equation as a covariant temporal composition, the CHSH inequality, and the attainable Tsirelson bound. In standard quantum mechanics, the complex field,theSchrödingerequation, and the Born rule are independent axioms. In our framework, they are unified as geometric corollaries of the PODA structure. Within this framework, the notorious quantum paradoxes are naturally dissolved: “wave-function collapse” is revealed as a conditional transition at the observation interface, and “decoherence” is understood as the objective geometric coupling between the physical and observation axes, rather than random environmental noise. Bell violation is thereby identified not as superluminal influence, but as the absence of a positive global answer section across incompatible observation states. Finally, the theory proposes a specific table-top experiment that predicts a sharp, parameter-free numerical constant: a geometric phase of (0.7π ≈ 2.1991) radians. This value directly distinguishes the dual-axis mechanism from standard quantum mechanics, which predicts exactly zero for the same configuration. Observing this constant would confirm the physical reality of the observation axis, whereas a null result falsifies the mechanism.