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The Tsirelson Bound in Absolute Frame Theory: Two Convergent Routes to the Quantum Correlation Ceiling

Patricio E. Valenzuela

Published Jun 9, 2026• 0 citations
Publisher

Abstract

A no-signaling theory may, in principle, be more nonlocal than quantum mechanics: the Popescu--Rohrlich box reaches the algebraic Clauser--Horne--Shimony--Holt value $4$, whereas nature stops at the Tsirelson value $2\sqrt2$. Absolute Frame Theory (AFT) produces nonlocal correlations---two systems entangled in $\mathcal{M}$ share a single point of the substratum $\mathcal{A}$ (Axiom II)---and forbids superluminal signaling (the channel is Bremermann-bounded). We ask where those correlations fall, and we establish, by two independent routes, that they cannot exceed the Tsirelson bound: AFT admits no super-quantum correlations. The operator route proves, inside the reconstructed Hilbert theory, the Tsirelson operator identity, with the local commutativity $[A_i,B_j]=0$ itself derived from AFT structure---the tensor factorization that local tomography supplies and the field microcausality of the constrained embedding---rather than assumed. The information route reduces the bound to a physical principle: AFT's shared resource is read out through a completely positive, trace-preserving channel of finite classical capacity (a theorem of the channel, founded on action quantization and the trace over the inaccessible fiber), this channel obeys the data-processing inequality, and the combination places AFT in the class that satisfies Information Causality, which singles out $2\sqrt2$ among the no-signaling theories. We state plainly what this does and does not achieve. It does not close the question of why the bound is $2\sqrt2$ rather than some other value: both routes, chased to their foundations, terminate at the same posited fact---the $\mathcal{M}$--$\mathcal{A}$ signature transition (the Wick rotation) that installs the complex-Hilbert structure---and finite capacity alone, stripped of that structure, does not exclude a Popescu--Rohrlich box. The two routes are thus the operator-algebraic and information-theoretic faces of one ceiling, and they reduce the selection of $2\sqrt2$ to a single substratum fact, with the residue an instance of the deepest open question of the programme rather than a problem specific to nonlocal correlations.

DOI
10.5281/zenodo.20613302

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Quantum Theory as the Operational Shadow of the $\mathcal{M}$--$\mathcal{A}$ Channel: a Substratum Reconstruction in Absolute Frame Theory

Patricio E. Valenzuela

The reconstruction program of quantum foundations asks why nature is described by complex Hilbert spaces with the Born rule, and why its correlations stop at the Tsirelson bound rather than saturating the no-signaling limit. Absolute Frame Theory (AFT) embeds the observable four-dimensional manifold $\mathcal{M}$ in a continuous Euclidean substratum $\mathcal{A}$ carrying a pilot wave $\Psi$, with an information-bottleneck channel that projects $\mathcal{A}$ onto $\mathcal{M}$ and traces out an inaccessible fiber. We propose that finite-dimensional quantum theory is the operational shadow of that channel---the channel viewed from within $\mathcal{M}$, restricted to its $\mathcal{M}$-accessible, identifiable part---and we carry out the reconstruction in that reading. This is a program, not a completed theorem: we mark every claim as derived, as a structural correspondence, or as an open frontier, and we state plainly that AFT is not yet formulated as an operational-probabilistic theory in full rigor and that the complex Hilbert structure is here a mechanism (the analytic-signal representation of the channel's sampling), not a posit. Within those limits the reconstruction reduces the why-of-quantum-theory to a single physical fact---the $\mathcal{M}$--$\mathcal{A}$ signature transition, the Euclidean-to-Lorentzian Wick rotation---of which the complex structure and the Born rule are the two faces. Concretely, a linear map carries both the complex generator and the conserved $|\Psi|^2$ density if and only if it is unitary. The same signature transition is, in AFT, the root of parity violation and of the arrow of time; quantum theory, parity violation, and time's arrow thus share one origin. The operator-level complex structure that makes the theory locally tomographic is supplied by the Osterwalder--Schrader reconstruction about that single Euclidean time---reflection positivity being a theorem for the free embedding sector and preserved, by a local Feynman--Kac argument, for static configurations---hence complex quantum theory is derived, not posited, in the static sector. The named open frontier is then the time-dependent and interacting reconstruction, while the Born measure $|\psi|^2$ is shown to be the equivariant density of the reconstructed unitary and generic macroscopic matter is derived to be classical, leaving only the strict Dürr--Goldstein--Zanghì uniqueness open. That the embedding yields the Lorentzian signature, and with it emergent local Lorentz invariance, is established in a companion paper on emergent Lorentz invariance.

•Jun 4, 2026•0 citations

From Primitives to the Bell–Tsirelson Bound

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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.

•May 12, 2026•0 citations

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Abstract Objectives: To derive assumptions of Bell Inequalities from one single postulate – Absolute Postulate of Universe – as mentioned by the author S.Bagaria. Method: It focuses on the assumptions – locality, reality, measurement independence and counter-factual definiteness - made in proving Bell’s Inequality and derive them from Absolute Postulate of the Universe. Then, its consequences are studied like PR Boxes being real, DI-QKD, quantum information theory, randomness and foundations of quantum mechanics. Findings: From Absolute Postulate of the Universe, it has been showed that there is absolute determinism in world, resulting in zero theoretical uncertainty. Thus, the algebraically maximal correlations (S = 4) are possible. Thus, the PR-boxes can be real. But, Tsirelson Bound can’t be violated in quantum computing as HUP can’t be violated in Quantum Physics. Novelty: The four assumptions of Bell Inequality are derived from Absolute Postulate of the Universe. The correlations in the universe can be maximal yet unverifiable experimentally. EPR paradox is resolved without assuming faster than light transfer of information as the world is shown to be “non-local with no-signalling”. Keywords: Bell inequality, Popescu–Rohrlich box, Tsirelson bound, Quantum nonlocality, No-signalling, CHSH inequality, Quantum violation, absolute determinism

•Oct 22, 2026•0 citations

A violation of the Tsirelson bound in the pre-quantum theory of trace dynamics

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The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value $2\sqrt{2}$, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than $2\sqrt{2}$. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.

arXiv:2208.02209•Aug 3, 2022•8 citations

A Finite Two-Site Route to the Tsirelson Bound: Exchange Symmetry, Shared Closure, and the Remaining Measurement Gate

Amit Saxena, Ashok Kumar Saxena, Ashwarya Saxena

Tsirelson’s 2√2 ceiling is a finite-dimensional operator theorem; it does not require acontinuum quantum field theory. We formulate the exact two-site result needed by the Treoshared-closure programme and separate it from the framework-specific physical identification.In the one-excitation subspace span{|01⟩, |10⟩}, imposing a definite exchange parity uniquelyselects (|01⟩ ± |10⟩)/√2, up to phase. These are maximally entangled Bell states. For any twodichotomic observables at each site, the CHSH operator obeys ∥B∥ ≤ 2√2; suitable local axessaturate the bound exactly. No-signaling follows from the tensor-product trace rule.Two shortcuts must be avoided. Equality of the two reduced states alone does not implymaximal mixing, and the printed + Bell state does not automatically have the singlet correlationE(a, b) = −a · b in the same local basis. A local unitary maps the Bell states into one another,so their maximal CHSH value is the same, but the observable convention must be transformedwith the state. The remaining Treo burden is therefore not the algebraic bound. It is to derivethe local two-level register, the one-excitation shared-closure constraint, the allowed detectorobservables, and their spacelike operational readout from the native finite dynamics. The C9successor is not a premise of this two-site theorem.

•Sep 21, 2026•0 citations

The Tsirelson Bound in Absolute Frame Theory: Two Convergent Routes to the Quantum Correlation Ceiling

Patricio E. Valenzuela

•Jan 1, 2026•0 citations
A no-signaling theory may, in principle, be more nonlocal than quantum mechanics: the Popescu--Rohrlich box reaches the algebraic Clauser--Horne--Shimony--Holt value 444, whereas nature stops at the Tsirelson value 222\sqrt222​. Absolute Frame Theory (AFT) produces nonlocal correlations---two systems entangled in M\mathcal{M}M share a single point of the substratum A\mathcal{A}A (Axiom II)---and forbids superluminal signaling (the channel is Bremermann-bounded). We ask where those correlations fall, and we establish, by two independent routes, that they cannot exceed the Tsirelson bound: AFT admits no super-quantum correlations. The operator route proves, inside the reconstructed Hilbert theory, the Tsirelson operator identity, with the local commutativity [Ai,Bj]=0[A_i,B_j]=0[Ai​,Bj​]=0 itself derived from AFT structure---the tensor factorization that local tomography supplies and the field microcausality of the constrained embedding---rather than assumed. The information route reduces the bound to a physical principle: AFT's shared resource is read out through a completely positive, trace-preserving channel of finite classical capacity (a theorem of the channel, founded on action quantization and the trace over the inaccessible fiber), this channel obeys the data-processing inequality, and the combination places AFT in the class that satisfies Information Causality, which singles out 222\sqrt222​ among the no-signaling theories. We state plainly what this does and does not achieve. It does not close the question of why the bound is 222\sqrt222​ rather than some other value: both routes, chased to their foundations, terminate at the same posited fact---the M\mathcal{M}M--A\mathcal{A}A signature transition (the Wick rotation) that installs the complex-Hilbert structure---and finite capacity alone, stripped of that structure, does not exclude a Popescu--Rohrlich box. The two routes are thus the operator-algebraic and information-theoretic faces of one ceiling, and they reduce the selection of 222\sqrt222​ to a single substratum fact, with the residue an instance of the deepest open question of the programme rather than a problem specific to nonlocal correlations.
Quantum Theory as the Operational Shadow of the M\mathcal{M}M--A\mathcal{A}A Channel: a Substratum Reconstruction in Absolute Frame Theory
The reconstruction program of quantum foundations asks why nature is described by complex Hilbert spaces with the Born rule, and why its correlations stop at the Tsirelson bound rather than saturating the no-signaling limit. Absolute Frame Theory (AFT) embeds the observable four-dimensional manifold M\mathcal{M}M in a continuous Euclidean substratum A\mathcal{A}A carrying a pilot wave Ψ\PsiΨ, with an information-bottleneck channel that projects A\mathcal{A}A onto M\mathcal{M}M and traces out an inaccessible fiber. We propose that finite-dimensional quantum theory is the operational shadow of that channel---the channel viewed from within M\mathcal{M}M, restricted to its M\mathcal{M}M-accessible, identifiable part---and we carry out the reconstruction in that reading. This is a program, not a completed theorem: we mark every claim as derived, as a structural correspondence, or as an open frontier, and we state plainly that AFT is not yet formulated as an operational-probabilistic theory in full rigor and that the complex Hilbert structure is here a mechanism (the analytic-signal representation of the channel's sampling), not a posit. Within those limits the reconstruction reduces the why-of-quantum-theory to a single physical fact---the M\mathcal{M}M--A\mathcal{A}A signature transition, the Euclidean-to-Lorentzian Wick rotation---of which the complex structure and the Born rule are the two faces. Concretely, a linear map carries both the complex generator and the conserved ∣Ψ∣2|\Psi|^2∣Ψ∣2 density if and only if it is unitary. The same signature transition is, in AFT, the root of parity violation and of the arrow of time; quantum theory, parity violation, and time's arrow thus share one origin. The operator-level complex structure that makes the theory locally tomographic is supplied by the Osterwalder--Schrader reconstruction about that single Euclidean time---reflection positivity being a theorem for the free embedding sector and preserved, by a local Feynman--Kac argument, for static configurations---hence complex quantum theory is derived, not posited, in the static sector. The named open frontier is then the time-dependent and interacting reconstruction, while the Born measure ∣ψ∣2|\psi|^2∣ψ∣2 is shown to be the equivariant density of the reconstructed unitary and generic macroscopic matter is derived to be classical, leaving only the strict Dürr--Goldstein--Zanghì uniqueness open. That the embedding yields the Lorentzian signature, and with it emergent local Lorentz invariance, is established in a companion paper on emergent Lorentz invariance.
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.
Abstract Objectives: To derive assumptions of Bell Inequalities from one single postulate – Absolute Postulate of Universe – as mentioned by the author S.Bagaria. Method: It focuses on the assumptions – locality, reality, measurement independence and counter-factual definiteness - made in proving Bell’s Inequality and derive them from Absolute Postulate of the Universe. Then, its consequences are studied like PR Boxes being real, DI-QKD, quantum information theory, randomness and foundations of quantum mechanics. Findings: From Absolute Postulate of the Universe, it has been showed that there is absolute determinism in world, resulting in zero theoretical uncertainty. Thus, the algebraically maximal correlations (S = 4) are possible. Thus, the PR-boxes can be real. But, Tsirelson Bound can’t be violated in quantum computing as HUP can’t be violated in Quantum Physics. Novelty: The four assumptions of Bell Inequality are derived from Absolute Postulate of the Universe. The correlations in the universe can be maximal yet unverifiable experimentally. EPR paradox is resolved without assuming faster than light transfer of information as the world is shown to be “non-local with no-signalling”. Keywords: Bell inequality, Popescu–Rohrlich box, Tsirelson bound, Quantum nonlocality, No-signalling, CHSH inequality, Quantum violation, absolute determinism
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value 222\sqrt{2}22​, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than 222\sqrt{2}22​. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.
Tsirelson’s 2√2 ceiling is a finite-dimensional operator theorem; it does not require acontinuum quantum field theory. We formulate the exact two-site result needed by the Treoshared-closure programme and separate it from the framework-specific physical identification.In the one-excitation subspace span{|01⟩, |10⟩}, imposing a definite exchange parity uniquelyselects (|01⟩ ± |10⟩)/√2, up to phase. These are maximally entangled Bell states. For any twodichotomic observables at each site, the CHSH operator obeys ∥B∥ ≤ 2√2; suitable local axessaturate the bound exactly. No-signaling follows from the tensor-product trace rule.Two shortcuts must be avoided. Equality of the two reduced states alone does not implymaximal mixing, and the printed + Bell state does not automatically have the singlet correlationE(a, b) = −a · b in the same local basis. A local unitary maps the Bell states into one another,so their maximal CHSH value is the same, but the observable convention must be transformedwith the state. The remaining Treo burden is therefore not the algebraic bound. It is to derivethe local two-level register, the one-excitation shared-closure constraint, the allowed detectorobservables, and their spacelike operational readout from the native finite dynamics. The C9successor is not a premise of this two-site theorem.