Description:A geometric analysis of GHZ–Mermin experiments that separates deterministic parity obstruction, classical Bell constraints, and independent attenuation mechanisms using the incidence structure of \(K_4\). The paper derives an exact edge-cone compatibility criterion, proves uniqueness of attenuation failure, establishes the sharp tradeoff \(W\le 3+e^{-\delta_A}\), extends the construction to \(K_n\), and develops a coarse-grained KL/Fisher framework for statistical separation and experimental design. Keywords:GHZ–Mermin experiments; Bell nonlocality; incidence geometry; edge cones; parity obstruction; attenuation models; Mermin inequality; contextuality; Kullback–Leibler divergence; Fisher information; experimental design; statistical separation; complete graphs.
Description:A geometric analysis of GHZ–Mermin experiments that separates deterministic parity obstruction, classical Bell constraints, and independent attenuation mechanisms using the incidence structure of (K_4). The paper derives an exact edge-cone compatibility criterion, proves uniqueness of attenuation failure, establishes the sharp tradeoff (W\le 3+e^{-\delta_A}), extends the construction to (K_n), and develops a coarse-grained KL/Fisher framework for statistical separation and experimental design. Keywords:GHZ–Mermin experiments; Bell nonlocality; incidence geometry; edge cones; parity obstruction; attenuation models; Mermin inequality; contextuality; Kullback–Leibler divergence; Fisher information; experimental design; statistical separation; complete graphs.
Description:A unified geometric analysis of GHZ–Mermin experiments connecting the binary parity obstruction, classical correlation polytope, independent attenuation cone, and finite-sample information geometry. The paper derives an exact multiplicative criterion for attenuation compatibility and a sharp tradeoff between Mermin violation and attenuation-model failure, W ≤ 3 + e⁻δ_A, then develops Fisher/KL diagnostics for statistically separating nonclassicality from noise-model mismatch. Keywords:GHZ–Mermin inequality; Bell nonlocality; quantum contextuality; parity obstruction; incidence geometry; attenuation cone; noise models; correlation polytopes; information geometry; Kullback–Leibler divergence; Fisher information; experimental design; quantum foundations; multipartite entanglement.
This paper establishes a local structural bridge between two recently developed information-geometric frameworks for observer-dependent probability representations on finite state spaces. Background. On the parametric side, the partition-adapted framework of Khomyakov (KLEO v3.24.1) defines observer entropy as the Kullback–Leibler divergence between a parametric distribution and the uniform lift of its coarse-graining, and shows that it scales quadratically with the resolution threshold, with a coefficient determined by the Fisher information matrix. On the empirical side, the Interference-First Reality (IFR) framework of Bianchi (2026) identifies multi-observer objectivity with the Fréchet barycenter of observer-induced distributions in the Jensen–Shannon metric, with almost-sure convergence guaranteed by metric barycenter theory. Principal results. The paper derives four main results: 1. Local Metric Bridge Lemma (Section 3): On the open probability simplex, the Jensen–Shannon divergence satisfies the local quadratic expansion D_JS(p ‖ p + εδ) = (ε²/8) I_p(δ) + O(ε³), where the explicit coefficient 1/8 arises from the midpoint structure of the symmetric divergence. 2. Multi-Observer Quadratic Aggregation Theorem (Section 4): For a partition-adapted parametric family satisfying conditions (A1)–(A3) of Khomyakov (KLEO v3.24.1), the empirical Jensen–Shannon Fréchet barycenter of an N-observer ensemble, centered on the within-class neutral locus, exists uniquely for small ε and coincides to leading order with the weighted Fisher–Rao barycenter of the within-class perturbation directions. The minimal Jensen–Shannon dispersion satisfies a quadratic scaling law governed by the within-class block I_ww(θ₀*) of the Fisher information matrix. 3. Leading-order coincidence of estimators (Corollary 6.1): The variational Fréchet minimizer p_F and the closed-form geometric overlap estimator p_G of Bianchi (2026) agree to order O(ε²) in Jensen–Shannon distance within the local regime. 4. Multi-observer Landauer corollary (Corollary 6.6): Under a per-channel irreversibility assumption and an additivity postulate for independent observer channels, the aggregate energetic cost of barycenter formation is bounded below by k_B T times the Jensen–Shannon dispersion. Scope. The bridge is local in ε (the within-class perturbation amplitude) and finite in N (the number of observers). It does not subsume Bianchi's almost-sure N → ∞ consistency theorem nor Khomyakov's ε → 0⁺ quadratic scaling result; it establishes their structural compatibility in the overlapping regime where both apply. The results are specialized to balanced exponential families (softmax), with explicit worked-example coefficients for four- and five-point state spaces. Domain-specific applicability is assessed against the cross-domain distinguishability hierarchy of Bianchi (2026), Table 14, identifying the low-dispersion regime (d_JS ≲ 10⁻² nats) as the primary domain of validity. Keywords: Jensen–Shannon divergence, Kullback–Leibler divergence, Fisher information, Fisher–Rao metric, multi-observer aggregation, Fréchet barycenter, probability simplex, partition-adapted parameterization, observer entropy, information geometry. MSC 2020: 62B10, 94A17, 53B12, 62F12.
We present a constraint-based axiomatization of quantum correla- tions that derives Bell statistics without invoking Hilbert space, wave functions, or causal mechanisms. Starting from five physically moti- vated axioms—rotational invariance, perfect anti-correlation, uniform marginals (no-signaling), smoothness, and extremality—we prove that the singlet correlation function E(ˆ a,ˆˆb) =−ˆa·b is uniquely deter- mined. The proof proceeds through functional equation analysis: the extremality axiom implies angle additivity, reducing the problem to Cauchy’s equation with boundary conditions. We derive the Tsirelson bound |S|≤2√2 directly from this correlation function without op- erator algebras, and show that CHSH violation follows as a theorem. The no-signaling condition emerges as a structural property of the con- straint, ensuring compatibility with relativistic causality. We extend the framework to tripartite systems, deriving GHZ correlations from analogous axioms and proving maximal Mermin inequality violation. The constraint interpretation dissolves the tension between entangle- ment and causality: correlations are neither caused nor explained by hidden variables—they are specified by the constraint structure itself. This framework recovers all quantum predictions while eliminating causation from the foundational ontology.
There exist numerous proofs of Bell's theorem, stating that quantum mechanics is incompatible with local realistic theories of nature. Here the strength of such nonlocality proofs is defined in terms of the amount of evidence against local realism provided by the corresponding experiments. Statistical considerations show that the amount of evidence should be measured by the Kullback-Leibler (KL) or relative entropy divergence. The statistical strength of the following proofs is determined: Bell's original proof and Peres' optimized variant of it, and proofs by Clauser, Horne, Shimony, and Holt (CHSH), Hardy, Mermin, and Greenberger, Horne, and Zeilinger (GHZ). The GHZ proof is at least four and a half times stronger than all other proofs, while of the two-party proofs, the one of CHSH is the strongest.
A local deterministic model can reproduce Bell-inequality violations only if the hidden state depends on the measurement settings. Hall found the minimum such dependence for the singlet state [Phys. Rev. Lett. 105, 250404 (2010)] and posed the corresponding problem for the Mermin state [Phys. Rev. A 84, 022102 (2011)]. This paper solves it, for the n-party GHZ-Mermin scenario at every n≥3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is Fmin(n)=R/[2(R+1)], with R=2⌊(n−1)/2⌋ the Mermin violation ratio: 1/3,2/5,4/9,8/17,…, always below 1/2. The lower bound follows from Mermin's inequality by a general overlap bound; the upper bound is an explicit model, uniform on the hidden-variable classes that violate the fewest constraints, with closed-form counts; a reduction theorem shows that for Mermin scenarios the marginals impose no additional dependence. The Mermin values are also certified by integer-arithmetic enumeration through n=22. The formula was stated as a conjecture in the first version of this paper and was first proved, by a different route, by Wang, Yao and Chen; the proof here was obtained independently and the two agree. Every minimal contradiction of size k among stabilizer constraints has correlator-only minimum 1/(k−1). With the marginals included the minimum is the same on every core of the C5 ring cluster, on the size-4 and size-5 cores of C4 and on every four-core of C6, but larger on the size-6 and size-7 cores of C4 and on 112 of the 1,920 five-cores of C6, with exact certificates on both sides in every case.
This work introduces a resolution-dependent geometric framework—Quantum Fractal Geometry (QFG)—in which the effective physical description of systems depends explicitly on observational scale. Within this framework, entangled quantum systems are modelled as unified geometric structures at deep resolution, characterised by a scale-dependent metric g_{μν}(ℓ) and a geometric phase field φ(x,ℓ) valued on S¹. Measurement is interpreted as a resolution-dependent projection of this unified structure onto detector-scale observables. A symmetry-constrained coarse-graining map is introduced as the minimal fixed point of resolution flow, identifying antipodal phase configurations (S¹ → S¹/ℤ₂). Under isotropy, continuity, and convex-linearity, the quantum correlation function E(a,b) = −cos(θ_a − θ_b) is recovered as the unique rotationally invariant statistic compatible with the projection structure, saturating the Tsirelson bound. Within QFG, Bell inequality violation arises not from nonlocal signalling or measurement dependence, but from the failure of scale-independent separability. Locality (in the operational no-signalling sense) and measurement independence are preserved. The framework further outlines a programme for deriving the Born rule from resolution-induced measure structure and identifies falsifiable predictions, including resolution-dependent deviations from ideal cosine correlations. This work should be understood as establishing the minimal geometric structure required for the QFG interpretation of Bell correlations, with full dynamical derivations deferred to future work.