Novel CHSH Bell inequality bound derived from H4 Coxeter geometry. Three independent algebraic proofs, zero free parameters.

The Pentagonal Prism Bell Bound A Golden-Ratio CHSH Inequality from H4 Coxeter Geometry Timothy McGirl · Independent Researcher, Manassas, VirginiaGitHub Repository · Geometric Standard Model (Zenodo) Smax = 4 − φ ≈ 2.381966... A novel CHSH Bell inequality bound derived from H4 Coxeter geometry. Three independent algebraic proofs, zero free parameters. Abstract We derive a novel CHSH-type Bell inequality bound S = 4 − φ (where φ = (1+√5)/2 is the golden ratio) from the geometry of a pentagonal prism inscribed on S². The prism height h² = 3/(2φ) is uniquely determined by H4 Coxeter root system structure via the relation h² = 6φ · det(GH3). Three independent algebraic derivations are presented: (i) from H4/H3 Cartan matrix determinants, (ii) from the Gram determinant hierarchy S = 1 + det(CH2), and (iii) directly from pentagonal prism geometry yielding S = (10φ − 7)/(3φ − 1). All three reduce to 4 − φ using only the minimal polynomial φ² = φ + 1. The bound 4 − φ ≈ 2.382 lies strictly between the classical CHSH limit (S ≤ 2) and the Tsirelson bound (S ≤ 2√2 ≈ 2.828), and is consistent with loophole-free Bell test measurements (S = 2.38 ± 0.14, Delft 2015). Three Independent Proofs I. Cartan Determinant Path γ² = det(CH3)/2 + det(CH4)/4 = (13 − 7φ)/4 S = 2√(1 + γ²) = 4 − φ Verification: (4−φ)² = 17 − 7φ = 4 + (13 − 7φ) ✓ II. Gram Determinant Path 16(det(GH3) − det(GH4)) = det(CH2) = 3 − φ S = 1 + det(CH2) = 4 − φ The Bell bound = 1 + the H2 Cartan determinant. III. Pentagonal Prism Path Prism height: h² = 3/(2φ) = 6φ · det(GH3) S = (10φ − 7)/(3φ − 1) = 4 − φ Cross-multiply: (4−φ)(3φ−1) = 10φ − 7 ✓ All three proofs use only the single algebraic identity φ² = φ + 1 and structures intrinsic to the H4 Coxeter group. No free parameters. Derivation Chain H4 geometry → H2 ⊂ H4 → pentagonal symmetry → prism with h² = 3/(2φ) → 10 directions on S² → Smax = 4 − φ Key Numerical Values Quantity Exact Value Numerical Golden ratio φ (1+√5)/2 1.6180339887... Bell bound S 4 − φ 2.3819660113... Prism height² 3/(2φ) 0.9270509831... Prism height h √(3/(2φ)) 0.9628348680... Geometric parameter γ² (13 − 7φ)/4 0.4184405197... det(CH2) 3 − φ 1.3819660113... det(GH3) (2 − φ)/4 0.0954915028... det(GH4) (5 − 3φ)/16 0.0091186271... Why the Prism, Not the Antiprism The pentagonal prism (not antiprism) is selected because H4 is a reflection group: Property Prism (D5h) Antiprism (D5d) Key symmetry Horizontal reflection σh: z → −z Improper rotation S10 Coxeter element? Yes — proper reflection No — improper rotation Max CHSH |S| 4 − φ ≈ 2.382 ≈ 2.222 The prism decomposes as (H2 reflections) × (ℤ2 reflection), matching the subgroup structure of H4. Uniqueness Theorem The function Smax(h²) for pentagonal prisms on S² is strictly monotonically decreasing in h². As h² → 0 (flat pentagon), Smax → 2.49. As h² → ∞ (poles), Smax → 2. Therefore h² = 3/(2φ) is the unique height at which Smax = 4 − φ. Experimental Proposal The 10 measurement directions are explicitly specified as the vertices of a pentagonal prism on S² with h ≈ 0.9628. A dedicated CHSH experiment using these directions could test whether nature saturates this geometric bound. The Delft loophole-free Bell test (Hensen et al., Nature 526, 682, 2015) reported S = 2.38 ± 0.14, with a central value strikingly close to 4 − φ ≈ 2.382. Numerical Verification Brute-force computation over all 8,100 vertex quadruples confirms: Check Result Quadruples achieving |S| = 4 − φ 80 of 8,100 Quadruples exceeding 4 − φ 0 Relative error < 10−15 Symmetry of optimal set D5h × ℤ2 Core identities formally verified in Lean 4 by the theorem prover Aristotle. Resources GitHub Repository e8-phi-constants E₈/H₄ Geometric Standard Model — source code, verification scripts, data Zenodo DOI The Geometric Standard Model E₈ × H₄ Unification of Fundamental Constants — full GSM framework References 1 J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, "Proposed Experiment to Test Local Hidden-Variable Theories," Phys. Rev. Lett. 23, 880 (1969).2 B. S. Cirel'son (Tsirelson), "Quantum generalizations of Bell's inequality," Lett. Math. Phys. 4, 93 (1980).3 B. Hensen et al., "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres," Nature 526, 682 (2015).4 A. Tavakoli and N. Gisin, "The Platonic solids and fundamental tests of quantum mechanics," Quantum 4, 293 (2020).5 T. McGirl, "The Geometric Standard Model: E₈ × H₄ Unification of Fundamental Constants," Zenodo (2025). doi:10.5281/zenodo.18261289 © 2026 Timothy McGirl · GitHub · Zenodo