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