A universal structural constant emerging from the information-geometric 2 ⊕ 3 symmetry

Overview This work establishes a new universal structural constant λ₃* ≈ 0.152 derived entirely from first principles—without free parameters or fitting. The constant emerges from the interplay of information geometry, group theory, and variational principles, suggesting a deep connection between quantum structure and cosmic organization. Key Discovery We introduce the Logos functional: J(λ; α) = −H(λ) + α L(λ) where: H(λ) is the Shannon entropy (information differentiation) L(λ) = −ln(λ² + (1−λ)²) is a logarithmic structural tension α = ln(5/2) is group-theoretically determined by the decomposition 5 = 2 ⊕ 3 The variational condition δJ/δλ = 0 yields a unique fixed point: λ₃* = 0.1520845... Three Independent Manifestations Context Value Origin Logos Functional λ* = 0.1520845... Variational principle with α = ln(5/2) Quantum GHZ States λ* = 0.152044... Odd-parity structural extremum (N = 3, 9...) Cosmological Baryon Fraction fb ≈ 0.156 Planck 2018 (~3% deviation from λ₃*) The Group-Theoretic Foundation: 5 = 2 ⊕ 3 The ratio r = 2/5 is not arbitrary—it is enforced by fundamental symmetry breaking patterns: SO(5) → SO(3) × SO(2) — Rotational symmetry decomposition SU(5) → SU(3) × SU(2) — Grand Unified Theory breaking Kaluza–Klein 5D gravity — 2 massless + 3 hidden geometric modes Gauge bosons — Massless (2 DOF) vs. massive (3 DOF) This universal 2/5 ratio determines α = ln(5/2) ≈ 0.916, which in turn fixes λ₃*. Five-Layer Derivation (Appendix A) The paper includes a complete continuity proof eliminating any "speculative leaps": Layer 1: GHZ odd-parity extremum → λ* ≈ 0.152 Layer 2: Axiomatic uniqueness of the Logos functional (4 axioms) Layer 3: Variational fixed point → λ* = 0.1520845... Layer 4: Group-theoretic origin of α from 5 = 2 ⊕ 3 Layer 5: Cosmological echo (fb ≈ 0.156 as macroscopic reflection) Scientific Rigor What we prove (Theorem): ✓ Unique fixed point λ₃* from the Logos functional ✓ Numerical coincidence with GHZ structural extremum What we conjecture (Hypothesis): ○ Connection to cosmological baryon fraction (~3% deviation acknowledged) ○ λ₃* as an asymptotic attractor for cosmic structure We distinguish rigorously between mathematical proof and physical speculation. Significance If correct, λ₃* joins the pantheon of universal constants—like π or the fine-structure constant α—that encode deep structural properties of reality. Unlike empirically measured constants, λ₃* is derived from first principles: information geometry + group theory + variational calculus. This framework suggests that fundamental parameters may arise from information-geometric necessity rather than dynamical accident. Keywords Universal constants · Information geometry · GHZ states · Baryon fraction · Symmetry breaking · Group theory · Logos functional · Shannon entropy · Grand unified theory · Kaluza-Klein