EXACT CLOSED COSMOLOGICAL BACKGROUNDS SOURCED BY AN ISOTROPIC HOPF TRIAD: Conformal-Direction Stability Bounds and Observability

This study develops an exact analytical framework for closed cosmological backgrounds sourced by an isotropic triad of Hopf maps on S³. Three geometrically coordinated fields nₐ: S³ → S² generate an isotropic stress-energy tensor compatible with the Friedmann–Lemaître–Robertson–Walker (FLRW) geometry. Within an explicitly specified Einstein–sigma-model action supplemented by a positive Faddeev–Skyrme interaction, the resulting energy density and pressure are ρ = 12κ/a² + 24λ/a⁴ and p = −4κ/a² + 8λ/a⁴. The construction provides an exact family of nonsingular bouncing solutions, with scale factor a(t)² = [C + √(C² − 4h²D) cosh(2h(t − t₀))] / (2h²), where h² = Λ/3, C = 1 − 32πGκ, and D = 64πGλ. Explicit inequalities determine the parameter domain in which a nondegenerate cosmological bounce exists. A central result concerns the distinction between topological conservation and dynamical stability. Nonzero Hopf charge alone does not guarantee a positive lower bound on the quadratic field energy. The quartic interaction introduces a stabilizing contribution, but its effectiveness is quantitatively restricted. The conformal matter-energy second variation is nonnegative precisely when 4λ/(κa²) ≥ 1. Compatibility with a regular bounce requires the sharper bounds 0 < η < 1/2 and η/2 − 3η²/4 ≤ ζ < (1 − η)²/4, where η = 32πGκ and ζ = 64πGλh². Crucially, this protection cannot persist throughout an indefinitely expanding universe with fixed finite couplings. These are exact results for a specified conformal variation, not a demonstration of complete Einstein–matter perturbative stability. The analysis also establishes important limits of cosmological observability. The quadratic Hopf contribution scales as a⁻² and is degenerate with spatial curvature in the background expansion, while the quartic contribution scales as a⁻⁴ and is degenerate with conventional radiation. Consequently, H(z) alone cannot uniquely identify the underlying Hopf-sector parameters. Additional geometric, perturbative, or topological observables are required to distinguish these contributions. A finite-sample quantum tomography construction further provides explicit statistical bounds for reconstructing the three Hopf classes, connecting topological information with measurable resource requirements. The resulting framework combines exact cosmological solutions, rigorous conformal-direction bounds, topological constraints, and operational observability, while keeping its physical assumptions and the limits of its predictions explicit.