ϡ (Sampi) Operator — Note IV: Spin, Sphere Eversion, Quantum Tunneling, Causal Structure, and the Dual Referential Operator at the Schwarzschild Event Horizon

Abstract:The ϡ (Sampi) Operator, defined as ϡ(r) = tan(πrₛ/2r), encodes the complete referential transition across the Schwarzschild event horizon in a single analytic function. This note develops four interconnected aspects of its structure. First, the physical grounding of ϡ in the Schwarzschild metric is made precise: Identity VII of the referential-transition algebra recovers the Schwarzschild time-dilation factor by algebraic inversion of ϡ, and a sub-identity establishes that the surface gravity κ = 1/(2rₛ) — and hence the Hawking temperature — is the exact derivative of the same geometric factor that defines f_grav(r). Second, spin is identified as the geometric generator of the tangent's poles, and the Dual Sampi Operator ϡ_dual = −cot(πrₛ/2r) is introduced: it models the smooth local crossing of the horizon in the free-fall frame and satisfies the referential duality identity ϡ · ϡ_dual = −1. Third, the Smale–Boy sphere eversion provides a topological model of horizon crossing formally grounded via π₁(SO(3)) = ℤ₂: the 720-degree closure of the eversion and the spinor double cover are the same topological object; the twist of 180° that inverts the poles is geometrically identical to the quadrant-reordering operation that produces ϡ_dual from ϡ. Fourth, quantum tunneling is the physical realization of the imaginary path; the tunneling at the horizon is spacetime tunneling, not merely spatial, because r assumes a temporal role inside the horizon; the exact pole-density formula ν(r) = rₛ/(2r²) is derived from the hierarchy rₙ = rₛ/(1+2n); and the conserved identity R + T = 1 is established for the reflective-refractive information partition.This revised version introduces six results not present in the initial release: (1) the Dual Sampi Operator and the duality identity; (2) the identification of the horizon pole as a spacetime rupture; (3) the formal topological grounding of the 720-degree eversion via π₁(SO(3)) = ℤ₂; (4) the exact derivation of ν(r) = rₛ/(2r²); (5) the conserved identity R + T = 1; and (6) the sub-identity connecting f_grav(r) to surface gravity κ and Hawking temperature.Keywords:Sampi Operator; Dual Sampi Operator; Schwarzschild metric; time dilation; surface gravity; Hawking temperature; event horizon; spin; sphere eversion; Riemann sphere; quantum tunneling; spacetime tunneling; Tunneling Operator; Whitney Index; causal inversion; referential duality; pole density; Unified Theory of Fields and Universal Dynamics.Pode copiar diretamente nos campos do Zenodo. O abstract vai no campo "Description" e os keywords no campo "Keywords", separados por vírgula ou um por linha dependendo da interface.