ϡ (Sampi) Operator — Note III: Referential Derivatives, Transition Algebra at the Event Horizon,and Accumulated Poles at r = 0

This note presents the third formal record in the series on the ϡ (Sampi) Operator, extending its mathematical treatment to the problem of referential transition between the external observer frame outside the event horizon of a Schwarzschild black hole (r > rₛ) and the freely falling frame (r < rₛ). The operator is defined as ϡ(r) = tan(πrₛ/2r), mapping the radial path r = ∞ → rₛ → 0 onto the poles and zeros of the tangent function. Six derivative forms are derived — with respect to r, rₛ, second derivative, cross-referential derivative, logarithmic/conformal form, and tensorial gradient along the geodesic — together with an algebra of seven referential-transition identities. It is shown that the cross-derivative dϡᵉˣᵗ/dϡᴵⁿᵗ = −1 at the horizon, and that the Cauchy principal value integral of ϡ(r) from 0 to ∞ is zero: the total referential-transition “account” closes. The structure of accumulated poles at r → 0 is then analyzed: the operator exhibits a non-sinusoidal train of +∞ → −∞ discontinuities with local frequency diverging as 1/r², structurally analogous to pair-creation events (Hawking radiation; Schwinger mechanism). A motivated hypothesis is presented connecting this pole structure to a causal inversion with apex at r = 0, relating the singularity to the holographic principle as a possible mechanism of causal projection. This hypothesis is explicitly marked as speculative, distinct from the directly derived results of Sections 1–6.