Exact Volume Formula of the Envelope Interface: Based on the Closed Surface of the Survey Line Spiral and Anti-Survey Line

Based on the geometric framework of the Survey Line Spiral and Anti-Survey Line Spiral, this paper derives the exact volume formula of the Envelope Interface. The Envelope Interface is formed by the closure of two spiral skeletons: the acceleration segment (south pole → equator line) follows the Survey Line Spiral, and the deceleration segment (equator line → north pole) follows the Anti-Survey Line Spiral. The growth rates b and b' of the two segments are independent and generally unequal, so the breathing rate R = L₂/L₁ is not necessarily equal to 1. This paper directly integrates with the true values of b and b' using the volume of revolution formula, starting from the geometric parameters of the spiral skeleton, and obtains the exact volume formula of the Envelope Interface. The formula applies to any projection layer and naturally reduces to the symmetric case when R = 1. Keywords: Envelope Interface, Survey Line Spiral, Anti-Survey Line Spiral, volume formula, breathing rate, projection layer