JMAT_V3_N3_RP1 James Osler II Journal on Mathematics 2277-5137 3 3 7 20 Analysis, Coextensive, Collaboration, Combinations, Complements, Coordinates, Equation, Formula, Investigation, Mathematical Model, Mathematics, Linear, Notation, Parallel, Positive, Proximal, Qualitative, Quantitative, Progression, Research, Statistics, Symmetry, Table, Tricolinear, Tricolinear Summative Squares, Tricolinearity, Trichotomous, Tri–Squared, United, Unlimited, Universal This monograph provides an epistemological rational for the Post Hoc Trichotomous Progression Analysis of the transformative process of qualitative data into quantitative outcomes through the Tri–Squared Test introduced in i-manager’s Journal on Mathematics, and further expounded and detailed in the Journal on Educational Technology, Journal on School Educational Technology, and in Journal on Educational Psychology. This particular advanced statistical metric is designed to measure the predictive outcomes of the outcomes of a significant Transformative Trichotomy–Squared [Tri–Squared] Test. To further explain the Trichotomous Progression Analysis mathematically, “Proximal Positive Parallel Notation” (a novel mathematical notation is used to define the relationship between values)is used to construct the Trichotomous Progression Line. The Progression Line describes the attributes of the Post Hoc significant outcomes as values that describe a specific set of Tri–Squared Test results. This novel approach to advanced Tri–Squared research as an advanced data analysis methodology adds additional value, breadth, and depth to the Tri–Squared mixed methods approach by further enhancing and adding to the rigor of trichotomous research design. Multiple sequential mathematical models are provided that illustrate the methodology used to determine advanced trichotomous progression inquiry. July - September 2014 Copyright © 2014 i-manager publications. All rights reserved. i-manager Publications http://www.imanagerpublications.com/Article.aspx?ArticleId=3106