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A structured approach for the Collatz conjecture is presented using just theodd integers that are, in turn, divided into categories based on the roles theyplay such as Starter, Intermediary and Terminal. The expression 4x+1 is used asa tool to expose all the hidden and significant characteristics of theconjecture that lead us to its proof. The mixing properties of the iterates areaddressed by showing that the Collatz iterates of half of all the odd integersthat are of the form 4m+3, on the average, increase by three times the value ofthe odd integer that was used to start with, while the iterates of those of4m+1, on the average, decrease by a factor of four. Further, expressions areprovided to generate all the sets of odd integers where the Collatz iterate ofall the integers in each set is an integer of the of form 6m+1 or 6m+5. Thesignificance of the Collatz net (tree) becomes obvious since it encompasses allthe Collatz trajectories.

Collatz Conjecture: Exposition and Proof Through A Structured Approach