## Slide Index for Slide Show

**Structure for Collatz Trajectories**

Collatz 3n+1 Conjecture Defined

General Predecessor Tree and Predecessor Formulas

General to Binary Tree Conversion

Features in Binary Predecessor Tree

State Transition Diagram for Predecessor Tree

Abstract Predecessor Tree Features

**Features in the Predecessor Trees**

Extensions, L.d.a.s Disjoint in Abstract Tree; Binary --> Abstract Mapping

Characterizing Set Elements (dn+c formulas)

Descent Portion of Formula Derivation

Ascent Portion of Formula Derivation

Deriving *d* of {*c[d]*} values for L.d.a. elements

Set Densities From Finite L.d.a. Element Sets

Unique Subsets in Abstract Tree

**All Integers Appear in the Predecessor Tree**

Use Densities to "Count" Integers in Abstract Tree

Densities in a Cardinality Table

Summation of the Densities ==> Integers Covered

**Finally, a Proof From Graph Theoretic Properties of the Predecessor Graph Nodes**

Graphical Properties of Nodes of Predecessor Trees

Binary Tree Constructed from Graph Elements

Summary of Observations Towards a Proof

**Some Slides in Slightly Off-Topic Categories**

** Paradoxes Appear and Show Useful Behaviors**

Paradox on (L.d.a. Header Values)/(L.d.a. Leaf Values)

Paradox on Odd/Even Integer Balance in Predecessor Tree

Paradox in Feinstein's Proof of Unprovability of Collatz Conjecture

**Alternative Proof Approaches**

L.d.a.s as Sieving Elements

Finding Any Integer in Abstract Tree

**A Sidelight: Homologous Collatz-Like Iterations**

(3n+3^j)/2^i Homologs of Collatz Conjecture

** What About Infinity?**

Comments on Metrics for Infinity

**Cemetary**
I just can't bear to throw anything away.

What Still Remains to Prove?

A Poorer L.d.a. State Transition Diagram

My Collatz Home Page
Index to Terms Used