  Set out here are some fundamental theories that may be regarded as newly
discovered metamathematics of the odd integers in relation to the Collatz
conjecture (also called the 3x+1 problem). Originally motivated by the
requirement to invent a new optimised integer factorisation method, this
foundational paper primarily focuses on the foundation, formalisation and
presentation of a new theoretical framework (schema or blueprint) of a Collatz
based number system. The proposed framework is based on metamathematical
theories meticulously derived through iterative analyses and reverse
engineering (i.e., by hand and mathematical computations) of many large subsets
of integers. A collation of the fundamental results from these analytical
attempts has led to the establishment of a completely deterministic model of a
generalised Collatz based number system that is fundamentally and strangely
associated with nonchaotic patterns. The proposed Collatz based number schema
comprises of both visual and theoretical representations of many hidden
patterns in Collatz sequences yet to be reported in literature. This novel
theoretical approach may be viewed as a new method to contemporary Collatz
conjecture research which may be connected to the proofs of many other
mathematical theorems in number theory and discrete mathematics.
