This paper proposes a zero-free expression framework for representing Collatz sequences. The standard Collatz map operates on positive integers using the rule that even values are divided by two and odd values are mapped by 3n + 1, The framework presented here does not alter the numerical value of Collatz terms, but rejects zero as a primitive digit in the written sequence. Any value containing the digit 0 is instead rewritten as an equivalent expression using nonzero digits and standard arithmetic operations. For example, "10 may be represented as 9 + 1, 20 as 19 + 1, and 100 as 99 + 1", This allows Collatz paths to be displayed through zero-free expression form while preserving their ordinary value. The paper also discusses a signed mirror extension in which positive odd terms use 3n + 1 and negative odd terms use 3n − 1. The framework is presented as a representational and philosophical structure for analyzing Collatz sequences, not as a completed proof of the original Collatz conjecture. Acknowledgement of chatgbt used for organizing and formatting support.
Rjd2 Rj Poeling (Mon,) studied this question.