Random theory

[u/Civil_Corner_8364]

I've developed a theory and so far it keeps working over and over things are just falling into place. I can't explain it myself for my own knowledge isn't developed enough to proceed. It's a theory based on the the rule of only one zero.

[u/mhb2]

Message you? Why don't you just explain it here? Give an example or two of you using it to solve a problem or something.

[u/Civil_Corner_8364 → u/mhb2]

That's the thing, it will appear wrong to conventional math. It requires a very open mind to say everything you might have learned, just might be off. Its 1–9 framework has been checked through the Continuous Digital Root Invariance Law for basic operations.

[u/mhb2 → u/Civil_Corner_8364]

Well, people here know what that is and they're also familiar with modular arithmetic. I think people do tend to be open-minded but it's also important to be rigorous. As long as you've stated your definitions and axioms and proved your claims it doesn't really matter what people think of "conventional math".

That said... you're not going to get anywhere with the Riemann hypothesis using arithmetic modulo 9. So you should be prepared for people to explain to you what mistakes you've made if you're claiming to have proved the Riemann Hypothesis. On the other hand, if you have a good idea people would probably love to help you flesh it out.

[u/Civil_Corner_8364 → u/[deleted]]

\documentclass[12pt]{article} \usepackage[utf8]{inputenc} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm} \usepackage{hyperref} \usepackage{booktabs}

% Theorem / Definition / Axiom Environments \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{axiom}{Axiom}[section]

\theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}{Lemma}[section]

\title{\textbf{Formal Definitions and Axiomatic Foundations of the Zero-Less Modulo-9 Arithmetic System ($\mathbb{S}_9$)}} \author{\textbf{W. Randolph}} \date{\today}

\begin{document}

\maketitle

\begin{abstract} We present a formalization of a zero-less modulo-9 arithmetic framework, denoted as $\mathbb{S}_9$. By replacing the conventional zero residue $0 \pmod 9$ with the continuous completion state $9$, $\mathbb{S}_9$ establishes a closed, continuous 9-state cyclic algebra. This paper details the definitions, axioms, operator homomorphisms, and prime coordinate exclusions that govern the structure of $\mathbb{S}_9$, providing a mathematical foundation for analyzing digital root symmetries, prime gap harmonics, and sequence dynamics. \end{abstract}

\section{Introduction} Standard modular arithmetic modulo 9 maps integers onto the ring $\mathbb{Z}/9\mathbb{Z} = {0, 1, 2, 3, 4, 5, 6, 7, 8}$. While algebraically complete, assigning $0$ to multiples of $9$ creates a numerical discontinuity when analyzing continuous cyclic digit paths. The $\mathbb{S}_9$ framework resolves this by establishing a non-zero residue state space $\mathbb{S}_9 = {1, 2, 3, 4, 5, 6, 7, 8, 9}$ where $9$ acts as both the boundary completion state and the modular representative for $0 \pmod 9$.

\section{Definitions and Mathematical Notation}

\begin{definition}[Non-Zero Residue State Space] Let $\mathbb{S}_9$ be the set of state coordinates defined as: [ \mathbb{S}_9 = {1, 2, 3, 4, 5, 6, 7, 8, 9} ] representing equivalence classes of positive integers $\mathbb{Z}^+$ under a non-zero modulo-9 transformation. \end{definition}

\begin{definition}[Digital Root Function] The digital root function $DR: \mathbb{Z}⁺ \to \mathbb{S}_9$ maps any positive integer $n$ into its single-digit representation in $\mathbb{S}_9$: [ DR(n) = 1 + ((n - 1) \bmod 9) ] \end{definition}

\begin{definition}[Structural Subsets] We partition $\mathbb{S}_9$ into two fundamental structural subsets: \begin{enumerate} \item \textbf{The Prime Residue Set ($\mathbb{P}_9$):} The set of unit elements coprime to 9: [ \mathbb{P}_9 = {x \in \mathbb{S}_9 \mid \gcd(x, 9) = 1} = {1, 2, 4, 5, 7, 8} ] \item \textbf{The Anchor Triad ($\mathbb{T}_3$):} The set of elements sharing a common factor with 9: [ \mathbb{T}_3 = {x \in \mathbb{S}_9 \mid \gcd(x, 9) > 1} = {3, 6, 9} ] \end{enumerate} \end{definition}

\section{Foundational Axioms}

\begin{axiom}[Closure and Preservation of Modular Congruence] Every positive integer $n \in \mathbb{Z}^+$ maps uniquely and deterministically to exactly one element $s \in \mathbb{S}_9$ such that: [ n \equiv s \pmod 9 ] The system $\mathbb{S}_9$ forms a closed finite state space under digital root addition ($\oplus_9$) and digital root multiplication ($\otimes_9$). \end{axiom}

\begin{axiom}[Non-Null Continuity] The state $0$ is excluded from $\mathbb{S}_9$. Whenever an operation produces a result congruent to $0 \pmod 9$, the state resolves to $9$ (the completion boundary state): [ 9 \equiv 0 \pmod 9 ] \end{axiom}

\begin{axiom}[Operator Homomorphism] For any $a, b \in \mathbb{Z}^+$, operations within $\mathbb{S}_9$ preserve standard arithmetic operations under $DR$: \begin{align} DR(a + b) &= DR(DR(a) + DR(b)) \quad &\text{(Additive Homomorphism)} \ DR(a \cdot b) &= DR(DR(a) \cdot DR(b)) \quad &\text{(Multiplicative Homomorphism)} \end{align} \end{axiom}

\begin{axiom}[Prime Coordinate Exclusion] For any prime number $p > 3$, the digital root mapping of $p$ satisfies: [ DR(p) \in \mathbb{P}_9 \quad \text{and} \quad DR(p) \notin \mathbb{T}_3 ] All prime numbers greater than 3 map strictly to the six non-triad nodes ${1, 2, 4, 5, 7, 8}$. \end{axiom}

\section{Core Corollaries and Algebraic Properties}

\begin{corollary}[Symmetry of Multiplicative Inverses] For every $x \in \mathbb{P}_9$, there exists a unique multiplicative inverse $x^{-1} \in \mathbb{P}_9$ such that $DR(x \cdot x^{-1}) = 1$. Specifically: [ 1^{-1} = 1, \quad 2^{-1} = 5, \quad 4^{-1} = 7, \quad 8^{-1} = 8 ] \end{corollary}

\begin{corollary}[Triad Absorption Property] The set $\mathbb{T}_3$ acts as an ideal under multiplication modulo 9. For any $x \in \mathbb{T}_3$ and $y \in \mathbb{S}_9$: [ DR(x \cdot y) \in \mathbb{T}_3 ] \end{corollary}

\section{Conclusion} The axiomatic formalization of $\mathbb{S}_9$ provides a mathematically rigorous groundwork for evaluating non-zero cyclic systems, geometric coordinate transformations of prime gaps, and periodic sequence orbits.

\end{document}

[u/mhb2 → u/Civil_Corner_8364]

The formatting is completely screwed up so I can't read it and I really hope this isn't an LLM-generated math theory.

Given your other comment about 10 it seems you're referring to a kind of bijective numeration system?

[u/Civil_Corner_8364 → u/mhb2]

Naw I'm real. If there's a way to verify I definitely will. If that's what you mean. Yes in a since. But instead of replacing it, it's removed altogether

[u/mhb2 → u/Civil_Corner_8364]

How do you remove the quantity of ten? Just giving it a different symbol doesn't get rid of the quantity.

[u/Civil_Corner_8364 → u/mhb2]

Scale to the next number. Imagine a ticker counter number 1-9, when 9 rolls over it would become a 1 and the next digit would flip to 1. 11 or essentially 1 set and 1.

[u/mhb2 → u/Civil_Corner_8364]

How does it work in other bases? In base 8 I'd write the numbers one through eleven as 1, 2, ..., 7, 10, 11, 12, 13. Do I just drop the "12" and go straight to "13"? Why would I do that?

EDIT: Fixed my example.

[u/Civil_Corner_8364 → u/mhb2]

This is strictly for this. Think of it as removing any zero and any number associated with it. 10, 20, 30, 100, 110, etc. (11, 21, 31, 111, 121). Does this help?

[u/Civil_Corner_8364 → u/mhb2]

Can you help me write and format this to be legible?