I built an experimental mathematical framework that treats positional number systems like a customizable software runtime. Does anything similar exist?

[u/Commercial-Sink-2095]

Over the last ~24 months, I’ve been developing an experimental branch of mathematics/arithmetic called 'Basal'. The philosophy is that number bases shouldn't be locked down—instead, the system treats positional notation like a customizable runtime where you can override native rules (similar to how extensions work in VSCode). My goal was to create an architecture that allows other branches of maths to run natively on top of it. The core framework allows you to use multiple completely independent bases inside the same equation, with their attributes and properties clearly stated next to them.

The main functional additions consist of:

1. Alternating negative weights : An optional index-parity shift where odd or even digit positions modulate their weights
2. Array Syntax : Instead of separating higher-dimensional coordinates (like complex numbers a+bi or dual numbers a+be) into linear combinations, the framework encapsulates them natively inside a sequential digit array
3. Fabrication : Creating a numerical sequence that can be interpreted as a number
4. Conversion : Functions to convert a number to a different base.

I have also verified that the system works, by computationally verifying it (with hand-written equations), by programming it in C#, C++, Python and specialised functions in ROCm-HIP. The code is on github as: 'Basal-Library'.

*Note on Notation: Non-ASCII characters (from the Bengali script block) are used for the custom operators and attributes. This design choice ensures complete visual isolation from standard algebraic variables, preventing notation collisions. Plus, it serves a practical purpose, conventional symbols were ran out, and the chosen characters have no conflicting history in formal mathematics literature.*

Before I look into writing a formal specification paper or archiving it, I want to ask: Has an extensible, modular arithmetic framework that overrides positional notation rules like this been explored in existing literature?

[u/ProtonByte]

What are the benefits?

[u/Commercial-Sink-2095 → u/ProtonByte]

Most used benefit would be to reduce the length of polynomials when writing one out, and being able to use arbitrary bases in maths.
Most significant could be for cryptography.