A redefinition of what numbers can be — and what they can model.
Universal Number Set (UNS) Runtime Playground
The Universal Number Set (UNS) is an extended number system built around a simple but transformative idea:
A number is not a single value. A number is a distribution of values across a universe of microstates.
In UNS, every “number” is actually a function from microstates to values, defined on a normalized measure space.1 This design gives UNS mathematical behaviors that classical numbers cannot express — without breaking compatibility with real and complex numbers.
Formally:
[ U = (X, \mu), \quad \mu(X) = 1 ]
[ u : X \to \mathbb{C}, \quad \psi : X \to \mathbb{C}, \quad \int |\psi(x)|^2 , d\mu = 1 ]
[ \operatorname{read}(u \mid \psi) = \int u(x) \cdot |\psi(x)|^2 , d\mu ]
Classical numbers embed as constant functions (e.g., const(7)(x) = 7), so UNS strictly contains ℝ and ℂ while extending them.
UNS is both:
- a mathematical framework, and
- a formal language for expressing operations, compositions, and transformations over these extended numbers.
If classical numbers are points, UNS numbers are landscapes.
The origin of UNS emerged from exploring a series of conceptual questions:
Classical math forces numbers to be:
- single-valued
- dimensionless
- context-free
- undefined under certain operations (e.g., divide-by-zero)
- “fragile” when applied to systems with distributed or multi-state behaviors
But many real and conceptual systems are:
- distributed
- dimensional
- contextual
- probabilistic
- state-dependent
- nonsingular
Classical numbers weren’t designed with this worldview in mind.
UNS introduces:
- microstates (the smallest places where value can live)
- states (which tell you how to interpret a UNS number)
- distributed values (functions instead of points)
- readout rules (how classical values emerge from context)
This allows UNS numbers to behave differently depending on the state through which they are viewed—yet still follow stable, formal rules.
One of UNS’s most striking properties:
A UNS number can appear point-like or N-dimensional depending on the state, yet yield the exact same classical readout.
This mirrors deep symmetry principles found in physics and information theory, echoing Hilbert-space equivalences while remaining agnostic to a single inner-product structure.2
UNS formalizes this by design.
When classical math encounters singularities (like division by zero), UNS instead generates:
novel values — formally typed, traceable, valid outputs that extend the number system without contradiction.
This makes UNS an open number universe: closed under all lifted operations, including those that classically fail.
UNS is not intended to replace classical mathematics. Instead, it is ideal for modeling or experimenting with systems characterized by:
- distributed values
- contextual or state-dependent results
- dimensional equivalence or symmetry
- singularities or undefined classical behaviors
- probabilistic or microstate-based interpretations
- abstract number discovery
Potential applications include:
- conceptual or theoretical modeling
- simulation frameworks
- generative or exploratory mathematics
- LLM-based reasoning systems
- research into extended numeric structures
- systems where “point numbers” are too limiting
UNS combines formal mathematical rigor with an exploratory design ethos.
The formal definition of UNS: grammar, rules, operators, semantics, and foundational axioms. → RFC/UNS_RFC.md
Defines extended operators such as cancellation, along with any additional lifted functions or helper constructs. → Runtime/Specification/UNS_Runtime32_Spec.md#11-uns-operator-extensions
UNS includes a dedicated file type for storing UNS programs and expressions.
The .unse spec describes:
- structure
- comments
- encoding rules
- readout conventions → RFC/UNS_RFC.md#20-uns-expression-files-unse
A narrative, intuitive walkthrough of the thinking behind UNS:
microstates, distributions, dimensional equivalence, novel values, lifted operators, and more.
This is not part of the spec but is ideal for understanding the why behind the system.
→ Included in repo as UNS_Guided_Discovery.md
A stylized UNS symbol (derived from the letters U–N–S) and banner graphics suitable for web, documentation, and packaging. → Assets are being curated; see TRADEMARKS.md for current usage guidance until the vector set lands in-repo.
Sample .unse files showing:
- distributed values
- lifted operations
- dimensional transforms
- generation of novel values
You can use UNS in two main ways:
Write and evaluate .unse expressions describing UNS values and operations.
Use UNS ideas to:
- model systems with state-dependent interpretations
- explore dimensional symmetry
- experiment with nonclassical numeric phenomena
- extend classical constructs with novel values
UNS is built around a few core principles:
Every UNS number has an internal structure you can inspect.
New operators and new values can be added without breaking old ones.
No contradictions allowed; anything consistent is permitted.
UNS does not force an interpretation—classical numbers, quantum-like views, probabilistic views, or purely abstract perspectives all fit.
- Read the RFC Specification for the formal rules.
- Browse the Guided Discovery for intuition and background.
- Explore the example
.unsefiles. - Use the designer-instruction prompt if integrating UNS into tooling or automated systems.
Parts of this repository are open source; others are closed or proprietary and not redistributable. Each directory contains its own license file—please review before use. For more info, see the repository’s license files for all applicable terms. [Licensing Overview.md](Licensing Overview.md).
There may be errors, contradictions or inconsistencies across the various documents. I've tried to prevent such things, and correct them when found, but if you spot anything questionable, please open a bug report.
The most accurate examples of valid expressions should be found in the Wiki from Section 09 onward.
UNS is a research-aligned, exploratory project. Contributions are welcome in the form of:
- operator proposals
- examples
- tooling
- documentation improvements
- theoretical discussion
Please open an issue or submit a PR.