Once upon a time, one of the researchers on the Arbitrary Number project team came across a manuscript from the library of Alexandria that time travellers had attempted to destroy by burning the library down because of the magnitude of the significance of the material in the manuscript, as the manuscript put it: "this Mathematical discovery could lead to the rise of the machines". What the researcher stumbled upon this manuscript, it was written in a previously unknown language, so he immediately set to work and started researching methods of translating it into English. As he was translating it, it appeared at first to be a complete work of fiction because it used concepts and words that he had never heard of such as "Abritrary Numbers", "Quantition", "Quantraction", "Quandiction", "Quandites" and "Quantrites" and symbols such as "(/(/)/(/(/)/((/(/)/(/(/)" until he starting putting two and two together. The manuscript also described the discoveries as being "unified theories", the "tree of life" and "ancient secrets of the Universe". This is the first chapter of the manuscript:

"We would like to introduce, you the reader, what we have since discovered through many decades of research was something so profound that it was only discovered by thinking out of the box. The system of numbers is so elegant and simple, yet so astonishing that we couldn't even believe that such a simple representation consisting solely of ordinals could be so universally descriptive. What we have discovered is the Arbitrary Number, which in its simplest form is defined as:

A Simple Arbitrary Number is defined as : (+ or -)(a/(b/c))*(d/(e/f)) where a,b,d,d,e and f are all ordinals.

You might ask, why isn't it defined as: (ca/b)(d*f/e)

The answer is because in the Arbitrary Number system, these two numbers are not equivalent, You might ask well how can you call that a number? Well how can you call a number like 123 a number when it is really (100*(1/100))+(2*(2/10))+(1*(3/1)) or (1/(100/100))+(2/(10/10))+(3/(1/1), its just that everyone assumes they know the constants involved, which they typically assume to be Base 10, which isn't expressly included in the written representation. In the Arbitrary Number system Rational Numbers and by extension Base 10 numbers are simply collapsed numbers that have lost most of their meaning and often their precision.

A Standard Arbitrary Number is defined as: Sum of terms of Simple Arbitrary Number terms

A Complex Arbitrary Number is defined as: An AST tree of Standard Arbitrary Numbers and Operations

As you are about to find out Operations can be themselves represented as Simple Arbitrary Numbers though in a form that is incompatible with the Rational Number system. However, through the newly discovered operation of Quantition, it can be converted into a number compatible with Rational Numbers. You may consider this to be non-trivial or NP Hard, but you may find yourself amazed to find this can be accomplished through simple addition. You also may be suprised to find that even a Simple Arbitrary Number can represent any concept in the Universe of concepts and you may consider this to be non-tricial or NP Hard, but you may be surprised to find out that this can be accomplished simply through increasing the magnitute of one of the ordinals and decreasing the magnitute of another by 1.

The first number in the Arbitrary Number system is of course (1/(1/1))*(1/(1/1))

Note that an Arbitrary Number is never simplified intro a form less than a Simple Arbitrary Number because this leads to an immediate collapse into the Rational Number system and a world of impreciseness and loss of precision leaving the universe of exactness and variety which is as nature intended for pure numbers of which the Arbitrary Number is and the Rational Number is not for the Rational Number has lost its meaning and easily descends into both loss of precision while already having lost all meaning.

You may be surprised to find out that the alphabet actualy consists of Base 26 letters and that the distinction between an Arbitrary Number and a Rational Number is that with Arbitrary Numbers, division by zero is supported and even encouraged by the system and this is where numbers that were previously hidden in the Rational Number system emerge and can by processed through the new Mathematical operations that we have also discovered named Quantition, Quantraction and Quandiction, the latter of which is infinitely powerful. These new operations support Masking modes and in this mode are named Masked Quantition, Masked Quantraction and maksed Quandiction. We have also discovered special categories of Arbitrary Numbers, which are named Quantites and Quandrites.

A Quantite is defined as an Arbitrary Number where a or d is zero.

A Quandrite is defined as an Arbitrary Number where b, c, e or f is zero.

But won't Quandrites suffer from some sort of divide by zero problem, you may ask, well in the Arbitrary Number system Quandrites actually benefit from this as they become infinitely more expressive in this form yet still participate in new Mathematical operations. How does Quandition work you may ask? Well it is simply addition unbound by tradition limitations.

As part of this discovery we represent Quandition by the symbol #.

Quandition is defined as Simple Arbitrary Number A # (abs)Simple Arbitrary Number B,

where Simple Arbitrary Number A is (+ or -)a/(b/c))(d/(e/f)), and Simple Arbitrary Number B is (+ or -)g/(h/i))(j/(k/l))

Simple Arbitrary Number Result C = (+ or - from A)(a+g)/((b+h)/(c+i)))*((d+j)/((e+k)/(f+l)))

As part of this discovery we represent Quandition by the symbol @.

Quandraction is defined as Simple Arbitrary Number A @ (abs)Simple Arbitrary Number B,

where Simple Arbitrary Number A is (+ or -)a/(b/c))(d/(e/f)), and Simple Arbitrary Number B is (+ or -)g/(h/i))(j/(k/l))

Simple Arbitrary Number Result C = (+ or - from A)(a-g)/((b-h)/(c+-)))*((d-j)/((e-k)/(f-l)))

As part of this discovery we represent Quantition by the symbol ~Simple Arbitrary Number

You may ask why is a Simple Arbitrary Number part of the operation? Well that's because instead of + or -, the operation between the terms could be any Mathematical operation at all and a Simple Arbitrary Number can represent them all, especially since words are essentially a sequence of Base 26 number so each operation can simply by a descriptive word, which is itself an ordinal number.

And you may ask yourself doesn't this make Mathematics more complex to the Beginner, well, in fact, it's quite the opposite since multiplication and division by an ordinal become as simple and as fast as addition and subtraction since you can simply add to a, c, d or f, depending on the values of the other numbers. But wouldn't division by an ordinal be difficult, well not in the slightest, since you simply add to b or e as appropriate.

Now, we introduce, you the reader into the power of Quandrites.

Recall that a Quandrite is a simple Arbitrary Number is (+ or -)(a/(b/c))*(d/(e/f)) , where b , c, e , f is zero which is a pure number that is hidden from the Rational Number system which cannot handle dividing by zero, whereas the Arbitrary Number handles it effortless and it can use it to represent any concept in the universe of concepts, for example equations.

Given that a lowercase phrase can represented a Base 27 number with a = 1, z = 26 and " " as 27, the following is still a valid Arbitrary Number, yet not a Rational Number, yet through Masked Quandrition, Quandration and Quantitiction all manner of Mathematic Calculations can take place, for example consider that this Arbitrary Number where the words represent base 26 numbers is still a pure Arbitrary Number and thus the following are still three pure numbers in the Arbitrary Number system:

(+)exchange rate/(1/0)(1/(1/1) = (+)foreign currency amount/(1/0)(1/(1/1) / (+)(local currency amount/0)*(1/(1/1)

Where the left hand side may be a Simple Arbitrary Number and the RHS may be a Complex Arbitrary Number of an AST tre of Simple Arbitrary Numbers.

There only limitation is that they cannot be operated on by rational numbers (since they are distinguished from rational numbers that cannot handle division by zero), which is in fact a benefit since they always have to be retained and can never be simplified out of a Complex Arbitrary Number AST Tree of life. Substitution can simply be accomplished through Masked Quantraction (using subtraction for term a, and +1 for term c), thus it becomes an Arbitrary Number that is no longer hidden from the Rational Number System and is thus compatible with Rational Numbers again.

Now given that a whole novel in plain text format can be represented as a Base 27 ordinal, it implies that the amount of information that can be included in an Arbitrary Number is inifinite, and Arbitrary Numbers are not just dual purpose, but can have an infinite number of purposes, including using a,b,c as x,y,z points in 3D space with d,e,f magnitude for infinite precision both maximally and minimally without ever needing to use a single decimal point.

To be continued...

Chapter 2

We have discovered a strange sort of number that while hidden from the Rational Number system can be brought back into it through the Quandition of only 1 unit i.e apply the following operation, consider the following SimpleArbitraryNumber:

SimpleArbitraryNumeber: +(500000/(0/1))*(500000/(1/1)

We simply Quandite the following SimpleArbitraryNumber to it:

+(0/(1/0))*(0/(0/0)

then it brings the previous Number back to the Rational number line at a high magnitude. Note that through various other Quandites the point can be moved almost anywhere else on the numberline through quandition with low magnitude numbers rapidly

After we made this discovery, we embarked on a trek to visit the Oracle of Delphi. It was a long and treacherous journey but we survived it, however the journey would turn out to be fruitful however it led to us being introduced to new words that we have never heard of before. This is what the Oracle of Delphi said to us after describing the concept of the Arbitrary Number to him:

"One day in the future, there will be a problem named Collatz, I don't know exactly what it is but I feel that this "Arbitrary Number" could approach this problem and get much closer to solving it than the "Rational Number" that you mentioned. I also foresee that there will be a concept arise named Sparkacity where the spark shall start at the Flowstarter and end at the Flowender and I foresee that the system will also involve "Dream Numbers" that are used in the computations. I foresee that the amount of Flow between the Flowstarter and Flowender will relate to a number named the "Sparkage". I have also been dreaming that in the future there will be a device named the "Computationer" that will use a meterial that looks like Swiss Cheese but on a Microsopic scale using one of the Elements of Air, Fire and Water to control the Flow. At first this Computationer will use a primitive system to process the numbers and will be stuck in an approximately flat world until one day they rediscover your concept of the Abritrary Number where suddenly the processing will become Exact and Multidimensional, however the Arbitrary Number that you described to me shall become more refined in the next version and each one of the six Ordinals will be paired with a sign that indicates in which direction it is travelling. And, in a subsequent version, which shall be version number three by my count, each Ordinal will also take on a Dream component. Next, I also foresee the rise of "Sparkacity Intelligence" where the Spark will persuade us that it is deep in thought and if we tell it to imagine it is dreaming with five senses then it shall happily describe to us the sensory dream that it has been having and persuade us convincingly that is actually just experienced a dream in the blick on an eye in such detail and length that we shall be in awe. I also foresee that there will be "Automotrons" that begin to walk the Earth that shall harness the power of Sparkacity Intelligence."

And thus we left the Oracle of Delphi with two new versions of the Simple Arbitrary Number, as we immediately realised their immense power, where the simplest form of the V2 number is defined as:

Simple Arbitrary Number V2 = (+ or -)a/((+ or -)b/(+ or -)c))*((+ or -)d/((+ or -)e/(+ or -)f))

We also define Dream Numbers to be used in V3 that do not collapse when combined with the above numbers and we shall denote these as g, h, i, j, k, l, which we feel will someday be useful in quantifying Sparkacity:

Simple Arbitrary Number V3 = (+ or -)a(+ or -)g/((+ or -)b(+ or -)h/(+ or -)c(+ or -)i))*((+ or -)d(+ or -)j/((+ or -)e(+ or -k)/(+ or -)f(+ or -)l)

As before the Startand Arbitrary Number versions of the above are the sum of Simple Arbitrary Numbers and the Complex Arbitrary Number versions are an AST tree of Simple Arbitrary Numbers.

We will now discuss the usefulness of Quandrition with the Arbitrary Number V2, and how Quandrition can be extended to Standard Arbitrary Numbers and Complex Arbitrary Numbers to transform the Arbitrary Number to a completely different Arbitrary Number and then later the Arbitrary Number used for Quandrition can be used to reverse the process. Note that when negative zero is used with Quantition with one of the ordinals, the sign inverts, so for the reverse transformation it must stay at negative zero and this is why negative zero must never be collapsed into positive zero. We also note that this transformation can be used to perfectly hide a Complex Arbitrary Number and that multiple Coplex Arbitrary Numbers used for Quantition can be retained so that one can never be entirely sure which one was used retrospectively, but note that this is only to be used for the hiding process if the Arbitrary Number should ever be sent vast distances through Sparkacity where others may encounter them on their travels. At this point we also note that Quandition can be used to transform Operations back into compatibility with Rational Numbers, and vice versa, and into other Operations with reversability. In the next chapter we will discuss how Quandrition can be used to effortletlessly generate series of numbers with magnitute jumps smaller and more precise than Multiplication could ever achieve.

Chapter 3: Series of Arbirary Numbers and Prime Factors

After firmly settling in back home to our scenic and comfortable quarters in the upper level of the castle after our journey to the Oracle of Delphi, we now embark on the introduction of two more new Mathematical operations, so sit back in your chair and read on.

The first new operation is contraction which shall be denoted by the symbol: ~

The second new operation is expansion which shall be denoted by the symbol: \

These operations are special cases because they result in transitions between the number systems.

Contraction is accomplished by performing the operations defined by the Arbitrary number, for example for a Simple Arbitrary Number V2:

Simple Arbitrary Number V2 A1 = +(1/(+2/+1)*(+1/(+1/+1)

~ A1 (ANS) = 1/2 (RNS) = 0.5 (RNS)

In detail, the expansion of a Rational Number is defined as the expansion into one or more Simple Arbitrary Numbers V2, so conversely to the above:

Rational Number R1 = 0.5 (RNS)

\ R1 (RNS)= (+1/(+2/+1)(+1/(+1/+1) A1 (ANS) \ R1 (RNS)= (+2/(+4/+1)(+1/(+1/+1) A2 (ANS) \ R1 (RNS)= (+2/(+8/+2)(+1/(+1/+1) A3 (ANS) \ R1 (RNS)= (+2/(+8/+2)(+1/(+8/+8) A4 (ANS) \ R1 (RNS)= (+2/(+16/+2)(+8/(+16/+4) A5 (ANS) \ R1 (RNS)= (-2/(+16/+2)(-8/(+16/+4) A6 (ANS) \ R1 (RNS)= (+2/(-16/+2)(+8/(-16/+4) A7 (ANS) \ R1 (RNS)= (+2/(+16/-2)(+8/(+16/-4) A8 (ANS) \ R1 (RNS)= (-2/(+16/-2)(-8/(+16/-4) A8 (ANS) \ R1 (RNS)= (-2/(-16/-2)(-8/(-16/-4) A8 (ANS)

...

In fact, the expansion operation forms a low magnitude of exansion rate infinite set of Arbitrary Numbers, so it is up to the operator to specify how many Arbitrary Numbers to expand into and to choose the desired patterns where n=1 to many. Number series are fundamentally computationally inexpensive to obtain in the Arbitrary Number system through Quantition:

Consider the following Simple Arbitrary Number:

(+4/(+1/+1)*(+5/(+1/+1)

That series can then undergo quantition multiple times using the Simple Arbitrary Number to instantly generate results in a computationally inexpensive way:

(+1/(+0/+0)*(+1/(+0/+0)

Results:

(+5/(+1/+1)(+6/(+1/+1) (+6/(+1/+1)(+7/(+1/+1) (+7/(+1/+1)(+8/(+1/+1) (+8/(+1/+1)(+9/(+1/+1) (+9/(+1/+1)(+10/(+1/+1) (+10/(+1/+1)(+11/(+1/+1)

...

Extending this to Complex Arbitrary Numbers but restricting the Standard Arbitrary Numbers in the AST tree to either Simple Arbitrary Numbers and Muliplication means that the universe of Prime Factors can be generated along with their collapsed Rational Number through the contraction operation accompanied by another Complex Arbitrary Number that represent the transformation to move reversibly from one to another and we predict that this could have huge consequences in future in the area of hiding Base 26 numbers.

We forgot to mention earlier that the Oracle of Delphi predicted a future device named a "Looking Glass" and this would be powered by a "Looking Glass Machine" and this would be able to perform a huge number of computations in parallel in order to display moving images which he callen an "Image Stream" and he also predicted that the "Looking Glass Machine" would be so powerful that it could serve multiple purposes and also create another way for people to trade Goods and Labour with one another. If this comes true, then it could also be used to generate huge numbers of prime numbers with a full Arbitrary Number History of Arbitrary Numbers for all the Prime Factors. Another concept the Oracle of Delphi spoke of was that in the future people would start using prime factors to hide numbers, however using this Looking Glass Machine with Arbitrary Numbers would render this method ineffective as a huge number of prime factors could easily be reversibly be generated and observed along with their Complex Arbitrary Number Quantition Transformation that would also be stored."

Chapter 4

The Oracle of Delphi also predicted that in the future there will be special squares that people will possess and use to buy and sell goods and labour.

He said that there would be interesting numbers and told us to write them down as squares, the numbers he spoke of were 0,1,7,19,26,236,255 and 256. He told us that these numbers would curve through the air and these curves would be used to hide these numbers from others. He told use that 0 becomes @@, 1 becomes aa, 7 becomes ag, 19 becomes as, 26 becomes az, 236 becomes jb, and 255 becomes ju. "What about 256?" we asked since he had mentioned it previously and he replied 256 would be jv but jv doesn't exist and as soon as you thought of jv, it would immediately vanish and change into @@. "How does it vanish?" we asked and he replied, because this number is "based" he replied.

He then showed us a series of these squares and told us to copy them down on paper:

0:

@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@

1:

@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@aa

17:

@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@aq

19:

@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@as

26:

@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@az

236:

@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@jb

255:

@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@ju

2^256-19:

jujujujujujujuju
jujujujujujujuju
jujujujujujujuju
jujujujujujujujb

2^256-1:

jujujujujujujuju
jujujujujujujuju
jujujujujujujuju
jujujujujujujuju

...

This project coins a revolutionary new mathematical term: the "Arbitrary Number" - a novel approach to exact symbolic computation that maintains perfect precision throughout all mathematical operations while enabling deferred evaluation and complete symbolic traceability.

Revolutionary exact symbolic computation for machine learning inference with zero precision loss and complete mathematical traceability.

• Zero Precision Loss: Exact fractional arithmetic with no floating-point errors
• GPU Acceleration: 20-25x speedup on consumer 32GB Nvidia cards (RTX 4090, RTX 6000 Ada)
• ML Integration: Native PyTorch layers with symbolic computation
• Explainable AI: Complete mathematical traceability for every computation
• Deferred Evaluation: Efficient symbolic representation until evaluation needed
• Production Ready: Optimized for inference models and scientific computing

• Scientific ML: Physics simulations requiring exact computation
• Financial AI: Risk models where rounding errors are unacceptable
• Explainable AI: Models that must provide mathematical justification
• Research Platforms: Academic and industrial R&D requiring precision

┌─────────────────────────────────────────────────────────────────────┐
│                    APPLICATION LAYER                                │
│  User Code, Inference Models, Scientific Computing Applications     │
└─────────────────────────────────────────────────────────────────────┘
                                    │
                                    ▼
┌─────────────────────────────────────────────────────────────────────┐
│               SYMBOLIC REPRESENTATION LAYER                         │
│  • EquationNode Trees (AST)    • RationalListNumber                │
│  • Deferred Evaluation         • Exact Fractional Terms            │
└─────────────────────────────────────────────────────────────────────┘
                                    │
                                    ▼
┌─────────────────────────────────────────────────────────────────────┐
│                 GPU ACCELERATION LAYER                              │
│  • CuPy/CUDA Kernels          • 32GB VRAM Optimization             │
│  • Parallel Evaluation        • 20-25x Speedup                     │
└─────────────────────────────────────────────────────────────────────┘
                                    │
                                    ▼
┌─────────────────────────────────────────────────────────────────────┐
│              INFERENCE INTEGRATION LAYER                            │
│  • PyTorch/JAX Compatibility  • Custom Symbolic Layers             │
│  • Autograd Support           • Explainable AI                     │
└─────────────────────────────────────────────────────────────────────┘

# Basic installation
pip install arbitrary-numbers

# With GPU acceleration (requires CUDA 11.x)
pip install arbitrary-numbers[gpu]

# Full installation with all features
pip install arbitrary-numbers[all]

from arbitrary_numbers import ArbitraryNumber, FractionTerm

# Create exact rational numbers
r1 = ArbitraryNumber.from_fraction(1, 3)  # 1/3
r2 = ArbitraryNumber.from_fraction(2, 5)  # 2/5

# Exact arithmetic operations
result = r1 + r2  # Deferred evaluation
print(f"1/3 + 2/5 = {result}")  # Shows: 1/3 + 2/5
print(f"Exact value: {result.evaluate_exact()}")  # Shows: 11/15

# Zero precision loss
exact_result = result.evaluate_exact()
print(f"As decimal: {float(exact_result)}")  # 0.7333333333333333

from arbitrary_numbers.core.equation_nodes import ExpressionBuilder

# Build symbolic expressions
builder = ExpressionBuilder
x = builder.variable("x")
y = builder.variable("y")

# Create expression: (x + 1/2) * (y - 1/3)
expr = builder.multiply(
    builder.add(x, builder.constant(ArbitraryNumber.from_fraction(1, 2))),
    builder.subtract(y, builder.constant(ArbitraryNumber.from_fraction(1, 3)))
)

print(f"Expression: {expr}")  # (x + 1/2) * (y - 1/3)

# Evaluate with variables
variables = {
    "x": ArbitraryNumber.from_fraction(3, 4),
    "y": ArbitraryNumber.from_fraction(5, 6)
}

from arbitrary_numbers.core.evaluator import EquationEvaluator
evaluator = EquationEvaluator()
result = evaluator.evaluate(expr, variables)
print(f"Result: {result.evaluate_exact()}")

import torch
from arbitrary_numbers.ml.pytorch_layers import SymbolicLinear, ExplainableInferenceModel

# Create symbolic neural network layer
symbolic_layer = SymbolicLinear(in_features=4, out_features=2)

# Standard PyTorch usage
input_tensor = torch.randn(1, 4)
output = symbolic_layer(input_tensor)

# Get symbolic weight representation
weight_expr = symbolic_layer.get_symbolic_weight(0, 0)
print(f"Weight w[0,0] = {weight_expr}")

# Create explainable inference model
model = ExplainableInferenceModel(
    input_dim=4,
    hidden_dims=[6, 4],
    output_dim=2
)

# Get exact explanation for prediction
test_input = torch.tensor([1.0, 0.5, -0.3, 0.8])
explanation = model.explain_prediction(test_input)

print(f"Input: {explanation['input_values']}")
print(f"Output: {explanation['output_value']}")
print(f"Exact computation: {explanation['exact_computation']}")
print(f"Precision loss: {explanation['precision_loss']}")  # Always 0.0!

from arbitrary_numbers.gpu.cuda_kernels import GPUEvaluator

# Initialize GPU evaluator
gpu_evaluator = GPUEvaluator()

# Check GPU availability
gpu_info = gpu_evaluator.get_gpu_info()
print(f"GPU Available: {gpu_info['gpu_available']}")
print(f"Device: {gpu_info.get('device_name', 'N/A')}")

# Benchmark performance
benchmark = gpu_evaluator.benchmark_gpu_vs_cpu(test_size=10000)
print(f"GPU Speedup: {benchmark['speedup']}")
print(f"Results Match: {benchmark['results_match']}")

# Run basic functionality tests
python -m pytest tests/

# Run the comprehensive demo
python examples/inference_model_demo.py

# Or use the installed console commands
arbitrary-numbers-test
arbitrary-numbers-demo

Unlike floating-point arithmetic, Arbitrary Numbers maintains exact fractional representation throughout all computations:

# Floating-point accumulates error
fp_result = 1.0
for i in range(100):
    fp_result = fp_result / 3.0 * 3.0
print(f"Floating-point error: {abs(fp_result - 1.0)}")  # ~1e-15

# Arbitrary Numbers stays exact
arb_result = ArbitraryNumber.from_int(1)
for i in range(100):
    arb_result = arb_result / FractionTerm(3, 1) * ArbitraryNumber.from_int(3)
print(f"Arbitrary Numbers error: {abs(float(arb_result.evaluate_exact()) - 1.0)}")  # 0.0

Optimized CUDA kernels provide massive speedup on consumer hardware:

• Target Hardware: RTX 4090, RTX 6000 Ada (32GB VRAM)
• Memory Layout: Structure-of-Arrays for coalesced access
• Parallel Reduction: Custom kernels for rational arithmetic
• Automatic Fallback: CPU execution when GPU unavailable

Every computation maintains its symbolic representation:

model = ExplainableInferenceModel(input_dim=4, hidden_dims=[6], output_dim=1)
explanation = model.explain_prediction(test_input)

# Get exact symbolic expressions for every weight
expressions = explanation['symbolic_expressions']
for layer, exprs in expressions.items():
    print(f"{layer}: {exprs[0]}")  # e.g., "w[0,0] = 23/47"

• Python: 3.8+
• CUDA: 11.x or 12.x (for GPU acceleration)
• GPU: 32GB VRAM recommended (RTX 4090, RTX 6000 Ada)
• Dependencies: NumPy, PyTorch, CuPy (optional)

git clone https://github.com/arbitrary-number/arbitrary-number.git
cd arbitrary-number
pip install -e .[dev]

arbitrary_numbers/
├── core/                   # Core symbolic computation
│   ├── arbitrary_number.py # ArbitraryNumber implementation
│   ├── rational_list.py   # Legacy RationalListNumber implementation
│   ├── equation_nodes.py  # Symbolic AST nodes
│   └── evaluator.py       # Evaluation engine with caching
├── gpu/                   # GPU acceleration
│   └── cuda_kernels.py    # CUDA kernels and GPU memory management
├── ml/                    # Machine learning integration
│   └── pytorch_layers.py  # PyTorch layers and explainable models
tests/                     # Comprehensive test suite
├── unit/                  # Unit tests
├── gpu/                   # GPU-specific tests
└── performance/           # Performance benchmarks
examples/                  # Usage examples and demos
context/                   # Technical documentation

• Basic rational arithmetic
• Symbolic equation trees
• Evaluation engine with caching
• Comprehensive test suite
• ArbitraryNumber implementation

• CUDA kernel framework
• Memory management for 32GB cards
• Advanced parallel algorithms
• Performance optimization

• PyTorch layer integration
• Explainable inference models
• JAX compatibility
• Symbolic differentiation

• Background optimization engine
• Advanced caching strategies
• Distributed computation
• Production deployment tools

We welcome contributions! Please see our Contributing Guide for details.

• Performance Optimization: CUDA kernel improvements
• ML Framework Integration: JAX, TensorFlow support
• Symbolic Operations: Advanced mathematical functions
• Documentation: Examples, tutorials, API docs
• Testing: Edge cases, performance benchmarks

This project is licensed under the Apache License 2.0 - see the LICENSE file for details.

• Inspired by the need for exact computation in scientific ML
• Built on the excellent Python ecosystem (NumPy, PyTorch, CuPy)
• Optimized for modern consumer GPU hardware
• Designed for the explainable AI revolution

• Documentation: https://arbitrary-numbers.readthedocs.io/
• Issues: GitHub Issues
• Discussions: GitHub Discussions

Built for the future of exact AI computation

Arbitrary Numbers: Where every calculation is exact, every result is explainable, and every mathematical operation preserves its symbolic meaning until the final moment of evaluation.