Draft:Coordinated Fixed-Point Domain Modular Arithmetic: Introduction into Cozy Math (Co-FPD-MA)

Coordinated Fixed-Point Domain Modular Arithmetic (Co-FPD-MA)

Coordinated Fixed-Point Domain Modular Arithmetic (Co-FPD-MA) is a computational framework developed by researcher Christopher Hill in 2025. It builds upon traditional Modular arithmetic by incorporating fixed-point quantization, segmented modular logic, and coordinated arithmetic scheduling. The system includes two subvariants: Modular Segmentation Arithmetic (MSA) and Modular Segmented Multi-Precision Arithmetic (MSMPA). Together, they form a hardware-friendly deterministic arithmetic system nicknamed Cozy Math.

Overview

Co-FPD-MA offers a bounded, deterministic alternative to floating-point math for real-time computing. The system is optimized for applications in GPU computing, cryptography, artificial intelligence, augmented reality (AR), and kernel module design. It is particularly well-suited for devices requiring precision with limited floating-point units.

Mathematical Definitions

Modular Arithmetic (MA)

$MA(a,b;M)=(a+b){\bmod {M}}$

Fixed-Point Domain Modular Arithmetic (FPD-MA)

$FPD\_MA(a,b;Q,M)=\left(\left\lfloor {\frac {a+b}{Q}}\right\rfloor \cdot Q\right){\bmod {M}}$

Coordinated Fixed-Point Domain Modular Arithmetic (Co-FPD-MA)

$Co\_FPD\_MA(C;Q,M)=\sum _{i\in C}\left(\delta (a_{i},b_{i})\cdot \left(\left\lfloor {\frac {a_{i}+b_{i}}{Q}}\right\rfloor \cdot Q\right){\bmod {M}}\right)$

Where:

• $Q$ is the fixed-point quantum
• $M$ is the modulus
• $\delta (a_{i},b_{i})$ is a dependency coordination function

Modular Segmentation Arithmetic (MSA)

MSA represents an integer $X$ in segmented form:

$X=\sum _{i=0}^{n-1}x_{i}\cdot B^{i}$ Where: $x_{i}=X{\bmod {B}}$, and $B$ is the segment base.

Modular Segmented Multi-Precision Arithmetic (MSMPA)

$R=\sum _{i=0}^{n-1}(A_{i}+B_{i}+c_{i-1})\cdot B^{i}{\bmod {M}}$

This enables multi-word arithmetic with carry and parallel coordination.

Sample C Implementation

uint32_t co_fpd_ma(uint32_t a, uint32_t b, uint32_t Q, uint32_t M) {
    uint32_t sum = a + b;
    uint32_t quantized = (sum / Q) * Q;
    return quantized % M;
}

Assembly Pseudocode (x86-64)

; Inputs: a in rax, b in rbx, Q in rcx, M in rdx
add     rax, rbx          ; a + b
div     rcx               ; divide by Q
mul     rcx               ; re-quantize
div     rdx               ; mod M

Applications

• GPU shaders: Deterministic pipeline stage math
• AR hardware: Low-power embedded processing
• Cryptographic engines: Efficient bounded modular calculations
• AI inference: Safe quantized arithmetic
• Linux kernel: Predictable scheduler routines and driver math

Historical Context

Co-FPD-MA was introduced in early 2025 by Christopher Hill and released under a customized MIT License with an optional moral clause, encouraging voluntary profit-sharing to support continued research.

• Donation Link

See Also

• Modular arithmetic
• Fixed-point arithmetic
• Multi-precision arithmetic
• Parallel computing

References

• Hill, Christopher. "Cozy Math: Coordinated Fixed-Point Domain Modular Arithmetic." (2025).
• D. E. Knuth. "The Art of Computer Programming, Volume 2: Seminumerical Algorithms."
• Intel Developer Reference Manual (2024)
• IEEE Symposium on Arithmetic Preprint (2025)

External Links

• Christopher Hill's Research Donation Link