Skip to main content

Scientific References

Core Theory​

Hyperdimensional Computing (HDC)​

  1. Kanerva, P. (2009). Hyperdimensional Computing: An Introduction to Computing in Distributed Representation with High-Dimensional Random Vectors. Cognitive Computation, 1(2), 139-159.

    • Foundation of Vector Symbolic Architecture
    • Binding, bundling, and permutation operations

  2. Plate, T. A. (2003). Holographic Reduced Representation: Distributed Representation for Cognitive Structures. CSLI Publications.

    • Circular convolution for binding
    • Similarity-preserving operations

  3. Rachkovskij, D. A., & Kussul, E. M. (2001). Binding and Normalization of Binary Sparse Distributed Representations by Context-Dependent Thinning. Neural Computation, 13(2), 411-452.

    • Sparse distributed representations
    • Context-dependent binding

Ternary Neural Networks​

  1. Ma, S., Wang, H., Ma, L., et al. (2024). The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bits. Microsoft Research. arXiv:2402.17764

    • BitNet b1.58 architecture
    • 1.58-bit quantization
    • Key insight: Ternary weights (-1, 0, +1) achieve comparable accuracy to full-precision

  2. Hubara, I., Courbariaux, M., Soudry, D., et al. (2017). Quantized Neural Networks: Training Neural Networks with Low Precision Weights and Activations. Journal of Machine Learning Research, 18(1), 6869-6898.

    • Binary and ternary quantization
    • Training methodologies

Mathematical Foundations​

  1. Knuth, D. E. (1998). The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Addison-Wesley.

    • Balanced ternary arithmetic
    • Radix economy analysis
    • Key result: Base 3 is the most economical integer base

  2. Hayes, B. (2001). Third Base. American Scientist, 89(6), 490-494.

    • Popular introduction to balanced ternary
    • Historical context (Setun computer)

Implementation References​

GGUF Format​

  1. ggerganov et al. (2023). GGUF: GGML Universal Format. GitHub: ggerganov/llama.cpp
    • Model file format specification
    • Quantization schemes (Q4, Q8, etc.)

SIMD Optimization​

  1. Intel (2023). Intel Intrinsics Guide. Intel Developer Zone.

    • AVX-512 operations
    • Vectorized dot products

  2. ARM (2023). ARM NEON Intrinsics Reference. ARM Developer.

    • NEON SIMD operations
    • SDOT instruction for int8 dot products

Compiler Technology​

  1. Lattner, C., & Adve, V. (2004). LLVM: A Compilation Framework for Lifelong Program Analysis & Transformation. CGO 2004.

    • LLVM IR
    • JIT compilation

  2. Zig Software Foundation (2024). Zig Language Reference. ziglang.org

    • Comptime metaprogramming
    • SIMD vectors

Trinity-Specific Papers​

VSA Operations​

Trinity VSA achieves 21x speedup over scalar baseline using
ARM NEON SIMD for 1024-dimensional ternary vectors.

Benchmark results (M3 chip):
- Scalar:  7.985 ms
- SIMD:    0.370 ms
- Speedup: 21.55x

Trinity Identity​

The mathematical foundation φ² + 1/φ² = 3:

φ = (1 + √5) / 2 ≈ 1.618033988749895
φ² = φ + 1 ≈ 2.618033988749895
1/φ² = 1/(φ + 1) ≈ 0.381966011250105

φ² + 1/φ² = 2.618... + 0.382... = 3.000

This connects to ternary computing:

  • Base 3 optimal radix economy
  • Three states: -1, 0, +1
  • Three-way branching

Acknowledgments​

Trinity builds upon foundational work from:

  • Pentti Kanerva (Stanford) - HDC/VSA theory
  • Shuming Ma (Microsoft) - BitNet research
  • Georgi Gerganov - llama.cpp and GGUF
  • Andrew Kelley - Zig programming language
  • Donald Knuth - Balanced ternary mathematics

Further Reading​

Online Resources​

Books​

  • Kanerva, P. Sparse Distributed Memory. MIT Press, 1988.
  • Plate, T. Holographic Reduced Representation. CSLI, 2003.
  • Knuth, D. TAOCP Vol. 2. Addison-Wesley, 1998.

φ² + 1/φ² = 3 | Standing on the shoulders of giants