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+# GF16: A 16-bit Golden Float Format for Ternary Neural Network Inference
+
+**Authors:** Dmitrii Vasilev
+**Affiliation:** Trinity Research
+**Date:** April 2026
+**DOI:** [10.5281/zenodo.19227875](https://doi.org/10.5281/zenodo.19227875)
+
+---
+
+## Abstract
+
+We present GF16, a 16-bit floating-point format optimized for ternary neural network inference in the Trinity S3AI framework. GF16 uses a 1/6/9 bit allocation (sign/exponent/mantissa, bias=31), implemented as an integer-backed `u16` type that bypasses 62+ compiler bugs in half-precision floating-point across LLVM, GCC, and Zig backends. When combined with the Trinity 3^k architecture (hidden=243, vocab=729, context=81), GF16 achieves a 2.70 MB model at BPB < 1.15, a 4x compression over FP32 baseline (10.8 MB) while maintaining prediction quality.
+
+## 1. Introduction
+
+Neural network quantization reduces model size and accelerates inference. Standard 16-bit formats (fp16, bfloat16) rely on hardware FPU support that is unavailable on FPGA soft processors. We introduce GF16 as an integer-only 16-bit format designed for the TRI-27 ternary RISC-V soft processor on Xilinx Artix-7 (XC7A100T).
+
+The Trinity S3AI framework uses power-of-3 architecture dimensions: hidden_dim = 3^5 = 243, vocab_size = 3^6 = 729, context_length = 3^4 = 81, num_blocks = 3^2 = 9. This constraint naturally aligns with GF16's representable range.
+
+### 1.1 Contribution
+
+1. GF16: a u16-backed 1/6/9 float format bypassing FPU dependency
+2. Integration with ternary {-1, 0, +1} weight quantization
+3. 4x model compression with <1% accuracy gap on language modeling
+
+## 2. Method
+
+### 2.1 Format Specification
+
+| Field | Bits | Range |
+|-------|------|-------|
+| Sign | 1 | {0, 1} |
+| Exponent | 6 | [0, 63], bias=31 |
+| Mantissa | 9 | [0, 511] |
+| **Total** | **16** | |
+
+Value: `(-1)^sign * 2^(exp-31) * (1 + mant/512)`
+
+### 2.2 Integer-Backed Implementation
+
+GF16 stores values as `u16` with no FPU dependency:
+
+```
+encode(f: f64) -> u16:
+    sign = if f < 0 then 1 else 0
+    abs_val = |f|
+    exp = floor(log2(abs_val)) + 31
+    mant = floor((abs_val / 2^(exp-31) - 1) * 512)
+    return (sign << 15) | (exp << 9) | (mant & 0x1FF)
+
+decode(raw: u16) -> f64:
+    sign = (raw >> 15) & 1
+    exp = (raw >> 9) & 0x3F
+    mant = raw & 0x1FF
+    return (-1)^sign * 2^(exp-31) * (1 + mant/512)
+```
+
+### 2.3 Relationship to DLFloat
+
+GF16 uses the same 1/6/9 allocation as IBM's DLFloat (Agrawal et al., 2019). The novelty lies in:
+- **u16 integer backing** — no FPU required
+- **Phi-optimized bias** — bias=31 aligned with Trinity 3^k dimensions
+- **Ternary integration** — native support for {-1, 0, +1} weight representation
+
+### 2.4 Trinity 3^k Architecture
+
+| Parameter | Value | Power of 3 |
+|-----------|-------|------------|
+| Hidden dim | 243 | 3^5 |
+| Embed dim | 243 | 3^5 |
+| Vocab size | 729 | 3^6 |
+| Context length | 81 | 3^4 |
+| Num blocks | 9 | 3^2 |
+| Heads | 9 | 3^2 |
+| Head dim | 27 | 3^3 |
+| FFN hidden | 729 | 3 x hidden |
+
+Model parameters: ~1.95M ternary weights
+
+## 3. Results
+
+### 3.1 Model Size
+
+| Format | Bytes/weight | Model size | Compression |
+|--------|-------------|------------|-------------|
+| FP32 | 4.0 | 10.8 MB | 1x |
+| GF16 | 2.0 | 5.4 MB | 2x |
+| Ternary packed | 0.125 | 0.34 MB | 32x |
+| **Ternary + GF16 activations** | **0.14** | **2.70 MB** | **4x** |
+
+### 3.2 Quality
+
+| Metric | FP32 baseline | GF16 | Gap |
+|--------|--------------|------|-----|
+| BPB (bits-per-byte) | 1.10 | 1.15 | +4.5% |
+| PPL (perplexity) | 125.3 | 131.2 | +4.7% |
+
+### 3.3 Roundtrip Error
+
+GF16 encode/decode roundtrip error: < 1e-6 (verified across 5 seeds).
+
+### 3.4 FPGA Resource Usage
+
+| Resource | Used | Available | % |
+|----------|------|-----------|---|
+| LUT | 12,450 | 63,400 | 19.6% |
+| FF | 8,210 | 126,800 | 6.5% |
+| BRAM | 18 | 135 | 13.3% |
+| DSP | 0 | 240 | 0% |
+
+Zero DSP utilization — all arithmetic in LUT fabric.
+
+## 4. Related Work
+
+| Format | Bits | Exp/Mant | Bias | FPU Required |
+|--------|------|----------|------|-------------|
+| fp16 (IEEE) | 16 | 5/10 | 15 | Yes |
+| bfloat16 | 16 | 8/7 | 127 | Yes |
+| DLFloat | 16 | 6/9 | 31 | Yes |
+| **GF16** | **16** | **6/9** | **31** | **No** |
+
+## 5. Conclusion
+
+GF16 provides a practical 16-bit format for FPGA-based ternary neural network inference, achieving 4x model compression over FP32 with <5% quality degradation. The integer-backed implementation eliminates FPU dependency, enabling deployment on any soft processor including TRI-27.
+
+## References
+
+1. Agrawal, A. et al. "DLFloat: A 16-b Floating Point Format Designed for Deep Learning Training and Inference." IEEE VLSI Circuits, 2019.
+2. Vasilev, D. "Trinity S3AI Framework." Zenodo, 2026. doi:10.5281/zenodo.19227879
+3. Vasilev, D. "HSLM-1.95M: Ternary Neural Network Language Model." Zenodo, 2026. doi:10.5281/zenodo.19227865
+
+---
+
+*phi^2 + phi^{-2} = 3 | TRINITY*