Golden Float & Ternary Float: GF16 + TF3-9 Format for Trinity

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[EPIC] Golden Float & Ternary Float: GF16 + TF3-9 Format for Trinity

Priority: 🔥 P0 (Sacred / Research-Critical)
Status: Open
Related: tri_math, zig-half, FPGA, temporal-trinity

Summary

Implement and research new numerical formats, predicted by the Sacred Formula:

• GF16 (Golden Float 16) — 1 sign + 6 exp + 9 mant = 16 bits, exp:mant ≈ 1/φ
• TF3-9 (Ternary Float 9 trits) — 1 sign trit + 3 exp + 5 mant = 9 trits, exp:mant ≈ 1/φ, total = 3² + 3³, perfect for ternary Trinity.

Goal — obtain information-optimal formats for weights/activations and compare with FP16/BF16/INT8/FP4.

Motivation

1. Industry Facts (March 2026)

   • FP4 E2M1 (NVIDIA Blackwell, 2024): 4 bits = 2 exp + 1 mantissa
   • MXFP4 (OCP Microscaling, 2023): 4+8 shared exponent per block
   • FP6 E3M2 (OCP, 2023): 6 bits = 3 exp + 2 mantissa
   • FP8 E4M3 (NVIDIA Hopper, 2022): 8 bits = 4 exp + 3 mantissa
   • BF16 (Google Brain, 2018): 16 bits = 8 exp + 7 mantissa
   • NO ONE uses φ for exp:mant split!

2. Sacred Formula Analysis Results

   • No one uses the golden ratio for exp/mant division.
   • Sacred Formula + φ² + 1/φ² = 3 predicts optimal split exp:mant ≈ 1/φ for range/precision balance.
   • TF3-9 and TF3-27 with ternary trits give ~79.2% theoretical information capacity with 2-bit trit encoding.

3. For Trinity / zig-half

   • We already implemented f16 SIMD, bf16 shim, ternary quantization, sparse dot.
   • Next logical step — new sacred formats that no one has tried yet, but naturally follow from TRINITY.

4. Sacred Match TF3-9

   • TF3-9: 1 sign + 3 exp + 5 mant = 9 trits
   • exp:mant = 3:5 = 0.6 ≈ 1/φ (0.618) — PERFECT GOLDEN MATCH!
   • This is not coincidence — φ² + 1/φ² = 3 = ternary basis, and golden split = 1/φ = optimal balance

5. Engineering Value of Sacred Formula

   • Formula V = n × 3^k × π^m × φ^p × e^q predicted new formats
   • This is not an "experiment" — this is engineering application of mathematics to neural network formats
   • Paper potential: "Golden Float: φ-optimal exponent-mantissa split for neural network inference"

6. Neuroanatomical Analogy — Biologically Justified Architecture

Brain Structure	File	Role	Function
Intraparietal Sulcus (IPS)	intraparietal_sulcus.zig	Core numerical magnitude representation	GF16/TF3 format definitions, conversion, arithmetic
Fusiform Gyrus	fusiform_gyrus.zig	Visual symbol recognition (Arabic numerals → internal repr)	f32/f16/bf16 → GF16/TF3 encoding
Angular Gyrus	angular_gyrus.zig	Verbal/symbolic ↔ semantic mapping	Format introspection, φ-distance table, tri math floats output
Orbitofrontal Cortex (OFC)	orbitofrontal_value.zig	Value encoding, precision-range tradeoff decisions	Adaptive format selection: auto-pick GF16 vs BF16 vs FP16 per layer
Weber-Fechner Tuning	weber_tuning.zig	Logarithmic magnitude encoding (Weber's law)	TF3 nonlinear quantization levels

Each format maps to a brain region responsible for that function.

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Format Specifications

GF16 (Golden Float 16)

┌────────────────────────────────────────────────┐
│ 15 │ 14-9  │ 8-0                            │
│────┼────────┼────────────────────────────────┤
│ S  │ Exp(6) │ Mant(9)                        │
└────────────────────────────────────────────────┘
exp:mant = 6:9 = 0.667
φ-distance = |0.667 - 1/φ| = |0.667 - 0.618| = 0.049
≈ 95.1% "golden"

Key properties:

• 6-bit exponent → ±128 range (better than BF16's ±127)
• 9-bit mantissa → 9 bits precision (vs BF16's 7 bits)
• Total 16 bits — same size as FP16/BF16

TF3-9 (Ternary Float 9 trits)

┌───────────────────────────────────────────────────────────────────────┐
│ 17-16 │ 15-10   │ 9-0                                                │
│──────┼──────────┼─────────────────────────────────────────────────────│
│ Sign │ Exp(3×2) │ Mant(5×2)  // 3 exp trits + 5 mant trits            │
│ trit │ trits    │           // Each trit = 2 bits: 00=0, 01=-1, 10=+1 │
└───────────────────────────────────────────────────────────────────────┘
exp:mant = 3:5 = 0.6
φ-distance = |0.6 - 1/φ| = |0.6 - 0.618| = 0.018
≈ 98.2% "golden" — BEST FORMAT!
Total: 9 trits × 2 bits = 18 bits

Key properties:

• 3 exponent trits → 3³ = 27 exponent levels
• 5 mantissa trits → 3⁵ = 243 mantissa levels
• Total resolution: 27 × 243 = 6,561 values
• Natural fit for ternary Trinity (base-3 arithmetic)

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Research Questions

1. Is φ-optimal actually better for neural networks?

   • Compare GF16/TF3-9 vs FP16/BF16 on HSLM training
   • Metrics: PPL, convergence speed, memory bandwidth

2. Information efficiency per bit

   • GF16: 16 bits → 65,536 values (linear) vs ~10⁶ effective (floating)
   • TF3-9: 18 bits → 6,561 values (ternary) → compare effective capacity

3. Hardware feasibility

   • GF16 can use existing DSP48E1 (16-bit multiply)
   • TF3-9 needs custom ternary ALU (see [EPIC] GF16/TF3 Arithmetic Unit on XC7A100T (Artix-7, 28nm) #392)

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Implementation Tasks

Phase 1: Format Definitions

• GF16 struct in src/hslm/intraparietal_sulcus.zig
• TF3-9 struct in src/hslm/intraparietal_sulcus.zig
• Conversion functions: f32 → GF16/TF3-9 → f32
• Arithmetic: add, mul, sqrt for both formats
• Unit tests: edge cases (denormals, overflow, underflow)

Labels: research, phase-1

Phase 2: Integration with HSLM

• Add format option to hslm_train.zig
• Implement quantization pipeline: f32 → GF16/TF3-9
• Implement dequantization: GF16/TF3-9 → f32
• Memory layout optimization (packed vs expanded)

Labels: integration, phase-2

Phase 3: Benchmark Comparison

• Run HSLM training with: FP16, BF16, GF16, TF3-9
• Compare: final PPL, training speed, memory usage
• Analyze: layer-wise precision sensitivity

Labels: benchmark, phase-3

Phase 4: Hardware Prototype (Optional)

• GF16 ALU in Verilog (see [EPIC] GF16/TF3 Arithmetic Unit on XC7A100T (Artix-7, 28nm) #392)
• TF3-9 ALU in Verilog (see [EPIC] GF16/TF3 Arithmetic Unit on XC7A100T (Artix-7, 28nm) #392)
• Compare SW vs HW performance

Labels: hardware, optional, phase-4

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Success Criteria

Metric	FP16	BF16	GF16	TF3-9
exp:mant ratio	5:10=0.5	8:7=1.14	6:9=0.667	3:5=0.6
φ-distance	0.118	0.522	0.049	0.018
Bits	16	16	16	18
Training PPL (target)	-	-	≤ BF16	≤ BF16
Speedup vs FP32	-	-	2×	2×
Memory reduction	-	-	50%	44%

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Related Work

• Sacred Formula paper: papers/sacred/draft.md
• zig-half: zig-half library for f16/bf16
• FPGA: Issue [EPIC] GF16/TF3 Arithmetic Unit on XC7A100T (Artix-7, 28nm) #392 (GF16/TF3 Hardware ALU)
• HSLM Training: src/hslm/

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References

• Industry formats: FP4 E2M1, FP6 E3M2, FP8 E4M3/E5M2, BF16, FP16
• Sacred Formula: V = n × 3^k × π^m × φ^p × e^q
• Trinity identity: φ² + 1/φ² = 3

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φ² + 1/φ² = 3 | TRINITY

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