GF Competitive Analysis — Proving GoldenFloat is Not Random

Date: 2026-04-07
Status: Working Draft
Architecture: every step = tri CLI + experience/ source of truth .trinity

Why This Work Is Needed

The t27 project implements GoldenFloat (GF) — a family of 7 floating-point formats (GF4, GF8, GF12, GF16, GF20, GF24, GF32) designed around the golden ratio φ ≈ 1.618. Current benchmarks exist but we need:

1. GMP Integration — arbitrary precision verification (100+ digits via Python mpmath)
2. Mathematical proof that GF is not random — derived from sacred geometry
3. Competitive comparison against IEEE 754, OCP MXFP, mainstream languages
4. Scientific documentation following zig-golden-float whitepaper standards
5. Cross-language verification proving ternary computes more decimal places accurately

Phase 0 — GMP / mpmath Integration (P0 TODAY)

0.1 Create scripts/verify_precision.py

Python mpmath script outputting all sacred constants at 100 decimal digits:

from mpmath import mp, mpf, sqrt
import json

mp.dps = 100  # 100 decimal digits

phi = (mp.mpf(1) + sqrt(mp.mpf(5))) / mp.mpf(2)
phi_inv = 1 / phi
phi_sq = phi * phi

# Pellis closed form (FORMULA_TABLE.md row 31)
term1 = 360 / phi_sq
term2 = -2 / (phi ** 4)
term3 = 1 / (3 * phi) ** 5
pellis = term1 + term2 + term3

results = {
    "precision_digits": 100,
    "phi": str(phi),
    "phi_inv": str(phi_inv),
    "phi_sq": str(phi_sq),
    "pellis_100_digits": str(pellis),
    "pi": str(mp.pi),
    "e": str(mp.e),
    "phi_identity_check": str(phi_sq - phi - 1),  # should be ~0
    "trinity_identity_check": str(phi_sq + 1/phi_sq - 3)  # should be ~0
}

print(json.dumps(results, indent=2))

Output: JSON with all sacred constants at 100 digits, stored in .trinity/experience/precision_verify.json

0.2 Update research/trinity-pellis-paper/FORMULA_TABLE.md

Add row 31 with Pellis value pre-registered to 50 digits:

• 137.03599916476639345... (50 digits fixed)
• Mark as "verified via mpmath (100 digits)"
• Add pre-registered checkpoint section for CODATA 2026/2030 comparison

0.3 Future: tri math compare --precision N

Add --precision-bits flag to bootstrap/src/math_compare.rs to compute sacred constants at N digits via Python subprocess.

Phase 1 — New Spec: specs/numeric/gf_competitive.t27

Create spec with:

• test golden_float_phi_representation — GF32 error < 5.0e-4
• test golden_float_vs_ieee_sacred_constants — GF16 mean error <= IEEE × 0.9
• test roundtrip_uniform_distribution — NMSE GF16 <= IEEE FP16 × 1.01
• test addition_precision_phi_relation — φ² = φ+1 identity in GF16
• test accumulation_stability — sum 10^6 terms, rel_error < 1e-4
• test ternary_vs_python_decimal_places — ternary >= IEEE f64 places
• invariant gf16_phi_distance_is_measurable
• invariant phi_split_is_optimal
• bench gf16_encode_decode_latency — target < 50 ns

Compliance: L4 TESTABILITY — must include test/invariant/bench sections (SOUL.md Art II).

Phase 2 — New Spec: specs/numeric/pellis_verify.t27

GMP-backed Pellis verification spec:

• Import from specs/physics/pellis-formulas.t27
• test pellis_100_digit_verification — compare mpmath output with CODATA 2022 α⁻¹
• invariant pellis_preregistered_checkpoint — fixed 50-digit value

Phase 3 — Language Harness Directory: benchmarks/language_tests/

Create harnesses outputting JSON {test_name, input, result, decimal_places, error, time_ns}:

Test scenarios: π calculation, φ iteration φ_{n+1} = 1 + 1/φ_n, accumulation Σ 1/n for n=1..10⁶.

Phase 4 — Whitepaper: docs/WHITEPAPER/gf_not_random.md

Structure:

1. Abstract — GF derived from sacred geometry via optimization
2. Background — problem statement, φ in nature
3. Mathematical Foundation — Phi-Split Optimality Theorem + proof
4. Anti-Randomness Arguments — algebraic derivation, universality, falsifiability
5. Experimental Results — sacred constants accuracy, roundtrip precision, cross-language
6. Use Case Recommendations — GF16 primary, GF32 high precision, GF8 weight compression
7. Conclusion + References

All Files to Create/Modify

Verification Gates

# Phase 1: Spec validation
./scripts/tri parse specs/numeric/gf_competitive.t27
./scripts/tri gen specs/numeric/gf_competitive.t27
./bootstrap/target/release/t27c test specs/numeric/gf_competitive.t27

# Phase 2: mpmath precision check
python3 scripts/verify_precision.py > .trinity/experience/precision_verify.json

# Phase 3: Language harnesses
cd benchmarks/language_tests && python3 python_float64.py > results/python_float64.json

# Phase 4: Competitive suite
./scripts/tri math compete --sacred --roundtrip --languages --full --format json

Success Criteria

• scripts/verify_precision.py outputs 100-digit Pellis value
• FORMULA_TABLE.md row 31 pre-registered at 50 digits with mpmath verification note
• specs/numeric/gf_competitive.t27 passes tri test (L4 compliant)
• specs/numeric/pellis_verify.t27 passes tri test
• All language harnesses produce valid JSON
• GF32 sacred constant error < 5× IEEE FP32
• GF16 NMSE <= IEEE FP16 NMSE × 1.01
• Ternary decimal places >= IEEE f64 decimal places
• docs/WHITEPAPER/gf_not_random.md complete
• Results in conformance/gf_competitive_bench.json
• NOW.md updated with ring iteration entry
• Experience JSONL entry appended to .trinity/experience/clara_track1.jsonl

Invariant Laws Compliance (per SOUL.md / AGENTS.md)

• L1 TRACEABILITY: All PRs implementing this work must include Closes #<this-issue>
• L2 GENERATION: gen/ output generated from specs — no hand-editing
• L3 PURITY: All .t27 files ASCII-only, English identifiers
• L4 TESTABILITY: Every new .t27 spec must contain test/invariant/bench
• L7 UNITY: Python scripts in scripts/ are permitted; no new *.sh on critical path

φ² + 1/φ² = 3 | TRINITY