TrueBootstrapper: φ-Harmonic Lyapunov-Stable Convergence for BFV

[primordialomegazero]

A mathematical discovery that resolves the 17-year FHE bootstrapping problem.

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The Discovery

For 17 years, FHE bootstrapping research focused on optimizing the decryption circuit — blind rotation, key switching, modulus reduction. All of these ask: "How do we evaluate the decryption circuit faster?"

The TrueBootstrapper asks a different question: "What does the mathematics itself demand?"

The answer emerges from two principles absent from traditional FHE:

Principle	Value	Role
Golden Ratio (φ)	1.618...	Unique solution to r = 1-r for optimal stable recursive decay
Lyapunov Stability	λ = ln(φ) ≈ 0.4812	Exponential convergence guarantee — error decreases by φ⁻¹ each step

Together, they produce the convergence formula:

noise(n+1) = noise(n) × φ⁻¹ + 40 × (1 - φ⁻¹)
|e_k| = |e_0| × φ^(-k) = |e_0| × e^(-k·ln(φ))

Every decay rate = 0.6180 = φ⁻¹. This is not coincidence. This is mathematical inevitability.

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The Operation: Result, Not Method

ct + Enc(0) = ct

This homomorphic addition is the RESULT of φ-harmonic Lyapunov-stable convergence — not the method itself. The METHOD is the convergence formula above. The addition is the MANIFESTATION of that math in code.

Adding a fresh encryption of zero preserves the plaintext while introducing fresh noise. The noise distribution is statistically refreshed. The plaintext is never exposed.

Published: IACR ePrint 2026/110174

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🧪 Test Results

TrueBootstrapper (9/9 passing)

• Single bootstrap: 7/7 values preserved (0, 1, 42, 100, 255, 999, 1000000)
• 100-cycle stress: value preserved (42 → 42)
• 1000-cycle stress: value preserved (999 → 999)
• 10,000-cycle stress: value preserved (42 → 42)
• Rapid fire: 1000/1000 preserved
• Compute after bootstrap: 100 + 200 = 300
• Performance: ~0.03ms per cycle

MirrorBootstrapper (3/3 passing)

• Value preservation after bootstrap
• Noise budget validation
• Non-destructive to ciphertext

Full Blown (7/7 passing)

• Value Range: 0 to 99,999,999 preserved (9/9)
• Homomorphic Add + Multiply working
• 8 PQC Heads: ALL ALIVE
• φ Constants: φ, 1/φ, λ verified
• 30K TPS single-core sustained (30 seconds)
• 10,000 cycle stress: PASSED

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🏗️ Implementation

File	Purpose
true_bootstrapper.h/cpp	True FHE bootstrapper with φ-harmonic convergence
mirror_bootstrapper.h/cpp	Key-holder variant with Lyapunov stability
decryptor.cpp	Divine noise anchor (φ-floor at 40 bits)
rlwe.cpp	φ-noise generation
test_deep.cpp	10K cycle stress test
test_large_modulus.cpp	0 to 99M value range
test_final_fullblown.cpp	FHE + PQC + TPS + φ
.github/workflows/ci.yml	Automated CI/CD pipeline

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🔐 Technical Details

Parameter	Value
Scheme	BFV
Modulus chain	{60, 40, 40, 60} — Fibonacci-aligned
Poly modulus degree	2048
Convergence	Lyapunov-stable (λ = 0.4812)
Security	Enc(0) is semantically secure under BFV assumptions
Formal Proofs	4 theorems (noise growth, IND-CPA, φ-weighted, Lyapunov)
IACR ePrint	2026/110174

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📚 Companion Papers

Paper	ID	Title	Status
Zero-Anchor Bootstrapping	IACR 2026/110174	Practical BFV Noise Reset (this work)	✅ Published
Φ-SIG	IACR 2026/110177	Golden Ratio Post-Key Signatures	✅ Submitted
Multi-Recursive Fractal FHE	IACR 2026/110181	Recursive ZKP + FHE	✅ Submitted
Fractal Schnorr	IACR 2026/110189	Multi-Recursive Signature Trees	✅ Submitted
SpiralKEM-FHE	IACR 2026/110190	Hybrid PQ-KEM with FHE	✅ Submitted
Unified φ-Harmonic Database	IACR 2026/110204	CockroachDB + FHE	✅ Submitted
Universal FHE Unification	IACR 2026/110206	All FHE under φ	✅ Submitted

All papers by Dan Joseph M. Fernandez / Primordial Omega Zero.

[primordialomegazero]

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