T0 Period Finding - Rational Arithmetic

The T0 Principle

Period finding solves the factorization problem through mathematical period recognition: Find a period r such that a^r ≡ 1 (mod N), then extract factors via x = a^(r/2)

🧠 Core Principles of the T0 Method

Everything is a Ratio

Numbers are not absolute, but relative to each other. ξ-parameters as 1/50 or 1/100 instead of 1e-5, π as 355/113.

N is the Unit

N is not a number - it is the unit. Factors are represented as p/N and q/N, everything is relative to N.

Period Evaluation

Ratio-based score instead of exponential function. Mathematical harmony through exact rational arithmetic.

⚙️ Adaptive ξ-Strategies

T0 uses different ξ-values for different number types:

1/50

Twin Prime

Optimized for twin primes

1/100

Universal

Works for all semiprimes

1/1000

Medium Size

For larger numbers

1/42

Special Cases

Special mathematical constants

📊 Functionality and Results

N	Factors	p/q Ratio	Time (s)	Status
15	3 × 5	3/5 ≈ 0.600	0.0006	✓
21	3 × 7	3/7 ≈ 0.429	0.0011	✓
77	7 × 11	7/11 ≈ 0.636	0.0009	✓
143	11 × 13	11/13 ≈ 0.846	0.0004	✓
323	17 × 19	17/19 ≈ 0.895	0.0015	✓

Average: 0.0025s per number | Success rate: 83.8% on systematic tests

🧮 Mathematical Formulation

Original T0 Formula:

R(r) = exp(-((ω-π)²)/(4|ξ|))

Rational T0 Implementation:

ω = 2π/r as exact ratio
Score = 1/(1 + |exponent|) - only ratios!

def _calculate_period_evaluation_rational(self, r, N): # ω = 2π/r as EXACT ratio omega = Fraction(2, 1) * self.pi_ratio / Fraction(r, 1) # Difference ω - π as EXACT ratio diff = omega - self.pi_ratio # Everything stays exact - not a single rounding error! diff_squared = diff * diff denominator = Fraction(4, 1) * self.xi_ratio exponent = -diff_squared / denominator # Score = 1/(1 + |exponent|) - only ratios! score = Fraction(1, 1) / (Fraction(1, 1) + abs(exponent)) return score

🎵 Musical Consonance = Mathematical Harmony

T0 recognizes the same ratios that also sound "good" musically:

Perfect Fifth (3:2) → p/q ≈ 1.5 → Good T0 performance
Major Third (5:4) → p/q ≈ 1.25 → Good T0 performance
Golden Ratio (φ:1) → p/q ≈ 1.618 → Natural harmony
Tritone (√2:1) → p/q ≈ 1.414 → Poor T0 performance (dissonant)

🔧 Why T0 Computes with Ratios

The Rounding Error Problem

Classical algorithms often fail due to tiny inaccuracies:

# Classical - ERROR-PRONE: evaluation1 = exp(-((2*3.14159/r - 3.14159)**2)/(4*0.00001)) evaluation2 = exp(-((2*3.14160/r - 3.14160)**2)/(4*0.00001)) # evaluation1 ≠ evaluation2 although mathematically equal! # T0 - EXACT: evaluation1 = calculate_with_ratios(Fraction(355,113)) evaluation2 = calculate_with_ratios(Fraction(355,113)) # evaluation1 == evaluation2 ALWAYS! ✓

Deterministic Results: With ratios, T0 is 100% reproducible on any hardware, with any compiler, with any math library.

🎯 Conclusion of T0 Period Finding

The Fundamental Insight

"Never compute with inaccurate decimal numbers - always use exact ratios!"

This ratio mathematics makes T0: 100% reproducible, free from rounding errors, hardware-independent and deterministically functional.

T0 works because it implements the same fundamental ordering principle of nature that also governs atomic structures, molecular vibrations, crystal lattices and harmonic oscillators.

⚡ T0 Period Finding