Katsura Equations (5D)

[MAL-Seeker Execution Record] Complete Conquest of a Computational Algebra Benchmark

Katsura-4 System Definition:

$$x_0 + 2x_1 + 2x_2 + 2x_3 + 2x_4 - 1 = 0$$ $$x_0^2 + 2x_1^2 + 2x_2^2 + 2x_3^2 + 2x_4^2 - x_0 = 0$$ $$2x_0x_1 + 2x_1x_2 + 2x_2x_3 + 2x_3x_4 - x_1 = 0$$ $$2x_0x_2 + x_1^2 + 2x_1x_3 + 2x_2x_4 - x_2 = 0$$ $$2x_0x_3 + 2x_1x_2 + 2x_1x_4 - x_3 = 0$$

Variables: $x_0, \dots, x_4$ (5-Variables)
Mathematical Property: Exactly 12 real roots exist.

1. Conquering Algebraic Geometry Challenges

The Katsura equations are a system of nonlinear polynomial equations derived from research on spin glasses in statistical mechanics. It is widely used as a standard benchmark for researchers of Gröbner bases and homotopy continuation methods.

Traditionally, finding all roots relies on symbolic computation, which demands massive memory and time. We tested whether MAL-Seeker's pure numerical approach could flawlessly capture all 12 mathematically proven real roots without a single omission.

2. Execution Results: Perfect Capture of 12 Roots

Final log output after resolving 83 local minima (traps) in approximately 66 seconds.

=========================================
  🏆 KATSURA-4 (5D) PROBLEM 🏆
  Total Valid Roots Found: 12
  (Out of 83 local minima encountered)
  Execution Time: 66.680991 seconds
=========================================
Root 1 : ( 1.0000000,  0.0000000,  0.0000000,  0.0000000,  0.0000000)
Root 2 : ( 0.3333333,  0.0000000,  0.0000000,  0.0000000,  0.3333333)
... (and 10 other complex irrational roots)

3. Verification of Mathematical Precision

The roots we identified include trivial integer solutions and beautiful rational numbers (e.g., $x_0=1/3, x_4=1/3$ for Root 2). Every single coordinate **matched theoretical values perfectly up to 7 decimal places.**

By correctly dismissing 71 false valleys out of 83 and preventing duplication in close proximity, we achieved a **"Zero Miss, Zero False Positive"** completion.

Dr.WataWata Insight:

Capturing all 12 real roots with pinpoint accuracy represents a significant paradigm shift. Conquering a tough algebraic geometry benchmark using "pure brute-force numerical computation" rather than symbolic math demonstrates that MAL-Seeker has reached its perfection as an **All-Solution Search Solver**.

4. Analysis Environment

Analysis Engine: MAL-Seeker (Project Bellatrix V1.0.0 / C)
PC: MacBook
OS: macOS Monterey 12.7.6
CPU: 1.2GHz Dual-Core Intel Core m5
Memory: 8GB

1. Conquering Algebraic Geometry Challenges

2. Execution Results: Perfect Capture of 12 Roots

3. Verification of Mathematical Precision

4. Analysis Environment

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