Multivariable Quadratic (MQ) Problem

【MAL-Seeker Execution Record】Analysis of Multivariable Polynomial Equation System (5D Toy Model)

$$f_i(x_1, \dots, x_5) = x_i^2 + 0.125 \sum_{j \neq i} x_j^2 - x_i + 0.05 \sum_{j \neq i} x_j - 1 = 0 \quad (i = 1, \dots, 5)$$

Variables: 5-Dimensional System (Dense Polynomial Coupling)
Search Bounds: $[-5.0, 5.0]$

1. The Target Equation (Dense Multivariable Quadratic System)

The MQ problem involves solving a system of polynomial equations of degree two or higher, where multiple variables are intricately intertwined. As the dimensionality (number of variables) increases, the number of candidate solutions grows exponentially, making it difficult to find analytical solutions even with supercomputers. Therefore, it is considered one of the "ultimate barriers" in next-generation (post-quantum) cryptography.

In this execution, to catch a glimpse of this mathematical difficulty, we set up a 5-dimensional toy model where all variables mutually interfere with each other.

2. Execution Results: An 11.5-Minute Endurance Race

Here is the final log produced by MAL-Seeker (Project Bellatrix V1.0.0) after an 11-minute fierce battle in an extremely brutal terrain swarming with 1168 local minima (traps).

=========================================
  🏆 MQ PROBLEM TOY MODEL (5D) 🏆
  Found 2 Unique Roots 
  (Out of 1168 local minima encountered)
  Execution Time: 691.140233 seconds
=========================================
Root 1: (-0.5922732, -0.5922732, -0.5922732, -0.5922732, -0.5922732)
Root 2: ( 1.1256066,  1.1256066,  1.1256066,  1.1256066,  1.1256066)

3. Mathematical Verification: Perfect Match with Theoretical Values

Focusing on the symmetry of this model, if we assume all variables are equal ($x_1 = \dots = x_5 = x$), the original system of equations is reduced to the following single-variable quadratic equation.

$$1.5x^2 - 0.8x - 1 = 0$$ $$\Rightarrow x = \frac{0.8 \pm \sqrt{6.64}}{3} \approx -0.5922732, \,\, 1.1256066$$

The two coordinates identified by MAL-Seeker perfectly matched the theoretical values down to the 7th decimal place. This proves an extremely high calculation precision that does not forgive even the slightest numerical deviation—the very core of cryptography.

4. Stance on Exhaustiveness: Are These Truly "All" the Roots?

What is particularly noteworthy in this analysis is that 1168 "false valleys (local minima)" were detected within a mere 5-dimensional space. MAL-Seeker evaded these traps one by one and ultimately reached the two true roots.

However, as a numerical analyst, there is a fact I must convey honestly. That is, "We do not know if these two are 'all' the roots that exist in this mathematical space."
Due to the nature of the MQ problem, it is extremely difficult to predict in advance how many solutions theoretically exist. MAL-Seeker dug through the space 1168 times this time and unearthed these two, but it cannot complete the devil's proof of "there are absolutely no other roots" in a mathematical sense through numerical computation alone. This result simply means "two roots were found."

Dr. WataWata Insight:

The figure of "1168 traps" encountered this time speaks volumes about the vicious "deceptiveness" inherent in this problem. Had the engine's precision been even slightly compromised, or had the computations broken down midway, we might have mistaken a false minimum for the truth and prematurely concluded our search.

More valuable than the "two roots found" is the absolute robustness of the engine, which relentlessly forged ahead despite being deceived over a thousand times. This sheer toughness—surviving an 11.5-minute traverse through a mathematical minefield without a single singular matrix error or NaN crash—will undoubtedly be our greatest asset as we continue our quest to uncover all roots in uncharted territories.

5. Analysis Environment

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

1. The Target Equation (Dense Multivariable Quadratic System)

2. Execution Results: An 11.5-Minute Endurance Race

3. Mathematical Verification: Perfect Match with Theoretical Values

4. Stance on Exhaustiveness: Are These Truly "All" the Roots?

5. Analysis Environment

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