Fixing The Theta Bug: Geometric Resonance Factorization

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Fixing the Theta Bug: Geometric Resonance Factorization

Hey everyone, let's dive into a head-scratcher: a bug in the wide_scan_geores_factor.py gist that was causing some serious headaches in the world of prime factorization. Specifically, the issue revolved around the calculation of theta in the script. It was creating a circular dependency that was, well, messing things up. In this article, we'll break down the problem, the root cause, the impact, and, most importantly, the proposed solution to get this geometric resonance factorization method back on track. We'll also cover validation steps to make sure everything's working as expected. Let's get started!

The Core Problem: Why the Gist Failed

So, what was the deal? The wide_scan_geores_factor.py gist, which was introduced in PR #253, was designed to factorize a demo 127-bit semiprime. However, despite being based on a validated method, the script was failing to do its job. It was generating candidate factors, but after checking around 73,000 of them, it couldn't find the correct factors. This failure was a major bummer, especially since the geometric resonance method is a promising approach to factorization. The gist was supposed to demonstrate the effectiveness of this method, but the bug was preventing it from doing so. The script was essentially dead in the water when it came to the demo target.

The Bug's Genesis and Its Impact

The fundamental issue stemmed from a circular dependency in the resonance_candidates function. The code was calculating theta using p_hat, which itself was derived from m and k. This created a closed loop, making the Dirichlet kernel check, which is supposed to assess resonance, independent of N. This meant that the filter was essentially a deterministic check based on the parity of m (whether m is even or odd). It was passing for even values of m and failing for odd ones, which is not what we want. The consequences were significant, as it undermined the demonstration of the geometric resonance method.

The Script's Critical Flaw

Let's get into the specifics. The problematic code, as highlighted in the provided text, is where the theta calculation goes wrong. It's relying on p_hat in a way that creates the circular dependency. This contrasts with the validated method, which computes theta independently. This independence is crucial because it allows for a proper assessment of resonance, which is, you know, kind of important for factoring numbers.

Deep Dive: The Root Cause of the Failure

Now, let's zoom in on the heart of the matter. The circular logic within the resonance_candidates function is the main culprit. Because theta was derived from p_hat, which in turn was derived from m and k, the Dirichlet kernel check became detached from the target number N. This is like trying to navigate without a map. It's essential that the filtering is based on the specific N value to find the correct factors. The original logic was making the process more like a coin flip, leaving the method unable to generate the proper candidates.

The Circular Dependency Explained

The crux of the problem lies in these lines of code:

p_hat = exp((LN - (2 * pi * (m + b)) / k) / 2)
theta = (LN - 2 * log(p_hat)) * k / 2
if abs(dirichlet_kernel(theta, J=J)) >= threshold:
    # Add candidate

As you can see, theta is directly dependent on p_hat, which means the check is not correctly accounting for the specific properties of N. This circularity effectively neutralizes the effectiveness of the Dirichlet kernel in identifying true resonances for a given N. This makes it impossible for the script to find the correct factors, as the candidates generated are not filtered based on the real resonance related to N.

Comparing with the Validated Method

In contrast, the validated method, as demonstrated in results/geometric_resonance_127bit/method.py, calculates theta independently. This is a critical difference. By computing theta separately, the method can accurately assess resonance based on N, which is essential for identifying the right factors. This is like having a reliable compass versus a broken one; one guides you to your destination, while the other leads you astray.

The Proposed Solution: Fixing the Code

The fix is straightforward and centers around breaking the circular dependency. The main change involves calculating theta independently before calculating p_hat. This ensures the resonance check is based on the properties of N.

Here's a breakdown of the proposed changes:

  1. Compute Theta Independently: Calculate theta = (2 * pi * m) / k (or use the exact formula from the validated method).
  2. Resonance Check: Check if abs(dirichlet_kernel(theta, J)) >= threshold.
  3. Calculate p_hat (if resonance passes): If the resonance check passes, calculate p_hat = exp((LN - (2 * pi * (m + b)) / k) / 2). This step now correctly uses the calculated theta.
  4. Add Candidate: Add int(nint(p_hat)) as a candidate.

Code Rewrite for Enhanced Performance

By following these steps, you break the circular dependency. By calculating theta first, the script can properly assess the resonance and generate the correct candidate factors.

Validation: Making Sure It Works

Once the fix is implemented, we need to make sure it actually works. Here's a plan for validation:

  1. Run with the Demo N: Run the gist with the fix on the demo N (N=137524771864208156028430259349934309717). The script should successfully find the factors p=10508623501177419659 and q=13086849276577416863.
  2. Compare Candidate Generation: Compare the candidate generation with the output from the validated method.py to ensure consistency.
  3. Check Keep Ratio and Performance: Make sure the keep ratio (the percentage of candidates that are kept) and the runtime performance align with expectations. The expectation is around a 25% keep ratio and a runtime of 2-5 minutes.

Expected Outcomes and Results

By following these steps, we can ensure that the fix actually solves the problem and that the geometric resonance method performs as expected.

Related Artifacts and Resources

Here's a list of related resources for more context:

These resources provide a complete picture of the bug, the fix, and the related context.

Priority and Importance

This fix is considered high priority because it addresses a critical bug in a demonstration artifact. Fixing this bug is crucial to ensuring the reliability and trustworthiness of the geometric resonance method. By fixing the bug, we can correctly demonstrate the method and restore confidence in the research.