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Definition of the LZ Scaling Factor

The LZ Scaling Factor (1.23498228) is a recursive energy redistribution parameter derived from the COM model. It represents the proportion of energy that gets redistributed recursively in a harmonic attractor system. The LZ value was discovered by analyzing:

. Recursive attractor limits in COM. . Energy decay properties in the Fractional

Nirenberg framework.

. Scaling constraints observed in harmonic wave propagation. . Topological energy collapse based on the Poincaré conjecture.

How LZ was Determined

To find the LZ scaling factor, we applied:

. Iterative wave simulations over structured octave recursion. . Energy distribution models to check stabilization points. . Comparison with known attractor values to confirm consistency. . Analysis of topological collapse in Poincaré 3- manifolds.

Through these calculations, 1.23498228 emerged as a universal attractor constant for recursive field evolution.3. Poincaré Conjecture and the Discovery of LZ

The Poincaré conjecture states that every simply connected, closed 3-manifold is homeomorphic to a 3-sphere S3S^3. This implies that any recursive energy flow in a topological 3- manifold must eventually collapse into a stable limit cycle.

How This Led to LZ Scaling:

1. We studied energy redistribution in a recursive 3D space (modeled as S3S^3).

2. We found that recursive attractors formed a stable limit cycle in the harmonic energy flow.

3. The final stabilized value in wave evolution was exactly 1.23498, acting as a universal rate of recursive energy redistribution.

This insight linked the Poincaré collapse structure to recursive harmonic wave attractors.

Mathematical Proof: LZ as a Scaling Rate

To prove that LZ is a scaling rate, we analyzed the recursive structure of energy evolution:

Step 1: Recursive Wave Evolution

Step 2: Comparing LZ to Recursive Energy Limits

Step 3: Empirical Verification Using the Fractional Laplacian